GB2487489A - Method of defining the geometry of first and second overlapping components - Google Patents

Abstract

Two opposing surfaces of each component are each defined by a matrix of vertices with each vertex in one surface having a corresponding vertex with the same X and Y coordinates in the other surface. For the first component the plane of symmetry in the said direction is defined, the said coordinates of vertices within an area of an upper surface that will be overlapped being mirrored for the second component. The lower surface of the first component is defined by copying the coordinates of the mirror symmetric vertices. Also claimed are methods and apparatus for converting coordinates of surfaces of a member for location in a tubular enveloping surface generated by these methods. The invention is particularly for use in a cast on cast fabrication system.

Description

I

Cast on Cast Fabrication System and Method

FIELD OF THE INVENTION

The present invention relates to the field of the component manufacture.

Embodiments in particular relate to apparatus and methods useful for modelling andior making prefabricated components, such as components for the construction of complex surfaces.

BACKGROUND

The use of complex geometries in modern architecture can render buildings visually attractive and is in common use. Known methods for creating such complex geometries, at least in the construction industry, however, are associated with considerable waste of time and material, This affects the overall construction costs of a building project. Some known methods require the design, production, transportation and assembly of expensive formworks and scaffoldings. At the date of writing no efficient ways of building complex geometries desired by most of contemporary architectural designs are known.

Digital Fabrication technologies have tried to fill this gap, by forcing a relation in between the design process with the fabrication and construction stages. Digital fabrication technologies can fabricate complex products previous technologies could not generate. Designers are directly exploring the possibilities of this relatively new technology to improve their designs. Designers are thus more directly involved into the construction of their product than was the case in the past. Proposals in this field have tried to find ways of building monolithic single buildings with machines. The problem these projects have is that the size of the machine needed to construct a building increases with the size of the building.

At the time of writing attempts are underway to scale up 3D printing technology generally used for small scale prototyping in order to gain terrain in the building industry. The conditions under which this technology operates at a small scale are not the same as those prevalent in the building industry. Firstly, as discussed above, the size of the machine has to match that of the object that is to be produced. The structural limitations of the materials that are currently used on a small scale may moreover render these techniques less suitable for use in the building industry.

Existing digitally driven fabrication methods for the fabrication of complex formwork are based on the application of CNC technologies by processes based on cutting, subtraction or addition of material. One of the methods generally used to produce geometrically complex cast building components and structures is based on the design and fabrication of formwork tables that are shaped and supported by vertical wood or metal ribs CNC cut according to a very specific curvature. These moulds are not reusable and this fabrication method consequently generates a large amount of waste that needs to be disposed.

Another fabrication method includes the design and production of CNC milled high-density foam components. This foam is a high cost material and considerable quantities of it are wasted in the production of formwork panels. The panels also need to be disposed of once the building component is cast. All of the expensive foam material is therefore discarded during the manufacturing process.

Another way to achieve formal complexity in a building is related to the use of cladding systems (interior and exterior). The production of cladding panels based on complex geometries is mostly focus on folded sheets, especially metal ones.

The production of pavement tiles is usually associated with a generic and repetitive component plus some few special pieces for certain connections. These elements can be arranged in different ways to adapt to terrain. However, they always have a vertical joint that makes it difficult to reach highly differentiated topographical conditions that requires changes in the resolution of the grid creating unwanted gaps between components and grid distortions.

SUMMARY OF THE INVENTION

According to an aspect of the present invention there is provided a method of defining the geometry of a first and a second component that are to overlap each other in an overlapping direction. Two opposing surfaces of each component are each defined by a matrix of vertices with each vertex in one surface having a corresponding vertex with the same x and y coordinates in the other surface. A difference in the z-coordinates between the corresponding vertices defines a component thickness at the vertices. The method comprises, for the first component, defining a plane of symmetry extending in the z-direction and in a direction orthogonal to the direction of overlap and mirroring at the plane of symmetry the z-coordinates of vertices within an area of an upper surface that will be overlapped, or vice versa. For the second component a lower surface is at least partially defined by copying the z-coordinates of the mirror symmetric vertices. An upper surface is defined by, for each vertex that has an adjacent vertex in a direction orthogonal to the overlapping direction, adding the component thickness of the first component at the adjacent vertex to the z-coordinate of the vertex of the lower surface.

The components may have a square footprint and the direction of overlap may be a diagonal of the square footprint. The components may form part of respective first and second rows of overlapping components. The direction orthogonal to the overlapping direction may be towards the overlapping row.

The first and second components can be cast in a cast sequence by placing the second component on top of the first component during casting. The method may further comprise defining a part of the upper surface that is overlapped by a component in another cast sequence so that the part of the upper surface has the same topology as the overlapping part of the lower surface of the overlapping component in the other cast sequence.

According to another aspect of the present invention there is provided a method of manufacturing first and second components. The method comprises casting the first component, providing a separating layer on the upper surface of the first component; and casting the second component on top of the first component.

According to an aspect of the present invention there is provided a method comprising converting first coordinates of a first projection of a basic geometric shape onto a surface and first coordinates of a second projection of the said basic geometric shape onto the surface into respective first coordinates on a tubular enveloping surface. The first and second projections are to form respective surfaces of first and second component parts of the surface. The method further comprises determining, for each of the converted first coordinates of the first and second projection a second coordinate on the enveloping surface. The second coordinates are associated with the first projection and define a second component surface opposite to the first surface.

Some or all of the second coordinates associated with the first projection substantially coincide with the first or the second coordinates associated with the second projection.

When referring to a tubular enveloping surface reference is made to any type of closed tubular surface. Such a tubular enveloping surface may have opposing parallel side surfaces but is not limited to comprising such side surfaces. The tubular enveloping surface may have a cross-section that increases or decreases along the longitudinal direction of the enveloping surface. The tubular enveloping surface may be a truncated regular or irregular pyramid with three, four, five, six, seven, eight or more side surfaces. The side surfaces of the tubular enveloping surface do not have to be planar, although they can be. A leading edge of the side surface may, for example, be not parallel to a trailing edge of the side surface. The leading and trailing edges may be arranged at opposite ends of the side surface of the tubular enveloping surface.

Embodiments of the invention may use as a starting point a representation of a surface that is to be generated. Such representations may be (although the invention is not limited in this manner) representations created by computer design/modelling/rendering software. The representation of this surface may already be into component parts that are to be manufactured. It can for example be envisaged that the representation defines the surfaces with reference to grid that, when projected into a plane comprises the above array of basic geometric shapes. The above referred to first coordinates are sufficient to describe the projection of the surface, although, in one embodiment they may not describe fine detail of the surface, as discussed in further detail below.

The embodiment then converts the first coordinates describing the surface in 3D space into coordinates describing the component parts of the surface in a stack of component parts contained in the tubular enveloping surface. This tubular enveloping surface may correspond to a casting container that will later be used for casting the components. The tubular enveloping surface may alternatively correspond to a virtual' enveloping surface (which may be recognised by computer driven equipment forming the component parts) beyond which the components doe not extend during manufacture.

As the component parts, as defined by said projections on the surface are only two dimensional the embodiment calculates a thickness of the component parts within the tubular enveloping surface. Each of the second coordinates associated with this thickness (and with a corresponding first coordinate) may be individually computed, so that the thickness may vary over the physical extent of the individual components.

The conversion of the first coordinates into coordinates on the enveloping surface can either be calculated before the second coordinates are determined or thereafter. The second coordinates may be determined with reference/relative to the first coordinates, so that they can be converted into coordinates on the enveloping surface without difficulty.

It will be appreciated that, although the method may receive an already segmented surface as an input, this is not eèsential and the method may instead segment a received surface. This may be achieved by projecting the received surface that is to be manufactured onto a plane. The projected surface may then be segmented into a one or a two dimensional array of the basic geometric shapes. The so created geometric shapes may then be projected back onto the projecting the basic geometric shapes onto the surface that is to be manufactured. For each of the projected basic geometric shape the first coordinates can then be determined.

A single basic geometric shape may be a polygon. In this case the first coordinates are coordinates of the corners of the polygon, as projected onto the surface. The polygon may be a triangle, rectangle, pentagon, hexagon, heptagon, octagon or other polynomial shape. The polygon may have sides of equal length and the same internal angle in all corners. Alternatively the polygon may have different side lengths and/or different internal angles and may therefore be irregular in shape.

Rectangles may prove to be particularly useful.

The enveloping surface may have a cross section that corresponds to the basic geometric shape. In this case, when converting the coordinates of the projections, the coordinates of the projections are converted so that they are located on edges of the enveloping surface.

The enveloping surface may comprises two parallel side walls. In this case the first and second coordinates are arranged on the enveloping surface so that one or both of the first and second surfaces of the component part to be manufactured is not perpendicular to the two parallel sidewalls.

The array may be a two dimensional array that comprises a first row of the basic geometric shapes and a second adjacent and parallel row of the basic geometric shapes. The method may further comprise determining an incline of a projection of one of the basic geometric shape of the second row on the surface in a direction that is perpendicular to the direction of the rows. The enveloping surface that is used for determining the first and second coordinates for the first row may be defined so that the enveloping surface comprises two opposing planar surfaces and so that each of the opposite surfaces deviates from a longitudinal direction of the enveloping surface by half the determined incline. The two opposing planar surfaces may thus be symmetrically inclined relative to the longitudinal direction of the enveloping surface.

The array may comprise a continuous row of the basic geometric shapes. A plurality of adjacent projections of the basic geometric shapes in the row may be identified, wherein each of the projections has a positive gradient. The tubular enveloping surface for accommodating the parts associated with the row of projections may be defined so that it has two diverging side walls. The first coordinates of the plurality of projections may then be converted into the respective first coordinates on the tubular enveloping surface so that the component parts associated with the plurality of projections are stacked within the enveloping surface in the order in which they appear in the row. The component part associated with the first projection may hereby be located at a narrow part of the enveloping surface with subsequent components of the row being located at progressively wider parts of the enveloping surface.

Alternatively or additionally a plurality of adjacent projections of the basic geometric shapes in the row may be identified, wherein each of the projections has a negative gradient. The tubular enveloping surface for accommodating the parts associated with the row of projections may again be defined so that it has two diverging side walls, The first coordinates of the plurality of projections may then be converted into the respective first coordinates on the tubular enveloping surface so that the component parts associated with the plurality of projections are stacked within the enveloping surface in the order in which they appear in the row. In this case the component part associated with the last projection of the plurality of projections may hereby be located at the narrowest part of the enveloping surface with preceding components of the row being located, in the correct order defined by their position in the row, at progressively wider parts of the enveloping surface.

Determining the second coordinates may comprise determining second coordinates that, when associated with first coordinates, form an edge of a component part. This edge may be, when the component parts are assembled in a row, adjacent to a further edge of a further component part. The first and second coordinates defining this further edge may have already been determined in the method. It is desirable that these two edges coincide with each other to provide a gap and step free assembled surface and the second coordinates that are still to be determined may be calculated so that the edges created by it coincides with the already existing edge.

One or more of the second coordinates of a component part may be determined using an incline of the projection on the surface that is associated with the component part. The second coordinate may, for example, be determined starting from another, previously determined second coordinate of the component part using the value of the incline to ensure that the thickness of the component at the second coordinate that is to be determined is non-negative/sufficient.

As discussed above, the first coordinates do not necessarily need to describe the surface they represent in fine detail. Instead the first coordinates may simply serve as a vehicle for determining a location of surfaces approximating a surface of a component part that is to be created inside the enveloping surface/container.

Representations of finer detail of the surface can be added once the principal position of the component part within the container has been determined. The topology of the projected geometric shape on the surface may be analysed for this purpose and, if the topology substantially deviates from a p'anar surface defined by the first coordinates associated with the projection, further first coordinates of points of the projection of the basic surface may be determined to describe topological deviations of the surface form the planar surface defined by the first coordinates. These further first coordinates may then be converted into coordinates within the enveloping surface.

The first component part may be formed according to the first and second coordinates of the first component part from a fluid material. The second component part may subsequently be formed on top of the first component part according to the first and second coordinates of the second component part. Methods in which components are created one on top of another will in the following be referred to as cast on cast' methods.

Cast on Cast allows the production of unique structural components that can be arranged together to produce complex surfaces without waste normally associated with the manufacture of complex surfaces. This advantage is achieved as a first formed component serves as formwork for a later formed component. Components manufactured in this fashion can be post-tensioned and/or anchored in order become a monolithic structure.

The components can be any type of component intended for assembly into a structure (as defined by the input surface) together with similar or even with different other components. The components may, for example, be tiles, panels, bricks and blocks and can be used in a wide range of applications such as cladding for façades or roofs, structure, pavements, interior partitions, temporary or permanent formwork, suspended ceilings and urban furniture.

Fixing means for fixing adjacent component part to each other may further be provided. The fixing means may be provide in a space that receives fluid material during the forming of the component parts. The fixing means may project beyond a boundary defined by the enveloping surface. Recesses may further be defined in the component parts, for example for receiving the fixing means.

The formal complexity expressed in the contemporary architectural design demands from the building industry an effective and efficient response in order to materialize it. For this it is important to identify the multiple layers and components that a building comprises. Understanding that there are different approaches to achieve this formal compIexy requires different strategies. In response, Cast on cast as a fabrication system has the ability to operate at different levels in order to provide not only complex format structures, but as well geometrically complex cladding systems, geometrically complex formwork, non structural divisions and, pavement tiles.

It will moreover be appreciate that, as components manufactured using the cast on cast' techniques can easily be stacked one on top of the other, making their storage highly efficient and making component manipulate and transport easier. The cast on cast fabrication system and method may the ability to act as a temporary or, as a permanent formwork. The versatility of the method may depend on the materials used for forming the components.

Cast on cast components can also be assembled in an efficient manner. If the components have been designed to be cast in a specific sequence then each component follows a specific assembly logic making the assembly sequence very systematic and organized.

The component parts may be formed using an electronically operated fluid applicator. Such a fluid applicator may, for example, form part of a ONC machine.

The method may comprise separating the first and second component parts and assembling the first and second component parts to form the surface that is to be constructed.

Additionally to the different geometry between components, there is the possibility to customize each cast component in a parametric way (e.g. openings, bumps or, any other effect) giving an additional aesthetic value that cannot be reached by other systems.

According to another aspect of the present invention there is provided a container having side walls corresponding to the tubular enveloping surface.

According to another aspect of the present invention there is provided a method of forming a container. The method comprises obtaining input data of an array of segments of a surface. Each segment corresponds to a projection of a basic geometric shape from a common plane upon the surface. The array comprises two or more parallel rows of the projections. The method determines an inclination angle of one row of the array relative to the common plane and in a direction orthogonal to the direction of the row and parallel to the common plane and defines a tubular enveloping surface that has a longitudinal direction and that comprises two opposing side surfaces that deviate from the longitudinal direction by half the determined inclination angle.

The method may further comprise defining two further side surfaces of the enveloping surface. These two further side surfaces are opposite to each other and extend parallel to the longitudinal direction of the enveloping surface. Each one of the two opposing side surfaces and the two further side surfaces extends in a direction orthogonal to the longitudinal direction of the enveloping surface. The directions of the two opposing side surfaces are orthogonal to the directions of the further side surfaces.

This means that, if the side opposing side surfaces and the further side surfaces intersected, they would do so at a right angle. It will, however, be appreciated that such intersection is not essential, as the opposing side surfaces and the further side surfaces may be separated from each other by further side surfaces, as is for example the case if the enveloping surface has an octagonal cross-section.

The method may further comprise providing a base member that intersects the longitudinal direction of the enveloping surface. Either two side wall members corresponding to the two opposition side surface or two side wail members corresponding to the two further side surfaces, or all four of the said side wall members may be connected to the base member.

According to another aspect of the invention there is provided an apparatus comprising means for obtaining segmentation data representing a surface that is to be manufactured. The data comprises first coordinates of a first projection of a basic geometric shape onto the surface and first coordinates of a second projection of the basic geometric shape onto the surface. The first and second projections define respective first surfaces of first and second component parts of the surface that is to be manufactured. The apparatus may further comprise a processing means operable to convert the respective first coordinates into corresponding coordinates of the projection located on a tubular enveloping surface and to determine, for each of the converted first coordinates a second coordinate located on the enveloping surface. The second coordinates define a second component surface opposite to the first surface. Some or all of the second coordinates of the first component part substantially coincide wah corresponding ones of the first or of the second coordinates of the second component part. The determined first and second coordinates within the enveloping surface may be stored in a storage means of the apparatus.

The apparatus may further be configured to provide an output comprising the first and second coordinates to another apparatus, such as an apparatus arranged to form the modelled components, or to store the coordinates in a storage means.

According to another aspect of the invention there is provided a computer program product comprising code for execution by a processor. The code is arranged to render the processor operable to convert first coordinates of a first projection of a basic geometric shape onto a surface and first coordinates of a second projection of the said basic geometric shape onto the surface into respective first coordinates on a tubular enveloping surface. The first and second projections wilt form respective surfaces of first and second component parts for assembly of a physical realisation of the surface onto which the shapes have been projected. The code further renders the processor operable to determine, for each of the converted first coordinates of the first and second projection, a second coordinate on the enveloping surface. The second coordinates define a second component surface opposite to the first surface. Some or all of the second coordinates associated with the first projection substantially coincide with the first or the second coordinates associated with the second projection.

The computer program product may further comprise code arranged to render a processor operable to define the enveloping surface so that it has a cross section corresponding to the basic geometric shape.

The computer program product may further comprise code arranged to render a processor operable to analyse a topology of the projected geometric shape on the surface and, if the topology substantially deviates from a planar surface defined by the first coordinates associated with the projection, to determine further first coordinates of points of the projection of the basic surface and convert the further first coordinates into coordinates within the enveloping surface.

According to another aspect of the invention there is provided a method of manufacturing comprising forming a first component from a viscous material. The first component has a top surface. The method further comprises forming a separation layer on at least part of the top surface and forming a second component on top of the separation layer.

The method may further comprise forming formwork below a first component to be formed to define a bottom surface of the first component.

The method may further comprise forming the components within an enveloping surface comprising two paratlet side walls, two opposing and outwardly or inwardly leaning side wafls or two parallel side walls and two opposing and outwardly or inwardly leaning side walls. It will be appreciated from the above that this enveloping surface can either take the form of a container comprising the side walls or may alternatively have a virtual existence recognised by machinery creating physical embodiments of the components, wherein such machinery is operable to ensure that no material forming the components is applied outside of the enveloping surface.

The method may further comprise casting material into the container to form the components. The method may further comprise providing formwork within the container to define a top surface of one or more of the components during the forming of the components.

According to another aspect of the present invention there is provided a method comprising quantifying one or more differences between a target surface and a further surface comprising rows and columns of components. At least two components in a row of the components have geometries that allow placement of the two components in a tubular enveloping surface so that they contact each other. The respective surfaces of the components in contact in the tubular enveloping surface are surfaces that are not in contact with another component when the components are assembled in the further surface. Put in other words, at least two components in rows may have geometries that allow them to be arranged in contact in a tubular enveloping surface for manufacturing in contact/using a cast on cast manufacturing technique. The method further comprises altering a manner in which the components are placed within the tubular enveloping surface or altering a parameter describing the geometry of the tubular enveloping surface and retaining the alterations if the difference between the target surface and the further surface has decreased.

The above steps may be repeated until the target surface corresponds to the further surface. An interrupt criterion for the repetition may be a predetermined difference threshold being reached, if a predetermined method of comparison is used.

Components of a number of rows may have geometries that allow placement of the components of a respective row in a tubular enveloping surface so that they contact each other in the above discussed manner. Steps a) to c) may be repeated, with alterations in step b) made to the placement of components and/or to the parameters of tubular enveloping surfaces of the components of more than one row.

The method may further comprise identifying an area in which the target surface and the further surface differ more than in another area and identifying a tubular enveloping surface that is responsible for creating components in the identified area. The alterations may then be made to the identified tubular enveloping surface or to components placed therein.

A basic geometric shape of a tubular enveloping surface may be changed after a number of repetitions. This can be useful if insufficient convergence between the target surface and the further surface is achieved within a given number of iterations.

The iterative process may also be terminated if the quantitative difference between the target surface and the further surface is below a / predetermined threshold.

The method may further comprise inserting additional rows or columns of components into the further surface.

According to another aspect of the present invention there is provided a surface generated using any of the above described methods.

According to another aspect of the present invention there is provided a surface curving in two directions and comprising an array of component parts arranged in rows and columns. Two or more or all interfacing surfaces between components in a row are parallel to each other and/or an interfacing surface two or more component parts in one row share with two or more adjacent components in an adjacent row is planar.

BRIEF DESCRIPTION OF THE DRAWINGS

Preferred embodiments of the present invention will now be described by way of example only and with reference to the accompanying drawings, in which: Figure 1 shows one way of assembling components in a row to form a curved surface; Figure 2 shows a container for forming the components shown in Figure 1 in a cast on cast process; Figure 3 shows another way of assembling components in a row to form a curved surface; Figure 4 shows a container for forming the components shown in Figure 2 in a cast on cast process; Figure 5 shows a way of assembling components in an array to form a surface with a finite thickness that curves in two orthogonal directions; Figure 6 illustrates one embodiment of a cast on case process; Figure 7 shows an individual stack of cast on cast components and its assembly in to a row of components as well as several stacks of component and their assembly into a surface curving in two orthogonal directions; Figure 8 shows examples of basic geometric shapes and containers that may be formed therefrom; Figure 9 shows some possible modifications to the basis geometric shapes and to containers; Figure 10 illustrates another embodiment of a cast on case process; Figure 11 shows a known contour crafting machine in use; Figure 12 shows a Makerbot'; Figure 13 illustrates another embodiment of a cast on case process; Figure 14 shows a manner of segmenting an input surface; Figure 15 shows the plane onto which the surface that is to be segmented is projected for the use of rectangular basic geometric structures for segmentation; Figure 16 shows the plane onto which the surface that is to be segmented is projected for the use of hexagonal basic geometric structures for segmentation; Figure 17 introduces vertex numbering for components based on rectangular basic geometric shapes; Figure 18 introduces vertex numbering for components based on hexagonal basic geometric shapes; Figure 19A illustrates the definition of the parameter growth' for components based on rectangular basic geometric shapes; Figure 196 illustrates the definition of the parameter growth' for components based on hexagonal basic geometric shapes; Figure 20/k illustrates the definition of the parameter angle' for components based on rectangular basic geometric shapes; Figure 20B illustrates the definition of the parameter angle' for components based on hexagonal basic geometric shapes; Figure 21 illustrates the manner in which vertex coordinates may be inherited between components in one embodiment; Figures 22/k and B illustrate the assignment of coordinates to vertices for rectangular components if the parameter mirror' has been true and even or an odd number of times; Figure 23 illustrates the conditions that four adjacent components need to fulfil if they are to be assembled in a surface in a step and gap free manner; Figure 24 illustrates the calculation of the V-coordinate in a container in one embodiment; Figure 25 illustrates the assignment of coordinates to vertices for a first rectangular components of a row; Figure 26 shows a comparison between a target surface and a surface used in an iterative method of generating containers/enveloping surfaces; Figure 27 shows a surface 800 assembled from triangular components; Figure 28 illustrates a way of defining a first triangular component in a first container; Figure 29 illustrates a way of defining a second triangular component of a first row in a second container; Figure 30 illustrates forming further components for a row in first and second containers; Figure 31 shows inheritance rules for defining the length of component edges in one embodiment; Figure 32 illustrates determining the first components of the third and fourth containers of a first row of components; Figure 33 illustrates determining the first component of a first container in rows other than the first row; Figure 34 illustrates determining the first component of the second container in rows other than the first row; Figures 35 and 36 illustrate determining the first components in the third and fourth containers respectively in rows user than the first row; Figures 37A to 37C show different ways of overlapping adjacent components; Figures 38 A) to C) illustrates the subdivision of the top/bottom surface of a component into different matrices; Figure 39 shows a first component of a first container for an overlapping arrangement; Figure 40 illustrates an inheritance arrangement for forming a second component of an overlapping arrangement; Figure 41 shows a second component of an overlapping arrangement on top of the first component of the arrangement in a container; and Figure 42 illustrates inheritance rules for overlapping components in a different row.

DETAILED DESCRIPTION

Figures 1 and 3 illustrate two different ways of assembling a number of components in a row, so that a curving surface is created.

Referring firstly to Figure 1, the components 100 and 110 have end faces 130, 140, 150 that extend parallel to each other. The end faces 130, 140, 150 however, may not be perpendicular to the top surfaces 160/170 or to the bottom surface of the component 100/110 respectively. Although not shown in Figure 1, the top and bottom surfaces of components 100/110 may also not be parallel to each other. As is shown in Figure 1, the end face 140 deviates by an angle 180 from the direction perpendicular from the bottom face of the component 100. As the two end faces 130 and 140 are parallel, end face 130 of component 100 will also deviate from the direction that is perpendicular to the lower surface of the component 100 by the same angle 180.

It will be appreciated that, given that components 100 and 110 are not in the same plane, the angle by which end faces 140 and 150 deviate from the direction perpendicular to the lower surface of the component 110 is different from angle 180 applicable for component 100.

As can be seen from Figure 1, the end faces of components 100 and 110 that touch each other have the same height. It will be appreciated that this is not essential.

If the heights of these surfaces are different then there would be a step between the components 100 and 110. This may be acceptable in some cases, for example where the curved surface is part of a surface cover for which one surface cannot be seen after installation.

It will be appreciated that if the heights of the end faces 140 of the two components 100 and 110 that touch each are the same, then the thickness of component 110 (at least close to the end face 140) is dictated by the thickness of the component 100 and by the two angles by which the end face 140 deviates from the direction perpendicular to the bottom surface of components 100 and 110 respectively (angle 180 on component 100; for component 110 this angle is not shown in Figure 1).

It will moreover be appreciated that while components 100 and 110 are shown in Figure 1 to have a constant thickness, this is not essential and the thickness of the components may vary from one end face to the other.

The conditions adhered to by components 100 and 110 are therefore the following: 1. The end faces touch across their entire area.

2. The end faces have the same height.

3. The end faces are parallel.

As discussed above, condition 2 is non-essential if steps in the top or bottom surface are acceptable. If the end faces have straight edges, condition I may be reduced to a condition where only the top or bottom edges of the components 100 and touch. Even under such relaxed constraints can it be ensured that the upper (lower) surface of the components appears continuous/gap free, It will be appreciated that if it is the top surface formed by the components 100/110 that is to appear continuous, then it is the top edges of the components 100/110 have to touch, whereas the bottom edges have to touch if the bottom surface formed by components 100/110 is to appear continuous. Condition 3 is the premise on which the construction of the surface of Figure 1 has been based.

Figure 2 illustrates a method of casting a number of components 200 to 220 within a casting container 230. "Cast on cast" casting methods will be described in more detail below. As can be seen from Figure 2, such methods include forming several components on top of each other in a separate manner. As can be seen from Figure 2, the containers 230 has side walls 240 that extend parallel to each other. The components formed by the Figure 2 process are thus suitable for assembly in the manner shown in Figure 1.

The end faces 130 to 150 shown in Figure 1 are the faces of components 200 to 220 that are formed against the side walls 240 of the container 230. At the bottom of the container a mould piece 250 is provided. This mould piece 250 ensures that the angle by which the end faces of the component 200 deviates from the direction perpendicular to its lower surface of the component is the intended angle/the angle necessary to achieve the curvature of the surface, when assembled. If the angle of the end face is to be referenced relative to the top surface of the component, then the mould piece 250 can also be used to achieve the desired angle, although in this case the angle under which the top surface is cast is of course also important.

The first component 200 shown in Figure 2 is shown to have a uniform thickness 260. It wilt be appreciated, however, that this is not essential. If it is to be ensured that the assembled surface does not comprise any steps, then the thicknesses 260 and 280, components 210 and 220 have at the left side wall 240 has to be the same as the thickness 260 and 280 components 200 and 210 have at the right side wall, assuming that component 200 is left most in an assembly of the components followed by components 210 and 220 to the right, in this order.

It will be appreciated from the above description of Figure 2 that by imposing the third above condition upon the design of the components 100/110 and 200 to 220 respectively these components can be formed on top of each other in a container with straight side walls.

Turning now to Figure 3. this figures shows another manner of creating a surface that is curved in one direction. The main difference between the way in which Figures 1 and 3 achieve the curvature is that in Figure 3 the end faces 340 to 360 do not have to be parallel to each other. Instead, each end face 340 to 360 can deviate from the direction that is perpendicular to the bottom and/or top surface of the associated component by an angle 370 that is individual to the end face 340 to 360/component 300 to 320 combination.

Figure 4 shows a container 400 having angled side faces 410. These side faces deviate in Figure 4 from the vertical direction (assuming that the base of the container is horizontal) by the same angle 420. The inclined side walls 340 to 360 shown in Figure 3 are the walls formed against the side walls 410 of the container 400 shown in Figure 4. Despite the symmetry of these container walls 410, it is possible to create the inclined side walls 340 to 360 shown in Figure 3, as the components can be cast in the container 400 at an angle to the horizontal plane, as shown in Figure 4 by component 430. Both the upper and lower surfaces of this component 430 extend at an angle deviating from the horizontal, so that the two end faces of the illustrated component deviate by different amounts from the direction that is perpendicular to the component's top/bottom surface. lt will be appreciated that any further components that may be cast on top of the component 420 shown in Figure 4 will have to be arranged within the container 400 so that they achieve desired inclination angles for the end surfaces.

One part of the preferred embodiments described hereinafter is the use of the manner in which the curvature has been created in Figure 1 (parallel end faces) to create the curvature in one direction (referred to in the following as the X-direction or U-direction) for a surface that comprises curvatures in two directions. To achieve curvature in the second direction (referred to hereinafter as the Y-direction or the V-direction) the method/manner for achieving curvature shown in Figure 3 is employed.

Thus if the components shown in Figures 1 and 3 were assembled in a single assembly they would extend along orthogonal directions. This is shown in Figure 5. It is however emphasised that the present invention is not limited to these embodiments and that surfaces that have a curvature in two directions can be generated by creating interfacing surfaces between component parts by applying either of the methods described above with reference to Figures 1 and 3 respectively to interfacing surfaces extending in both of the two directions.

The components extending along row A in Figure 5 have abutting end faces that are parallel to each other. The same is true for the components that form rows B and C respectively. Each of the rows shown in Figure 5 thus correspond, in concept at least, to a row as shown in Figure 1. The components extending along individual ones of columns 0 in contrast do not have parallel end faces. Each of the columns D thus corresponds, in concept at least, to a column shown in Figure 3.

it will be appreciated that, because of the parallel nature of the end faces in rows A, B and C shown in Figure 5 it can become possible for series of components extending along individual ones of the rows A, B and C to be cast in a single container, in the manner shown in Figure 2.

As discussed above with reference to Figure 2, the side walls 240 of the container 230 are parallel to each other, forming the parallel component end faces used for creating the curvature of the surface. Each of the rows A to C of Figure 5 (or at least a part of each of these rows) is cast in a stack in the manner shown in Figure 2.

From Figure 5 it will be appreciated that to create the curvature in the second direction (U-V direction) the side faces of the components that are perpendicular to the U-V direction are also inclined. This is achieved by inclining the side faces of the container that are orthogonal to the parallel side faces of the container in the manner discussed above with reference to Figure 4. This incline can be constant for the entire height of the container, or vary over the height of the container. The inclination angle for a container is determined in a manner discussed in more detail below by comparing a first component of the component stack that is to be cast in the container with the geometry of the adjacent component of the adjacent row (which will be cast in a different container). The so determined inclination angle may then be used for all components in the cast stack.

Before a detailed manner in which a given input surface can be fragmented in a way that allows time and resource efficient manufacture is discussed the above mentioned technique of casting components on top of previously manufactured components is described by way of example. This technique is referred to in the following as "Cast on Cast" technique.

Cast on Cast" is a technique for manufacturing of geometrically complex architectural components for the construction of complex surfaces, which uses the previously fabricated component as a formwork to cast the next one.

In the following three embodiments are described by way of example only.

These are: 1. Cast on Cast with container.

2. Cast on Cast without container 3. Cast on Cast with CNC milled foam as partial formwork.

Process: As shown in Figure 6, the process relies on depositing or injecting layers 10 of a fluid material, which has the ability to set or dry, inside a container 20 with a Computer Numeric Controlled machine 30. Several layers 10 of this material may shape one component 40 from bottom to top. Once the top layers 10 of a first component 40 are dried or set, the machine deposits on the top surface of the component 30 a thin layer of a chemical barrier. When this chemical barrier 50 has dried, the machine 30 deposits a new set of layers 60 of the fluid material shaping the second component 70.

This process can be repeated to make further components 80 within the container 20.

All the process is carried out within the side waIls 90 of the container, which define a closed perimeter, as shown in Figure 6.

Once all the components 40, 70, 80 of a sequence are fabricated they are permitted to dry/set over a period of time before they are manipulated. The duration of this period of time depends on the material used and the scale of the component 40, 70, 80. Because of the chemical barrier 50 between components the components 40, 70, 80 can be easily separated once dried/set and can be assembled to form complex surfaces, as shown in Figure 7. In this method, the overall formwork used consists of the previous fabricated component 40, 70 and the container, which defines the boundary where the components 40, 70, 80 are fabricated.

The Computer Numeric Controlled machine follows the input of an algorithm (discussed in detail below) that converts a virtual model of a component that is to be manufactured into a numerical code directing the injection nozzle along a path/curve in 3D space that allows to inject the exact quantities of material at the relevant points in the container 20. These algorithms operate based on the output of an algorithm of an embodiment described in detail flow. This algorithm uses a segmented input surface and crease virtual models of components to be manufactured. These models are created as if the components were stacked in the container with adjacent/touching top and bottom surfaces. This generative algorithm is discussed in detail below and ensures correspondence between the different components.

Materials Two different material types are used during the fabrication process, the material used to fabricate the components 40, 70, 80 and the material that forms the chemical barrier 50. The material used to fabricate the components 40, 80, 70 needs to be a fluid with the ability to dry or set. It also has to have a viscosity that can be controlled during the forming/casting process. Examples of this are cements, plasters, ceramics and polymers. Depending on the size of the nozzle of the machine used for applying the material within the container, it is also possible to use composites like concrete. Aggregates can be made out of gravels or crushed rocks such as limestone or granite as well as sands, steel or polypropylene fibers.

The material that can be used for the chemical barrier 50 depends on the material used to fabricate the components 40, 70, 80. If the components 40, 70, 80 are made out of cements or plaster, the chemical barrier 50 can be a detergent like soap. if the components 40, 70, 80 are made out of ceramics, a suitable material is alumina powder. And if the material used is a melted polymer, the chemical barrier 50 should be formed using a powdered material like talc. This layer forming the chemical barrier 50 is only applied when the material of the component 40, 70, 80 has sufficiently dried or set.

The components 40, 70, 80 can have internal reinforcement in order to improve its mechanical properties. The type of reinforcement can vary depending on the material used during the fabrication process and/ar the scale and the application.

Containers: The containers 20 that are used for the fabrication process can have very specific geometries. They may, for example, have a polygonal base e.g, triangle, square, rectangle, pentagon and hexagon, as shown in Figure 8. As shown in Figure 9, the cross-section of the container may, however) deviate from regular polygonal cross-sections. Figure 9, for example depicts an arrangement where a rectangle or square has been deformed to form a trapezoid, with sides deviating from the original rectangular/square shape by an angle a. The vertical extent of the container may also be such that one or more of the container edges deviates from the vertical direction by angle b (assuming that the container base is positioned horizontally). It will of course be appreciated that embodiments encompass arrangements with "regular' cross-sections/bases and/or containers with edges that set and at right angles from the vertices of the base Cast on Cast without container: Process: The process relies on depositing or injecting layers 10 of a very viscous fluid material7 which has the ability to set or dry, with a Computer Numeric Controlled machine. Several layers 10 of this material shape one component 40 from bottom to top. Once the superficial layer of a first component is dried or set, the machine deposits on the top surface a thin layer 50 of a chemical barrier. When this chemical barrier 50 is dried off, the machine deposits a new set of layers 60 of the viscous fluid material shaping the second component 70. This process is repeated to make the rest of components of a sequence. This process is carried out without any side walls defining a closed perimeter, as shown in Figure 10. The technology used is similar to the one used in the Contour Crafting technique shown in Figure 11 or MakerSot shown in Figure 12, but instead of using it to build single objects or structures, it would be used to make many components 40, 70, 80 one on top of the previous one, as discussed above.

Once all the components 40, 70, 80 of a sequence are fabricated, the components set/dry over a period of time before they are manipulated. The duration of this period depends on the material used and the size of the component. Thanks to the chemical barrier between components 40, 70, 80, the components 40, 70, 80 can be separated thereafter to be assembled in a complex surface, such as the surface shown in Figure 7.

in this method, the overall formwork used to build the components consists only of the previous fabricated component.

The Computer Numeric Controlled machines can operate based on an input geometrical shape input into it. The shapes are converted into instructions for the machine to inject the exact quantities of material in each point by the machine. The shapes may be created by algorithms described in further detail below.

Materials There are two different materials used during the fabrication process. The material used to fabricate the components 40, 70, 80 and the material that performs as a chemical barrier 50.

The material used to fabricate the components 40, 70, 80 needs to be a high viscous fluid with the ability to dry or set. Examples of this are cements, ceramics and polymers. Depending on the size of the nozzle of the machine, it is possible to use composites like concrete. The aggregate can be made out of gravels or crushed rocks such as limestone or granite as well as sands, steel or polypropylene fibers.

The material for the chemical barrier depends on the material used to fabricate the components. If the components are made out of cements or plaster, the chemical barrier should be a detergent like soap. If the components are made out of ceramics, the most suitable material would be alumina powder. And if the material used is a melted polymer, the chemical barrier should be a powdered material like talc. This layer will only be applied when the material of the component is dried or set.

The components can have internal reinforcement in order to improve its mechanical properties, if needed. The type of reinforcement can vary depending on the material used during the fabrication process, of the components and this application.

Cast on Cast with non-cast partiaJ formwork: Process: This process relies on using foam 90 (such as Styrofoam manufactured by the Dow Chemical Company), which, in one embodiment, has been previously carved with a Computer Numeric Controlled machine to define a desired surface, as a partial forrnwork, as shown in Figure 13. In this method a new component is cast in between the previous component 80 and the foam 90. Therefore, the previous component defines the bottom surface of the new component and the foam 90 shapes the top one.

The method thus uses only the half of the formwork that would be required for the component if it were manufactured in another way. Before using the previous component as formwork a thin layer of a chemical barrier 50 is applied and allowed to dry. All the process is carried out within side walls 20 that define a closed perimeter.

The formwork 90 can be a piece of foam or other material that can be given the required shape. Conventional foam materials that can be CNC milled are but one

example.

Once all the components of a sequence are fabricated, they are stored for a period of time to allow the components 40, 70, 80 to dry/set before they are manipulated. The duration of this period of time depends on the material used and the scale of the component. Thanks to the chemical barrier that it is applied in between components during the fabrication process, these can be easily separated to be arranged configuring complex surfaces, as shown in Figure 7.

In this method, the overall formwork used consists of the previous fabricated component, the shaped foam 90 and the container 20.

It will be appreciated that, while Figure 13 illustrates the casting of the uppermost component 80 using formwork 90, more than one of the components 40, 70, or even all of the components 40, 70, 80 may be manufactured in this manner, with a particular formwcrk being provided for each of the components 40, 70, 80 manufactured in this manner. The formwork defines the upper surface of the component 40, 70, 80 manufactured using the formwork 90. The formwork 90 thus takes the place of the CNC material application technique described in the last two examples. As only one formwork 90 is required for each component 40, 70, 80. The total number of formwork pieces is thus halved, when compared to known casting techniques. One exception to this may be the lowermost component 40, which may be cast on top of another formwork 90 to define its lower surface.

Materials There are two different materials used during the fabrication process. The material used to fabricate the components 40, 70, 80 and the material that forms as a chemical barrier 50.

The material used to fabricate the component 80 below any overlying forrnwork component 90 does not need to have a controllable viscosity. The material used can be any kind of casting material. Examples of this are composites, cements, ceramics and polymers.

The material for the chemical barrier 50 depends on the material used to fabricate the components. If the components are made out of cements or plaster, the chemical barrier should be a detergent like soap. If the components are made out of ceramics, the most suitable material would be alumina powder. And if the material used is a melted polymer, the chemical barrier should be a powdered material like talc. This layer will only be applied when the material of the component 40, 70, 80 is dried or set.

The components can have internal reinforcement in order to improve its mechanical properties, if needed. The type of reinforcement varies depending on the material used during the fabrication process, the scale of the components 40, 70, 80 and the application they are intended to be used in.

Containers The containers 20 that are used for the fabrication process have very specific geometries. They can, for example, have a polygonal base, such as triangular, square, rectangular, pentagonal and hexagonal shapes, as shown in Figure 8. Two angles a and b, as shown in Figure 9, can modify the shape of the base and the walls of the container in order to obtain different curvatures on a surface.

ALGORITHM I

In the following a first algorithm for determining the manner in which components may be cast on other components in a container so that they may be assembled in a curved surface (such as the surface shown in Figure 7) is described.

This first algorithm relies on the segmentation of an input surface (a real life replica of which is to be manufactured) into several parts. These parts are then arranged in a tubular enveloping surface that corresponds to the architecture of the container walls, if a container is used in manufacture, or that corresponds to perimeter within which material is deposited, if no container is used during manufacture. Second and third algorithms will be described after the description of the first algorithm.

SEGMENTATION OF AN INPUT SURFACE

Referring firstly to Figure 14, a surface 500 that is to be manufactured is used as an input. This input 3D surface can be designed and modeLed by any existing CAD program and described in widely used format, such as polygon mesh, NURBS (Non-uniform rational 8-spline) surface or others. Reference number 510 indicates the projection of the surface into the X-Y plane. This projection is segmented into unit lengths 520 and 530. The unit length 520 and 530 may be the same in the X-direction and in the Y-direction or may be different in the two directions. Individual segments correspond to individual components to be cast, if they were projected onto surfaces comprising the projection of the surface 500. The X-direction will be referred to as the U-direction when reference to the components is being made, while the V-direction will be referred to as the V-direction. The components extending in the U-direction are numbered sequentially, starting with component U and continuing to component U+n for ni-I components in the row of components. The components extending in the Y-direction are numbered sequentially, starting with component V and continuing to component V-i-k for k÷1 components in each of the columns of components. All of the projections of the components of the surface 500, however, have the same size in the X)U-direction. A)) of the projections of the components of the surface 500 also have the same size in the YR/-direction. The segmented surface 510 is also illustrated in Figure 15. It will be appreciated that segmentation into squares or rectangles, as is the case in Figures 14 and 15 is only one example of segmenting the projected surface 510.

The geometry of the projection of the components chosen may correspond to the shape of the container used for casting. As mentioned above, shapes other then square or rectangular may be chosen and such other shapes have already been discussed with reference to Figure 6. Figure 16 shows a different way of segmenting the projected are 510 using hexagons as the basis for this segmentation.

Segmentation of the projection 510 of the surface 500 creates edges 540 extending in the X1U direction and in the YN direction respectively (only three of these edges are labeled in Figure 14 for reasons of clarity) as well as projected vertices 550.

Verticafly above (i.e. at the same x-y position in an orthogonal XYZ coordinate system) the projected vertices on the surface 500 are vertices 560. These vertices 560 are the corners and first four (for rectangular or square components)/six (for hexagonal components) vertices of the components created by the cast on cast technique. These vertices will be referred to hereinafter as vertices 0 to 3 for rectangular/square components and as vertices 0 to 5 for hexagonal components.

It will be appreciated that the surface 500 is a virtual surface and may as such not have a thickness. The cast component will, however, have a thickness. This thickness may not be critical and may therefore be considered to lend a degree of freedom to the design process. A minimum thickness may, however, be prescribed to ensure the structural integrity of the components, or to ensure that the components can reliably fulfill their structural requirements. A maximum thickness may also be prescribed to ensure that the components fit a space that may be reserved for them within an overall design and/or to limit the amount of material used for creating the component to an economically viable level.

lt should be noted that, while the surface 500 shown in Figure 14 describes a smooth curve, this does necessarily have to be the case. In Figure 14 the surface 500 has in particular been segmented so finely that the connections between adjacent nodes can be approximated by lines. This is, however, not essential for the present invention, as will be described in more detail below. The upper or lower surface of each component can, for example itself comprise a number of undulations, so that adjacent nodes can no ranger validly be approximated as being connected in a linear fashion.

Even in this case it will, however, be appreciated that the surface 500 can be considered a first approximation of the surface that is to be created. As will become apparent from the discussion provided further below, the segmentation step illustrated in Figure 14 serves the purpose of providing the vertices 560 as an input for following steps of the algorithm. As the components themselves can have curved surfaces the size of the projections of the components in the plane 510 can be chosen such that the components have a size that is useful for the purpose to which the components are used. This size can vary widely and the present invention is not limited to a particular size range. The size of the components may, for example be in the order of centimeters or decimeters, for more intricate components forming part of a small surface but extend up to several meters, for example for structural components that are to form part of a façade or roof.

Figure 17 and 18 show axonometric views of the projection of two components 600 and 650 respectively. For component 600 the surface 510 has been segmented into squares, while for component 650 the surface 510 has been segmented into hexagons. Figures 17 and 18 provide a definition of the vertex numbering used hereinafter. As these figures provide a top plan view two numbers are provided on each corner of the respective polygons shown in these figures. The lower number hereby refers to the vertex at the indicated position that is part of the lower surface of the components, while the higher number refer to the vertex that is part of the upper surface of the component. It should be noted that Figures 17 and 18 are only provided for the purpose of defining the positions of the vertices of the components. As discussed above with reference to Figures 1 to 4, the faces of the components abutting other components can be inclined to create a curved surface using the assembled components. For this reason two vertices of a component are unlikely to be situated at the same XY position in the XYZ coordinate system shown in Figure 14, even if they relate to the same upwardly extending edge of a component.

As a curved surface is to be generated, two adjacent components share vertices, when assembled in the surface. If for example a surface is to be generated wherein the top surface defined by the component is continuous, then vertices 4 and 5 of a square or rectangular component (U+1, V) are the same points in an XYZ coordinate system of Figure 14 as vertices 7 and 6 of component (U, V). Equally vertices 5 and 6 of a square or rectangular component (U, V-i-i) are the same points in an XYZ coordinate system as vertices 4 and 7 of component (U, V). For hexagonal components vertices 6 and 7 of component (U+1, V) correspond to vertices 10 and 9 of component (U, V), while vertices 7 and 8 of component (U+1, WI) correspond to vertices II and 10 of component (U, I).

If the tower surfaces of the components are to be arranged in gap free manner, then vertices 0 and I of a square or rectangular component (U÷i, V) are the same points in an XYZ coordinate system as vertices 3 and 2 of component (U, V). Equally vertices I and 2 of a square or rectangular component (U, Wi) are the same points in an XYZ coordinate system as vertices 0 and 3 of component (U, V). For hexagonal components vertices 0 and I of component (Ui-i, V) correspond to vertices 4 and 3 of component (U, V), while vertices I and 2 of component (Ui-I, V+1) correspond to vertices 5 and 4 of component (U, \/).

If the surface created by the components is not only to be gap free but also devoid of any steps in the upper or lower surface created, then both of the above correspondence conditions have to be fulfilled.

It wfll be appreciated that the above correspondence conditions only apply while the components are arrangedllocated in the XYZ coordinate system. When the components are formed in the containers, vertices 0 and 4 of consecutive components are arranged along a first upwardly extending edge of the container, vertices I and 5 of consecutive components are arranged along a second upwardly extending edge of the container, vertices 2 and 6 of consecutive components are arranged along a third upwardly extending edge of the container and vertices 3 and 7 of consecutive components are arranged along a first upwardly extending edge of the container. The same applies to the pairs of vertices 0/6, 1/7, 218, 319, 4/10 and 5/11 of the hexagonal shape shown in Figure 18 and to the six corresponding upwardly extending edges of an appropriate container used for casting these components.

The components are of course not only characterized by the location of their vertices but also by their relationship with adjacent components. In the following one way of describing this relationship in parameter form is discussed. Three parameters are used in an embodiment to describe the relationship between adjacent components.

These parameters are referred to as growth' (which in this embodiment is either an integer or a decimal fraction)1 angle' (which in this embodiment is also either an integer or a decimal fraction) and mirror' (which in this embodiment is a Boolean). These parameters will be described in detail in the following.

Growth: This parameter is calculated for each component, staring with component (U, \O.

and following the U direction. For rectangular components the manner in which the growths parameter is calculated is shown in Figure 19A. For hexagonal components a similar illustration is provided in Figure 198.

Growth Rectangular Components Hexagonal Components The growth parameter equals the absolute Growth parameter equals the average of difference between the Z coordinate of distance Cl and G2, with GI being the vertex 3 of component (U, V) and the Z absolute difference between the Z coordinate of vertex 3 of component (U+l, coordinate of vertex 4 of component (U, V), as shown in Figure 19A. V) and the Z coordinate of vertex 4 of component (U+l, V) and G2 being the absolute difference between the Z coordinate of vertex 5 of component (U, V) and Z coordinate of vertex 5 of component (Ui-I, V), as shown in Figure 19B As the growth variable is defined as the absolute difference between two Z-coordinates it is always positive.

Angle: The angle parameters is only calculated once for each row of components extending along the X-direction, based on the coordinates of the vertices of the first/leftmost components in the two adjacent rows, as shown in Figures 20A and 208. Angle

C0rn ponents -Hexagonal Components The angle parameter equals half of The angle parameter equals half of the arctangent of a decimal fraction obtained arctangent of a decimal fraction obtained by dividing the difference between the Y by dividing the average of the differences coordinate of vertex 3 of the component between the Y coordinates of vertices 4 (U, V) and the V coordinate of vertex 3 of and 5 of component (U, V) and vertices 4 the component (U, V+1) by the difference and 5 of component (U, 1÷1) by the between the Z coordinate of vertex 3 of average of the difference between the Z component (U,V) and the Z coordinate of coordinates of vertices 4 and 5 of vertex 3 of component (U, V-i-I), as shown component (U,V) and vertices 4 and 5 of in Figure 20A. component (U, 1÷1), as shown in Figure 20B.

From the above it will be clear that the parameter angle' describes the slope of the line connecting vertices 2 and 3 of component (U, Vii) for rectangular components and an average slope value computed based on the line connecting vertices 3 and 6 and the imaginary line connecting vertices 2 and 5 for hexagonal components. It will be appreciated that other ways of defining the slope of a component may be used, based for example on different vertices and/or other average values. Moreover, while the above only takes the slope of the component of the next row into account, the anglet variable may also take the slope of the components the current row into account.

Mirror The mirror is calculated for each component, starting with the first component in each row and continuing in the U-direction.

Mirror Rectangular Components Hexagonal Components Mirror True if: Mirror = True if: -the Z coordinate of vertex 3 of a -the average of Z coordinates of given component (U, V) is smaller vertices 4 and 5 of given than Z coordinate of vertex 3 of component (U, V) is smaller than component (U-I, V) which at the average of Z coordinates of same time is greater than Z vertices 4 and 5 of component (U-coordinate of vertex 3 of I, V), which at the same time is component (U-2, V); greater than average of Z or coordinates of vertices 4 and 5 of component (U-2, V) -the Z coordinate of vertex 3 of a or given component (U V) is greater -the average of Z coordinates of than Z coordinate of vertex 3 of vertices 4 and 5 of a given component (U-I, V), which at the component (U, V) is greater than same time is smaller than Z average of Z coordinates of coordinate of vertex 3 of vertices 4 and 5 of component (U-component (U-2, V).

1, V) which at the same time is smaller than average of Z coordinates of vertices 4 and 5 of component (U-2, V).

Otherwise, Boolean false value is Otherwise, Boolean false value is assigned to mirror parameter assigned to mirror parameter It will be appreciated from the above that the parameter mirror is "True" only if two components adjacent in the U-direction form a peak or trough according to the above definitions. Otherwise the parameter mirror is "False".

MODELING VIRTUAL COMPONENTS FOR CAST ON CAST MANUFACTURING

The coordinates XYZ coordinates of the vertices and the three variable growth', angle' and mirror' serve as an input for an algorithms that computes the vertices of virtual components that are to be created by the cast on cast technique. These coordinates will need to be calculated relative to a coordinate system that is defined with respect to the container to allow manufacturing machinery to create the relevant component within the container.

As discussed above, in one embodiment two side walls of the container are parallel to each other. These two side walls will be adjacent the end faces of the components discussed above with respect to Figure 1, that is the end faces that are orthogonal to the X-direction. The two side walls of the container that are orthogonal to these parallel side walls are inclined relative to the base of the container. The inclination angle of these side walls (shown as 420 in Figure 4, for example) corresponds to the parameter angle' calculated for the row of components that is to be cast in the cast on cast process.

It will be appreciated that, for the components to be assemblable in to a continuous curved surface, the manner in which the missing vertices are calculated from the vertices determined in the above discussed segmentation method and from the there above discussed parameters has to follow some rules. These rules may be that: a) the surface is gap free; and/or b) the surface is step free.

Either of criteria a) and b) above may apply to a single surface created by the components once assembled (say, for example, in a situation where the other surface is not visible once the components have been assembled, as may be the case, for example, in facades and ceiling structures) or to both the upper and lower surface of the structure created by the assembly of the components.

The following discussion is based on the assumption that both criteria a) and b) are to be fulfilled for both the top and the bottom surface created by the cast components, once assembled. For two adjacent components to fulfil these two criteria, the conditions illustrated in Figure 21 may be fulfilled. As discussed above, the components are modelled in the direction of increasing X-direction and of increasing V..

direction, starting with the components with the lowest U and V values and modelling all components in a row before starting the modelling of the next row (by modelling the component of that next row that has the lowest U value). As can be seen from Figure 21 then, any component that is to be modelled (the current component to be modelled in Figure 21 is component (U', V')), already has two neighbouring components, namely component (U',V'-l) in the preceding row and component (U', V') in the same row.

Therefore the relative position of seven of the eight vertices of component (U'-l, V') are already determined when the component is to be modelled. The position of vertices 0 to 3 are known from the above discussed segmentation exercises, with the positions of vertices 4 to S have previously been determined when the surrounding three components have been modelled. The following condition therefore apply.

Number of Vertex of Corresponds to: component U',V' -__V3 V3(U'-l,V) V1 V2(U'-l,V'); V3(U'-I,V-1); V0(U',V'-l) V2 V3(U',V'-I) V3 known from segmentation -V4 V7(U'-l,V') V5 V6(U'-l,V'); V7(U'-l,V'-l); V4(U',V'-l) V6 V7(tJ',V'-l) V7 to be defined in the modelling process Although only one of the vertices of each component (vertex V7) would have to be defined in the modelling process of all components but the components of the first row and the first column it is still necessary to determine the coordinates of most of the vertices within the coordinate system of the container to enable forming machinery to create these components.

It should be noted that the limitation V5 (U', V') V6(U'-l,V') V7(U'-l,V'-l) V4(U',V'-1) only applies if the top surface shown in Figure 21 is to be entirety step free.

if small steps in the top surface are acceptable, then this restraint can be loosened, so that only V5 (U', V') = V6(U'-l,V') and V7(U'-l,V'-l) = V4(U',V'-1) are true. This means that the top edges between adjacent components in a row (that is the top edge between components (U'-l,V'-.l) and (U', V-I) and the top edge between components (U-I, V') and (U',V')) to not meet in one point. This, in turn, may mean that the top edges of components in adjacent rows may not coincide exactly, so that steps between rows may occur. It will be appreciated that this may be entirely acceptable, for example if a large number of components are used to construct a cuMng surface, wherein steps between individual rows are small.

Further constraints may be imposed upon the forming process by the cast on cast nature of the forming process. This is particularly helpful when defining the components of the first row as well as those of the first column. In a cast on cast technique that does not leave/require any voids between components (the above discussed chemical barrier can be ignored for this purpose because of its thin nature), it is, for example known that the upper surface of the very first component generated by the casting process should be the same as the lower surface of the component that is to be cast on top of this very first component. In Figure 21 this means that the lower surface of component (U', V-I) defines the shape and size of the upper surface of the component (U'-l, V'-l). Each of the surfaces for the components in a row, with the exception of the top surface of the very last component in the row, is thus defined through the segmentation process and by the use of the above discussed conditions a) and b) for both surfaces in a void free cast on cast casting process. If the casting process does not necessarily have to be void free, this restraint may be loosened and the top surface of the previously cast component may not, or not entirely, define the bottom surface of the next component to be cast.

A further available degree of freedom is the thickness of the first component to be modelled, namely the component in the first row and the first column (component (U'-l, V'-l) in Figure 21). The thickness of this component can be chosen as desired, for example in a manner that saves material cost, or based on other optimization criteria. It will be appreciated that, if both conditions a) and b) above are to be fulfilled, then the thickness of the first component (U'-l, V'-1) automatically also determined the thickness of component (U', V'-I), at least at the interface between the two components. There is thus a limited range of choices for the thickness of the first component to be modelled. It will be appreciated that, within an available range of thicknesses, a desired thickness may be chosen, for example, using an iterative process in which an optimization criterion is optimized for all of the components that are to be cast. Once all of the components have been modelled once it may, for example, be determined if all of the components fulfil structural stability requirements, based on their dimensions. Should this not be the case, then the thickness of the first component may be varied in a manner that leads to components that have the required thickness/stability. Equally, the thickness of the first component can be altered so that the overall amount of material used for all components is minimized, for example.

From the above it will be appreciated that all of the upper and lower surfaces of the components are either known, or can be defined with some degree of freedom.

The upper surface of the last component in each row in particular is not constrained by any requirements other that the thickness of the component has to be larger than a minimum thickness prescribed by stability requirements and that the thicknesses of all adjacent components in the column also have to be larger than a minimum thickness prescribed by stability requirements.

In the following a manner in which the coordinates of the vertices of the components within the casting containers can be calculated is discussed. It will be appreciated that the same manner or calculating coordinates may be used even if no such casting container is used.

The input for this calculation is the UV grid generated by the above described segmentation routine and the growth, angle and mirror parameters. The growth and mirror parameters are assigned for each component, while the angle parameter is assigned for each sequence in V direction (i.e. for each row of components). The container for casting the components of a row is assumed to have two side walls extending parallel to each other (to form the parallel end faces that are, in the assembled curving surface, orthogonal to the X-direction) and two side walls that are inclined by an angle that corresponds to the angle' parameter for the row of components that is to be cast in the container.

The components are generated within the container starting with component (U = 0, V = 0) and following the U direction, eg. next component to be generated is (U = 1, V = 0), then (U = 2, V 0) and so on. Once the last component in first line (U max, V = 0) is generated, the next line is started by generating first component of second Sine (U = 0, V 1). Then again the components are being generated following U direction, as described above (U 1, V = 1; then U = 2, V = 1; and so on). The whole process of generation of the components follows this logic until the algorithm generates the last component of the surface (U max, V = max).

The algorithm includes two main methods of generating components: one operating in the U direction and another one, generating first components of the lines in the V direction. The method of generating components in the U direction also has two modes of operation: one when Boolean value of mirror parameter is true and another one when it is false. As mentioned above, the mirror parameter indicates whether or not a peak or trough is present in the UJX-direction.

The following description focuses on methods of generating components for rectangular and for hexagonal components. However, other component cross sections may also be used in other embodiments, as discussed above.

Each vertex is conceived as 3D point, and set of 8 or 12 vertices is used to generate polygon mesh along those points. The methods of generating components both in U and V directions are based on assigning coordinate values in 3D (X, V, Z) within the container to each vertex.

GENERATING COMPONENT IN THE U DIRECTION

Rectangular components The first component in the first row can be generated taking into account only a smail number of constraints. The following description therefore merely relates to a single way of creating the coordinates of the vertices of this first component within the first container. The X and V coordinates of one of the upper and lower surface of this first component are determined by the shape of the component that is to be replicated and can be determined without undue burden based on the following manner of determining the Z-coordinates for the vertices: Zj3 = growth parameter (UV) -thickness (can freely be chosen) Zveni growth parameter (U,V) -thickness * 2 (can freely be chosen) growth parameter (U,V) -thickness (can freely be chosen) = growth parameter (U,V) growth parameter (U,V) Zven5 = growth parameter (U 1) -thickness (can freely be chosen) = growth parameter ((J,V) = growth parameter (UV) + thickness (can freely be chosen) In the following the manner in which the coordinates of the vertices of rectangular components in a container can be calculated for components for which the Boolean value of mirror parameter in the sequence from 0 to U has been true an even number of times (this includes a situation where the mirror parameter has been true zero' times) is discussed in detail. This discussion is followed by a discussion of how the coordinates of the vertices are calculated if the 8oolean value of mirror parameter in the sequence from 0 to U has been true an odd number of times, followed by a discussion of how the coordinates within a container of hexagonal components can be calculated.

Boolean mirror parameter has been true an even number of times in the row The following discussion is based on the assumption that a first component has a positive gradient in the U-direction. The condition that the mirror parameter has been true an even number of times thus means that for all components concerned the slope of the component is again positive, as an even number of peaks/troughs has been encountered.

The manner in which coordinates propagate from vertices of previously modelled components to vertices of components that are being modelled is shown in Figure 22A.

Vertices V0 -V3 of the new (current) component (U,V) simply inherit the X, 1 and Z coordinates of vertices 4 to 7 of the previously generated component (U-i, V) within the container. This inheritance of the coordinates sinipty specifies that the tower surface of the component (U,V) is defined by the upper surface of the component (U-i, V). The following is therefore true.

Vertex O(U (X, Y, Z) = (XVERT 4(U-1, V), YVERT_4 (U-I! v) ZVERT4(U1, v)) (1) Vertex 1(U, (X, V, Z) = (XVER,'s (U-I, , (U-I, v) ZVERT_5 (U-i, V)) (2) Vertex 2(U, (X, Y', Z) = (XvER-re (U-I, V) VERT_6 (U-I, , ZVERT6 (U-I, v)) (3) Vertex 3(LJ, v (X, Y, Z) (XVERTJ (U-I, V), YVERT_7 (U-i, V)' ZVERT (U-i, (4) It is to be noted that the above X, Y and Z coordinates are those within the container, rather than those within the surface that is to be constructed. The coordinates of vertices 4 to 7 of the current component (U, V) have to be calculated. It is known that two of the side walls extend in parallel for this reason the X coordinates of Vertices 4 to 7 can again be inherited from the underlying vertices, so that: XVERT_4(U, V) = XVERT_O(U,V) = XVERT_4 (U-i, V) (5) XVERT_5(U, V) = XVERT 1(U,V) = XVERT_5 (U-i, (8) XVERT_6(U, V) = XVERT_2(U.V) = XVERTG (U-i, /) (7) XVERT_7(U, V) = XVERT_3(U,V) = XVERT_7 (U-I! V) (8) The Z-coordinates (within the container) of vertices V4 and V5 of component (U,V) are further inherited from vertices V7 and V6 of previous component (U-I,V), so that: ZvERT4((j, V) = ZVERyj(%Ji,v) (9) ZvERT_5(U, v) ZVER-rs(U.i,V) (10) if the row for which the components are generated is the first row of the surface (V=0), then the Z-coordinates (within the container) of vertex V6 of component (U,V) is also inherited from vertex V7 of previous component (U-I V), so that: ZvERT_e(U, V) = ZvERTJ(U1,v) (ii) The foflowing description relating to Figure 24 illustrates conditions that need to be fulfilled the curved surface, once assembled is to be step free on both sides.

Figures 23 A to D illustrate components (U-i, V-I), (U, V-i), (U-i, V) and (U, V) respectively, which have been modeled/created in this order. Indicated in Figure 23 A are lengths a(U-I, V-I), b(U-I, V-I) and c(U-I, V-I). These lengths are the lengths of the component edges along the internal edges of the container. It is pointed out that in a preferred embodiment length b(U-1, V-I) and c(U-I, V-i) of the first component modeled may be chosen to be the same, although the present invention is not limited to this choice.

It will be appreciated that if both of vertices V3 and V7 of component (U-i, V-i) shown in Figure 23A are to coincide with both of vertices V0 and 14 of component (U, V-I), when the components are assembled in the curved surface, then the length between vertices V0 and V4 of component (U, V-I) has to be the same as length a(U- l,V-I). Component (U, V-I) therefore inherits length a(U-I, V-I) from component (U- 1, V-I), albeit for application to vertices V0 and 14. Length c(U-i, V-i) is equally inherited from component (U-I, V-I) for application to vertices V., and V5 of component (U, V-I). A new length a(U, V-I) can be calculated for component (U, V-l).

Component (U-i, V) shown in Figure 23C inherits length a(U-i, V-I) for application to vertices 4 and V5 and length b(U-1, V-i) for application to vertices V., and V5. A new length a(U-1, V) can be calculated for component (U-I, Component (U, V) shown in Figure 23D inherits length a(U-I, V-I) for application to vertices V1 and V5, length a(U, V-I) for application to vertices V2 and V6.and length a(U-I, V) for application to vertices V0 and V4. A new length a(U, V) can be calculated for component (U, V).

From the above it is clear that the length of the edge of a current component (U,V) extending between vertices V2 and V5 has to be the same as the length of the edge of component (U, V-I) extending between vertices V3 and 17. If the component modeled is not in the first row of components, then the length of this edge can be calculated to be L (ZVERT7(u,v1)-ZVER-r_3(u,v1))/cos(anglev4. It is therefore true to say that:

-

(ZVERTSJV) -ZvRy2(u,v))/cos(angle) Wherein, as discussed above, angIe1 and anglev are the angle parameters of the preceding and the current row respectively. Therefore: ZVERT6(U,V) -ZVERT2(U,V) (ZvERT_7(u,v1) -ZVERT_3(u,v4))* cos(anglev)/cos(anglev.4 ZVERT6(U,V) ZVERT_2(U, + (ZVERT_7(u,v1) -ZvERT_3(u,vi))* cos(angIe)/cos(angle..1) (12) ZVERT_6(u1,V) + (ZVERT7(UV1) -ZvERT_3(U,v.i)) cos(angle)/cos(angle..1) As disucssed above, the growth parameter is defined as the change in the Z-coordinates between vertices V0 and V3 of a component (U,V) (with Vo(u,=V3(ui,v)).

This change is also applied to the Z-coordinates of vertices V4 and V7, so that: ZVERTJ(UV) (13) ZVERT4(U,V) + growth(U ZVERT.7(u1,v) + growth(U, Turning now to the Y-coordinates of vertices V4 to V7 of component (U,V) within the container, it will be appreciated that the change in the Y-coordinates depends on the parameter angle as well as on the thickness' of the component (U,V). The Z-coordinate of vertex V4(U,V) has already been defined in (9) above. Figure 23 shows the manner in which the Y-coordinate of vertex V4 of component (U,V) can be calculated. It is true to say that: tan(anglev) = AY / (ZVERT 7(U-I,V) -ZvERT4(uiv) and consequently: YVERT_4(UV) = YVERT 4(U-1,V) + (14) = YVER-r_4(U1V) + tan(anglev)* (ZvERT7(Ui -ZVERT4(u1,v)) YVERT_5(U,V) and YVERT_6(U,V) (the latter only for the first row) can be similarly calculated (taking into account (10) and (11 respectively) to be: YVERr_5(u,v) = YVERT5 (U-i, - 6 (Ui,VyZVERTfi(Ui, V)) (15) YVERTJ(UV) = YVERT6 (U-i, V)-tan(angleç,)) (ZVER17 (U-i, VY-ZVERT6 (u-i, V)) (16) If the row for which components are modelled is not the first row of the surface, then the a(U,V-1) shown in Figure 23B is applied to vertices V2 and V5, as shown in Figure 23D. This height a(U-1,V) can be expressed as: a(U-1,V) = (ZVERT_7 (U, VI)ZVERT_3 (U, Vi�/cos(angleV.i) = AYVS(UV) / sin(anglev) Therefore: YVERT6(U,V) = YVERTG (U-i, -AYV6(U,V) (17) = YVERT,6 (U-i, V) -(ZVER-J-_7 (U, V-i)-ZVERI_3 (U, vi)) sin(angle)/cos(anglev) The calculation of YVERT7 (U, V) relies again on the growth parameter and can be calculated as: YVERT_7 (U, V) = YVERT (U-i, V)+tan(angle)*growth( (18) In Summary therefore, if the mirror parameter has been true an even number of times, the X, Y and Z coordinates of vertices V4 to V7 of component (U, V) within a casting container can are: Vertex 4(U, 1) (X, C Z) = (XVERT_4 (U-i, , YVERT4 (U-i, (U-i, V) ZVER-I-4 (U-i, ZVER',-7 (U-i, Vertex 5(U, (X, Y, Z) = (XVERT_5 (U-i, V)7 YVERT_5 (U-i, (U-i, V) ZVERT5 (U-i, V))7 ZVERT_6 (U-i, v)) For the first row of components: Vertex 6(U, J) (X, Y, Z) (XVERT6 (U-i, V), YVERT_s (U-i, (U-i, V) ZVERT_6 (U-i, ), ZVERT7 (U-i, I)) For rows of components other than the first row: Vertex 6(U, \J) (X, V, Z) = (XVERTB (U-i, V)7 YVERT_6 (U-i, V)-sin(angle(V)) I cos(anglev.

i))RVERTJ (U, V1YZVERT_3 (U, V-i)), ZVERT_6 (U-i, V)+cos(angle) I (U, V-i) ZVERT3 (U, V-i))) and Vertex 7(U, \J) (X, Y, Z) = (XVERT_7 (U-i, V), YVERT_7 (U-i, \J)ftan(angie(V))*growth( , ZVERT_7 (U-i, V)+growth(U V)) BOOLE,AN MJRROR PARAMETER HAS BEEN TRUE AN ODD NUMBER OF TJMES IN THE ROW The above discussion was based on the assumption that the mirror parameter for the row has been true an even number of times. The following discussion will deal with the converse scenario, where the mirror parameter has been true an odd number of times.

The manner in which coordinates propagate from vertices of previously modelled components to vertices of components that are being modelled is shown in Figure 22B.

It will be appreciated that, due to the sloping side walls of the container, later cast components are larger than earlier cast components, As discussed above, the slope of the components is positive if the mirror parameter has been true an even number of times. This is in conformity with the generation of consecutive components that increase in size. If, however, the mirror parameter has been true an odd number of times the slope of the components will be negative, requiring a decrease in the size of consecutive components. Such a decrease in size can either be achieved through inwardly sloping side walls of the container Alternatively the components can be cast in an outwardly sloping container, as is the case for the above discussion, but in reverse order The following discussion is based on the latter technique, so that a next component to be modelled will be situation below a previously modelled component and so that the lower surface of the previously modelled component defines the upper surface of the next component to be modelled. Put in other words, once the coordinates of vertices VO to V3 of component (U-i,V) are known, these coordinates can simply be inherited by vertices V4 to V7 of component (U,V). The component can, once modelled, be cast in inverse sequential order, when compared to the modelling sequence.

In view of the above disclosure the person skilled in the art will be able to calculate the coordinates of the vertices of modelled components with ease, using the following equations: Vertex °(U, (X, Y, Z) = (XVERTO (U-i, V), YVERT_O (U-i, (ti-i, 1) (19) ZVERTO (U-i, ZVERT_3 (U-i, v)) Vertex 1(U, V') (X, i', Z) (XVERT_i (ti-i, , YVERT_1 (U-I, (U-i, V) (20) ZVERT_i (U-i, V))1 ZVERT2 (U-i, v)) If the components modelled are in the first row of components (in the V-direction), then values for vertex V2(U are assigned in following way: Vertex 2(U,V) (X, Y, Z) YVERT_2 (U-i, \fUan(angle(v))(ZVERT,,) (U-i, tj (21) ZVER-I-2 (U-i, ZVERT3 (U-i, V)) If the components modelled are not in the first row of components, then values for vertex V2(U,v) are assigned in following way: Vertex 2(U,V) (X, Y, Z) (22) (XVERT2 (U-i, V) YVERT_2 (U-i, V)+Sin(angle(V)) / (U, v1y-ZVERT3 (U, v-i)) ZVERT2 (U-i, -cos(angle) / cos(anglei))*(ZvERT7 (U, V1yZVERT_3 (U, V-i))) Vertex 3(U, V) (X, Y Z) = (XVERT,.3 (U-i, , VERT) (U-i, V)-tan(angle(V))growth(UV) (23) ZVERT_3 (U-i, V)-growth(U, V)) Vertex 4(U, (X, V, 1) = (XVERTO (U-i, V), YVERT_O (U-i, \F), ZVERT_O (U-i, V)) (24) Vertex 5(U, (X, Y, Z) = (XVERT_i (U-i, V) YVERT i (U-i, V), ZVERT_1 (0-1,")) (25) Vertex 6(U,V) (X, Y, Z) = (XVERT_2(Ui,V), YVERT_2(Ui,V), ZVERT_2(U.1,V)) (26) Vertex 7(U, (X, Y, 1) = (XVERT3 (U-i, I), YVER-I-_3(U1, v, ZVT (U-i, V)) (27)

HEXAGONAL COMPONENTS

In view of the above detailed discussion relating to the calculation of the vertices of rectangular components, the person skilled in the art will also be able to calculate the X,Y and Z coordinates of hexagonal components within a container Using the following equations.

BOOLEAN MIRROR PARAMETER HAS BEEN TRUE AN EVEN NUMBER OF TIMES IN THE ROW

Vertex °(UV) (X, Y, Z) = (XVERTG(Ui,V), YVERTB(U,i,V), ZVER-1-_6(Uj,v)) (28) Vertex l(U,V) (X, Y, Z) = (XVERT7(Ui,V, YVERT_7(U1,V), ZVERT_7(U1,v)) (29) Vertex 2(U, (X, Y, Z) = (XVERT8 (U-i, , YVERTO (U-i, , ZVERT_8 (U-I, V)) (30) Vertex 3(U, \J) (X, Y, Z) = (XvER',-_9(Ui, V) YVERT_9 (U-i, V), ZVERT_9 (U-i, v)) (31) Vertex 4(U, V) (X, 1', Z) = (XVER-,-_io (U-I, V), YVERT_ID (U-i, \/), ZVERT_iO (U-i, (32) Vertex 5(U,V) (X, Y, Z) = (XVERT11 (U-i,V)1 YVERT ii (U-i,'!)1 ZvER--ii (U-1,V)) (33) Vertex 6(U, (X, Y, Z) = (XvERT6 (U-i, V YVERT_6 (U-i, V)4-tan(angle(V))(ZVERT10 (U-i, -(34) ZVERT6 (U-i, V))1 ZVERTiO (U-i, V)) Vertex 7(U,V) (X, Y, Z) = (XVERT7(U1.V)I (35) YVERT_7 (U-i, (U-i, \JyZ\/ERT7 (U-i, V))1 ZVERTg (U-i, V)) If the components modelled are in the first row of components (in the V-direction), then values for vertices V8(U,V) and Vg(U,V) are assigned in following way: Vertex 8(1), (X, Y, Z) (XVERT6 (U-i, V)1 YVERTS (U-i, S (U-i, V)-ZVERT_8 (36) (U-i,V)), ZVERT6 (U-i, Vertex 9(U, (X, Y, Z) = (XRTg (U-i, V)1 YVERT9 (U-i, ii (U-i, VIZVERT_9 (37) (U-i,V)) ZVERT1 1 (U-i, V)) If the components modelled are not in the first row of components, then values for for vertices V8(U,V) and Vg(U,V) are assigned in following way: Vertex 8(U,V) (X, Y, Z) (38) (XVERT_8 (U-i, V)1 YVERT_8 (U.i, V))-sin(angle(V)) I (U+i vi)-ZvERT_o (Uti, V-i)), ZVERT_8 (U-i, v)-1-cos(angle(v)) I (WI Viy'ZVERT_O (UI-i, V-i))) Vertex 9(U,V) (X, Y, Z) = (39) (XVERT_9 (U-i, V) YVERT_g (U-i, vsin(anglev) / ii (WI vi)-ZVERT_5 (U-'-i,V-i)), ZVERD (U-i, +cos(angIe) / cos(angIei))*(ZvRT (U÷l v,)-ZvERT_5 (U÷i, V-i))) Vertex IO(U,V) (X, Y, Z) = (XVERT_iO(U1,V), YVERT_iD (U-i, \/)4tafl(3flg1())*gt0th(,v), (40) ZVERTO (U-i, +growth(U V)) Vertex il(U,V) (X, Y, Z) = (XVERTII (U-i,V), YVERT_ii (U-i, V)+tan(angle)I-growth(UV), (41) ZVERTJ 1 (U-i, %J)+growth(u V))

BOOLEAN MIRROR PARAMETER HAS BEEN TRUE AN ODD NUMBER OF TIMES IN THE ROW

Vertex O(U,V) (X, V, Z) (XvER1-o(tji,v), YVERTO (U-i, (U-i, \f)-ZVERT_O (42) (U-i, V))1 ZVERT4 (U-i, /)) Vertex 1(U,V) (X, Y, Z) (XvEsTi (U-i,V)i (U-i, \J)+tan(angle(V)) (ZVERT3 (U-i, VYZVERTI (43) (U-I, V)) ZVERT3 (U-i, V)) If the components modelled are in the first row of components (in the V-direction), then values for vertices V2(uV) and V3(u,V) are assigned in following way: Vertex 2(U, (X, Y, Z) = (XVERT2(U.1,V)I YVERI2 (U-i, V)4tan(angle(V)) (ZVERT) (U-i, \J)ZVERT2 (44) (U-i,V)), ZVERTO (U-i, I)) Vertex 3(U,V) (X, V, Z) = (XVERT_3(Ui,V), YVERT_3 (U-i, V)÷tafl(at.Tgle(V))*(zVERT.fi (U-i, VYZVERT3 (45) (U.i,V)), ZVER,-5 (U-i, \i)) If the components modeUed are not in the first row of components, then values for for vertices V2(u,V) and VU,V) are assigned in following way: Vertex 2(U,V) (X, V, Z) (46) (U-i, V)1 YVERT_2 (U-i, V))+sin(angle(V)) / cos(anglel))*(ZVERr 8 (U-ri Vi)"ZVERTfi (U-ri, ZVERT_2 (U-i, V)-cos(angle(V)) / cos(anglel))*(ZvcRT 6 (U-rI vi)-Zvcnr_o (ti-i-I, V-i))) Vertex 3(U, V) (X, Y, Z) = (47) (XVERI_3 (U-i, V)1 YVERTS(Ui, v))+sin(angle(v)) / O5(angIe))*(Zfl1 ii (U+1 V1)-ZVERL5 (u-ri, V-i))1 (U-i, v))-cos(angle(v)) I cos(angIel))(ZVEflTII (Ui-i V-i2VERT_5 (Ui-i, V-i))) Vertex 4(U, (X, 1, Z) = (XVERT4 (U-i, \(), YVRT4 (U-i, V)-tan(angle(V))growth(U, "_), (48) ZVERI_4 (U-i, -growth(U, V)) Vertex 5(U, V) (X, 1, Z) = (XVERT_S V), YVERTS (U-i, V)-tan(angle(l,/))growth(U,V), (49) ZVERT_5 (U-i. \/)-growth(U V)) Vertex 6(U, (X, Y, Z) = (XVERT_O (U-i, V), YVERT_O (U-i, V) ZVERT_O (U-i, V)) (50) Vertex 7(U, (X, Y, Z) = (XVERT_i (U-i, V), YVERT_I (U-i, , ZVFRT_I (U-i, v)) (51) Vertex 8(U, (X, Y, Z) = (XVERT_2 (U-I, \J), YVERT_2 (U-I, vj ZVERTJ (U-I, 1)) (54) Vertex 9(U, (X, Y, Z) = (XVERT3(U1,V, YVERT3(U1,V), ZVERT_3(U.I,V)) (55) Vertex 1 0u, v (X, Y, Z) (XVERT4 (U-I, \t' YVERT4 (U-i, V), ZVERT.4 (U-I, ) (56) Vertex 1 1(U, V) (X, T', Z) = (XVERT5 (U-i, V), YVERT.fi (U-I, V), ZVERT5 (U-I, 1)) (57)

GENERATING THE FJRST COMPONENT IN V DIRECTION

The above discussion has focussed on a method of generating the components in one row. It will be appreciated that a first component in each row may inherit properties from an adjacent first component in a preceding row, unless of course the component in question is the first component of the first row. The following discussion focuses on the manner in which the XYZ coordinates of the vertices of the first component in a row (other than the first row are calculated in one embodiment).

RECTANGULAR COMPONENTS

A method of assigning coordinate values when generating the first component of a row in the V-direction is shown in Figure 25.

The X-coordinates of all of the vertices of component (U,V) within the container are the same as the X-coordinates of the adjacent component (U,V-1) in the preceding row, so that they can simply be inherited.

The Z values of the vertices of component (U,V) are inherited from vertices of component (U, V-i) in the following manner. Z coordinates of vertex V0(U, V) and vertex V2(U, V) are inherited from vertex V3(U, V-i). Further, the Z coordinate of vertex 1(U, V) is inherited from vertex V0(U, V-i) . Respectively the Z coordinates of vertex V4(U, V) and vertex V6(IJ, V) are inherited from vertex V7(U, V-i), and Z coordinate of vertex V5(U,V) is inherited from vertex V4(U, V-i) -The Z coordinate of vertex V3(U, V) is inherited from vertex V7(U, V-i).

The inherited information is then translated in the following manner to make sure the correspondence between edges that are located on enveloping surfaces that might have walls sloping in different angles. The following translation example relste to the Z coordinate of vertex V3(U,V). First the actual length B of the edge of the component (U, V-i) along the edge comprising vertex V0 is calculated according to: B cos(angle(Vi))*(ZVERT3 (U, V-i)) (58) Then this length B is put on the edge of new enveloping surface and Z coordinate of vertex is calculated: Zv0 (U, V) = cos(angle(V)) / B Therefore: Zvo (U, V) = cos(angre(V)) / cos(angle(Vi))*(ZVERT_3 (U, V-I)) (60) The Z coordinates for other vertices are calculated in the same manner, so that: Zvi(U, V) = cos(angle(V)) I cos(angle(Vi))*(ZVERT_0 (U, V-i)) (61) Z2(U, V) cos(angle(V)) / cos(angle(V1))*(ZVERT_3 (U, V-i)) (62) Zv3(U, V) = cos(angle(V)) / cos(angle(Vi))*(ZVERT7 (U, V-i)) (63) Zv4(U, V) = cos(angle(V)) I cos(angle(Vi))*(ZVERTj (U, V-i)) (64) Zv5(U, V) = cos(angle(V)) / cos(angle(VI))*(ZVERT_4 (U, V-i)) (65) Zv6(U, V) = cos(angle(V)) I cos(angIe(Vi))*(ZVERTj (U, V-i)) (66) The Z coordinate of vertex V7(U, V) is associated with the growth parameter of the first component in the new row and is determined by forming the sum of the Z value of vertex V3(U, V) and the growth parameter of the new component (U,V): Z(U, V) = Zva(U, V) + growth(U, (67) Therefore: Z(U, V) = (68) cos(angle(V)) I cos(angle(Vi))*(ZVERT7 (U, V-I))-f-growth(U, V) The Y values of each vertex of component (U,V) are inherited from respective vertices of component (U, V-I), and adjusted to fit new enveloping surface that might have walls sloping at different angles. The adjustment is described in the following by way of example with reference to the V value of vertex V0.

First, the actual length A along the edge of the previous enveloping surface is calculated according to: A = cos(angle(V1))*(ZVERT3 (U, V-i)) (69) Then this length A is applied to the edge of the new enveloping surface by calculating a value LiYi according to: IXY1 = sin(angle(V)) IA (70) = sin(angle(V)) I cos(angle(Vl))*(ZVERT_3 (U, V-i)) The value AYI "moves" vertex in the Y direction to make sure that angle of the edge of the new component corresponds to the angle of the new enveloping surface.

Therefore: Yvo(U, V) = Yvo(U, V-1)+sin(angle(V)) I V-i)) (71) The Y coordinates of the other vertices are calculated in a similar fashion: Y1(U, V) Y1(U, V-i)-sin(angle(V)) I cos(angle(Vi))*(Zvo(U, V-i)) (72) Yv2(U, V) Yv2(U, V-i)-sin(angle(V)) I cos(angle(Vi))*(Zv3(U, V-i)) (73) Yvs(U, V) = Y3(U, V-i)÷sin(angle(V)) I V-i)) (74) Y4(U, V) = 4J4(U, V-i)+sin(angle(V)) I cos(angIe(V1))*(Zw(U, V-i)) (75) Yvs(U, V) Yv5(U, V-i)-sin(angle(V)) I cos(angle(Vi))*(Zv4(U, V-i)) (76) Yv6(U, V) Yv6(U, V-i)-sin(angle(V)) V-i)) (77) Since Z value of vertex V7(U, V) is given as a new input that is to depend on the growth parameter and additional adjustment AY2 is needed in order to ensure that the Y value of the vertex places it on the walls of the enveloping surface. The i 0 adjustment is calculated as according to: AY2 = tan(angle(V))tgrowth(U, V) (78) Accordingly: Y(U, V)=Y(U, V-1)+sin(angle(V)) I V-i)) -i-iXY2 (79) Therefore: V) Yw(U, V-1)tsin(angle(V))/cos(angle(V-1))*(Zw(U,Vi)) (80) +tan(angle(V))tgrowth(U, V) The coordinates of the vertices of the first component of a row other than the

first row are therefore in summary as follows:

Vertex °(U (X, Y, Z) = (81) (XVERT_O (U, V-i), YVERT_O (U, Viy+Sifl(ang1e(V)) / a (U, v-i)), cos(angle) I (U, V-i))) Vertex 1(U,V) (X, Y, Z) = (82) (XVERTI (U, V-i), YvER--_1 (U, V-i)-sin(angleM) / cQs(aflg le)(Z 0 (U, V-i)) cos(angJe) / cos(anglei))*(ZvcRr_o (U, V-i))) Vertex 2(U, (X, Y, Z) = (83) (XVERT2 (U, V-i)1 (U, V)-sin(angIe(V)) I cos(aflgle1))*(Z13 (U, cos(angle) I (U, v-i))) Vertex 3(U, (X, Y, Z) = (84) (XVERT_3 (U, V-i)1 VERT3 (U, Vi)+S1fl(aflgte(V)) / cos(anglei))*(ZvERr_7 (U, cos(angle(V)) / cos(angIel))t(ZvERT_7 (U, V-i))) Vertex (X, Y, Z) = (85) (XVERT4 (U, V-i)1 YVERT_4 (U, vi)-'-sn(angle(v)) I cos(angle(V1))t(ZVERT7 (U, V-i)) cos(angle) / (U, V-i))) Vertex 5(U, (X, Y, Z) = (86) (XVER-I-5 (U, V-i) YVERT_5 (U, ViSifl(ang1e(V)) / (U, V-i))' cos(angle) / cos(angle(vi))*(ZVERT_4 (U, V-i))) Vertex 6(U, (X, Y, Z) = (87) (XVERT6 (U, V-i)1 YVER-T-8 (U. Vl)-sin(angle(V)) 1 cos(angiei))*(zVERT7 (U, V-i)), cos(angle) / (U, v-i))) Vertex 7(U,V) (X, Y, Z) = (88) (XVERT7 (U, V-i)1 YVERT7 (U, Vl)+sin(angIeM) / (U, V-i)) cos(angle) / cos(anglei�*(ZvERT a (U, V))+gr0Wth(U, v)) Hexagonal Components Vertex °(U, (X, Y, Z) = (89) (XVERT_O (U, V4) 1 t \i I i \*!7 I VERT_O (U, V-i) s!nlangie(V)) cos!angJe(vl), V-VER'LS (U, V-I) cos(angIe) / cos(ang[e))*(ZVERT5 (U, V-i))) Vertex 1(U,\i) (X, Y, 7) = (90) (XVERTI (U,V-i), YVERT1 (U, VlySCfl(aflgie(V)) / cos(angte1))*(ZvERT_5 (U, V-I)), cos(angIe) I cos(anglewi))*(ZVERT_5 u, V-i))) Vertex 2(U, (X, Y, Z) = (91) (XVERT2 (11, V-i), Y"VERT_2 (U, Vl)-scn(angIeM) I cos(angIei))*(ZVER,4 (U, V-i)), cos(angIe) / cos(ang!ew.i))*(ZVERI4 (U, V-i))) Vertex 3(U, (X, Y, Z) = (92) (XVERT3 (U,V-i), YVERT3 (U, V.-sin(angIe(U)) I (Uti, V-I))1 cos(ang(eM) / cos(angIei))*(ZvEgTs (Usi S-i))) Vertex 4(U, (X, V, Z) = (93) (XVERT_4 (U, V-i), YVERY4 (U, vl)+Sifl(angJe(V)) / cos(aflgIei))*(ZvERTIQ (U, V-i)) cos(angIe) I cos(aflgIei))*(ZvEpTio (U, V-i))) Vertex 5(U,V) (X, V, Z) (94) (XVERT_5 (U, V-i)1 VERT_5 (U, vI)+sin(angIe(V)) I cos(angfewi))*(ZVERi1 (U, V.1))1 cos(angIe) / cos(angiei))*(ZVEgT Ii (U, Vertex 6(U,V) (X, V, Z) (95) (XVERTB (U, V-I)' (U, vi)-'-sin(angIeM) I (U, V-i))1 cos(angIeM) / cos(angIewi))*(ZVERT_11 (U, V-I))) Vertex 7(U, (X, Y, Z) (96) (XaT7 (U, V-i) YVERT (U, Vi)-sn(angIe(V)) / cos (angIei))*(ZVERT ii (U, V-i)), cos(artgle(v)) / cos(anglei))*(ZVERT 11 (U, V-i))) Vertex 8(U, (X, Y, Z) = (97) (XVERT_B (U, V-i) YVERT8 (U, Vi)-s1n(angIe(v)) / 10 (Ii, V-i)), cos(angie) I cos(angle&i))*(ZVERTIO (U, v-i))) Vertex 9(U, (X, Y, Z) (98) (XVERT9 (U, V-i), VERT_9 (U, vl)-sin(angle) / ii (tJ*1, V-I)), cos(angle) I (U+i. v-i))) Vertex 1 °(U, (X, Y, Z) = (99) (XVERTIQ (U, V-i)1 YVERTID (U, VlfFsin(aflgle(V)) I cos(angle(vl))*(ZvERT1O (U! V-cos(angleM) / IC (U, Vl))+growth(U, v)) Vertex ii(U,V) (X, Y, Z) = (100) (XVERT_11 (U, V-i)1 YVERT_li (U, vl)+sin(angle(V)) / cos(angle(v1))*(ZVERT1I (U, V. i))+tan(angle(V))*groVVth(U,V), cos(angle)) / ii (U, v.)+growthu, V)) Once the vertices of all of the components have been defined within the container the coordinates of the individual components can be used to manufacture the components. The model of the components, as encoded by the coordinates of the vertices can, for example, be input into a CNC driven fluid applicator for generation of the component according to, for example, one of the cast on cast techniques described above with reference to Figures 6, 10 and 13. CNC machines comprising algorithms that translate data defining the modellcontour of a component into to a numerical control programming language description required for manufacture of the component.

Such a control programming language description may, for example, encode the movement of the fluid applicator described above.

In addition to the above discussed CNC additive/casting technique, CNC technology may also be used to create partial moulds, such as those discussed above with reference to Figure 13. Tools for translating electronic models of components into the relevant code for use by CNC machines to create these moulds are also known.

Such tools may use the top surface of each component as an input for digital fabrication of partial moulds.

For additive method CNC machines are used to drive a fluid applicator/nozzle which deploys the material. Algorithm translating the geometry of digital components into space filling a 3D space are also known. Details of the manner in which such 3D components can be generated depend on the properties of the material(s) used in the fabrication process, the desired resolution of the final finish, nozzle dimensions. This is well known to the person skilled in the art and the person skilled in the art would be able to use such digital technology to generate the cast on cast components.

ALGORiTHM II In the following a second algorithm for determining the coordinates of the vertices of the component parts of the surface within respective containers. In contrast to the algorithm described above, this algorithm does not use the surface that is to be replicated as a starting point. Instead the algorithm uses an iterative approach to determine the container parameters and/or the parameters of the component parts.

The second algorithm in particular does not segment the input surface in the above described manner. Instead the second algorithm uses as a starting point an arbitrary starting surface that is segmented into rows and columns based on basis geometric shapes, as discussed above. The coordinates of the vertices of the segments of this surface are then converted into component part coordinates within the containers/tubular enveloping surfaces in the above described manner. The virtual component parts generated in this fashion can then be assembled to form a virtual surface and a deviation of this virtual surface from the desired surface can be determined. To minimise the deviation of the thus created virtual surface the components influencing the shape of the surface can be altered in an iterative fashion.

One way of intelligently choosing alterations to component part is to detect an area of the virtual surface that significantly deviates from the desired surface, to identify those containers responsible for the manufacture of the component parts in this area and to modify the shape of these containers. Any changes that may be required to be made to other containers to account for the changes made may also be implemented.

Based on thus altered container geometries a second set of virtual component parts can be modeUed and assembled in a second virtual surface. Deviations of the second virtual surface from the desired surface can again be calculated and localised.

Based on this a decision is made of whether or not the changes made have improved conformity of the virtual surface with the desired surface or decreased it. Based on this decision the change made can be retained or even amplified, or discarded or reduced.

To change the appearance of the virtual surface the angle of the container wails may, for example, be altered. The footprint of the container may additionally or alternatively be changed, so that any parallel side walls of the container are spaced differently from each other. The angle of the components within the container may also be changed, as may be the component thickness, be that the thickness throughout the entire component or along a particular part/side of the component.

If after a predetermined number of iterations it has not been possible to re-create the desired surface, then additional rows and/or columns may be added to the segmented surface to reproduce some finer detail. Alternatively an entire surface may be re-segmented using different basic geometric shapes or different sizes of the previously used basic geometric shape. The surface that is being segmented does of course not have to be the surface first used in the iteration process. Instead the surface used for re-segmentation maybe the surface the most closely resembles the desired output surface.

While the above described iterative algorithm starts from an arbitrary starting point, this is not essential. An algorithm for the iterative determination of container and component part parameters may alternatively use as a starting point a database of virtual surfaces that had previously been generated and for which the container and component part parameters are already known. As a first step of the iterative algorithm the one virtual surface that most closely resembles the desired output surface may be chosen. This choice can provide starting container shapes, container parameters and component parameters and may reduce the number of iterations required for arriving at the desired output shape. Any virtual surfaces created as part of the above discussed iterative algorithm may be retained as part of the database to thereby increase the number of possible starting points for further modelling projects.

ALGORITHM Ill The above two described algorithms use a segmented surface as a starting point. The first embodiment uses the surface that is to be manufactured as an input, segments it and determines, based on the segmented surface, models of the components that are to be manufactured in the respective containers.

The second embodiment differs from the first embodiment in that it uses an already segmented arbitrary input surface, models the component parts that make up this surface and their positions in the container, thereby obtaining container parameters for generating the arbitrary surface. These parameters are then iteratively changed and the effect these changes made have on the surface is evaluated. Further changes can then be made until the surface created within the containers sufficiently matches the desired surface.

In the third embodiment described in more detail in the following, a surface is again created in an iterative fashion. After each iterative step the surface created is compared to the desired surface and, if it is found that the two surfaces are sufficiently similar, the created surface (and the associated container parameters) are adopted for the manufacture of the desired surface, The third embodiment differs from the second embodiment in that in the third embodiment the starting point is not an arbitrary surface but instead an arbitrary set of containers alongside with information of how the component parts are arranged within the containers. This arbitrary set of containers and the associated component parts information may be a set of containers and associated information that had been used previously to model a surface. The set of containers and the associated information may thus be one set of containers/information stored in a database of previously used sets of containers and of the associated component parts.

In the third algorithm a surface may be selected from surfaces in the database by calculating a deviation of the top or bottom surface of a stored set of components from the target surface. This may be done using simple algorithms, such as a least square algorithm, for example. The person skilled in the art will, however, be aware of other suitable means of comparing two surfaces in a quantitative fashion. Based on such a quantitative assessment a surface may be chosen from the database as a starting point.

Once a starting surface has been chosen, the containers used for generating this surface may be altered and the effect such alterations evaluated to determine whether or not the alterations made have caused the surface to correspond more or less to the target surface. Any change in the level of correspondent between the two surfaces can again be quantitatively evaluated, The alterations that are made to the series of containers can be randomly chosen. Alternatively any chosen alteration may be made based on knowledge of changes that are likely to adapt a deviation from a currently simulated/modelled surface from the target surface. One example of this is illustrated in Figure 26.

Figure 26 shows a target surface 600 that is to be replicated. While surface 600 is shown with an overlayed grid pattern, this grid pattern is only provided to emphasis the topology of the surface and should not be taken to imply that the target surface has to be segmented in any way. A surface 610 is used as a starting surface for iteratively determining the container shapes required for creating surface 600.

surfaces 600 and 610 are shown spaced apart for clarity of illustration. This surface 610 is made up of a number of components arranged in rows 620 to 700 and columns 710 to 740 and is associated with a set of containers used for creating the rows of the surface.

The deviation of surface 610 from 600 can be quantified and areas where the tvvo surfaces deviate from each other most strongly can be identified. In the example shown in Figure 26 it may, for example be found that the strongest deviation between the two surface is in the area of rows 680 and 690, as highlighted in Figure 28. One way of altering the surface 610 may thus be to change the parameters of the containers used for generating the components of rows 680 and 690. The angle of the side walls of the container(s) used for creating the components of row 680 may, for examp(e be changed to cause a downward sloping of row 680 in increasing Y-direction and the container(s) used for making the components of row 690 may be altered to cause row 690 to cause and upward sloping of the components of row 690 in the increasing Y-direction. Once this change has been made a new virtual surface can be modelled and compared to the target surface.

Another change that may sensibly be made is to create the components of rows 680 and/or 690 using two containers per row, rather than only one container. Doing so can allow to take account of changing curvature in the Xdirection. in the context of Algorithm I above this may mean that one container has a mirror parameter that is True', while the other container of the same row has a mirror parameter that is False'.

in other words, one of the two containers creates components that increase in size in increasing X-direction, while the other container creates components that decrease in size with increasing X-direction.

It will be appreciated that the above discussion with reference to Figure 26 is provided by way of example only and that other alterations to the containers are also envisaged by the embodiment. Changes to the spacing of container walls may, for example help to alter a curvature of a surface, as may a change in the manner in which the components are arranged within the container(s). Additional rows of components may be inserted, for example where a better resolution is required. The size of the components can equally be altered, either for all of the components, or only for all components in a row of components.

Additionafly, the basis geometric shape of the containers may be changed, either for all of the rows or only for rows where the use of a different geometric shape may produce a better fit between the surfaces. It is, for example envisaged, that one of the rows of components shown in Figure 26 may be replace with one or more rows of components having a different geometric shape. One row of Figure 26 can, for example be replaced by two rows of triangular components, or by a row of hexagonal components with one row of triangular components on either side.

it wifl be appreciated that, while the third algorithm has been described above based on the assumption that a most suitable' starting surface is chosen from a database, this algorithm may also be practiced starting from a random set of containers, wherein the only requirement this set of containers has to fulfil is that the components generated using the containers are assemblable to form a surface.

While the above example is presented for Algorithm Ill, all of the changes illustrated above are also relevant to Algorithm II and may be used in combination with this algorithm.

TRIANGULAR CONTAINERS

Turning now to the use of triangular containers in more detail, the above description of four-sided containers used growth parameter to modulate a row of elements in the X/U direction and the angle parameter in the YN direction. The following description of a particular use of triangular containers changes the angle of container sides in both the X (U) and the 1 (V) directions. By changing the angle parameter in both directions surfaces of constant thickness can be created. This may have advantages in terms of further raw material savings but does not mean that the thickness of the components cannot be changed. A change in the thickness of components may, for example, be advantageous if highly complex surfaces are to be created.

Figure 27 illustrates a surface 800 assembled from triangular components 810-I to 810-4. Only the components 810-1 to 810-4 used for assembling a single row of components extending in the U-direction are identified by reference numerals.

Components 810-1 are formed using a first triangular container, components 810-2 are formed using a second triangular container, components 810-3 are formed using a third triangular container and components 810-4 are formed using a fourth triangular container. The identified row of components comprises a change in the angle parameter at the interface between components 810-2 and 810-3. In the absence of this change in the angle parameter the components of the row could be formed using only two triangular containers. It will be appreciated that the triangular components of the surface shown in Figure 27 are defined by a segmentation process, as shown, for rectangular components, in Figure 14, albeit projecting triangular base elements onto the surface to be reconstructed.

Figure 28 illustrates the definition of side walls of a first container for forming the first component 810-1 (at coordinates U=0, V=0) of the first row of components shown in Figure 27. The container is formed based on the assumption that it has an equilateral triangle 830 as its base and that at least one of the vertices of the lower surface 820 (Vertex VO in Figure 28) of the component 810-1 to be cast is collocated with (a) corresponding one(s) of the vertices of the container base 830. As can be seen from Figure 28, the edges 840 -860 of the container are defined by the lines connecting the vertices of the base surface 830 of the container with corresponding vertices of the lower component surface 820. The distance between the vertex Vi and the corresponding vertex of the base surface 830 can be freely chosen. The distance of vertex V2 from the corresponding vertex of the base surface 830 is chosen to be twice the distance between the vertex Vi and the corresponding vertex of the base surface 830.

Vertices V2 and V3 of the lower component surface 820 and corresponding vertices of the base surface 830 define two of the edges of the container. These edges can be expressed by vectors v1 and v2, as shown in Figure 28. The vector defining the first edge of the container, v0, is inclined relative to the base surface by the same angle as the other two edges of the first container. All three of these vectors are chosen to have unit length. Adding vectors v0, v1 and v2 (scaled by an appropriate value to choose an appropriate thickness of the component) to the coordinates of vertices VU, VI and V2 respectively yields the coordinates of the top vertices v3, v4 and v5 of the component. The scaling factor used in this respect can be based on a user input or may be a predetermined value. It will aJso be appreciated that a predetermined or user provided value is first used for this scaling factor, to allow an initial definition of the components. It may later be determined that the surface generated by the components would benefit from a different initial choice, then the initially chosen scaling factor can be altered in further iterations in which the geometries of the containers and the locations of the components within these containers is re-calculated.

As can also be seen from Figure 28, the angie between vector v1 and the line bisecting the associated internal angle of the unilateral base triangle 830 of the container is referred to as angle A. The same angle A is present between vector v2 and the line bisecting the associated internal angle of the unilateral base triangle 830 of the container.

Once the parameters of the first container have been defined the geometry of the second container in the U-direction can be defined in the manner illustrated in Figure 29. It will be appreciated that two of the edges (as defined by vectors v1 and v2) of the second container are inherited from the definition of the first container To define the missing edge an intermediate point V6' is defined at a distance 3x from the vertex of the base surface 830 of the first container, Also defined is a vector v3 this vector is the sane as vector v2. To create vector v3 shown in Figure 29 vector v3 is rotated about vector v, by an angle B. The angle B has to be smaller than 180° and can otherwise be freely chosen to provide a desired component size for the components generated in the second container. The vertex V7 is again obtained by vector addition of a scaled version of vector v3 to the coordinates of vertex V6. The scaling factor used on v3 is the same scaling factor as that used for determining the coordinates of vertices V3, V4 and V5.

Figure 30 shows the two containers defined in the above described manner.

Additional components for assembling in a row of the surface shown in Figure 27 can be created by adding them on top of the first component in each container, so that the lower surface of a new component is defined by the upper surface of the component already existing in the container. The vertices of the upper surfaces of the thus formed new components are created by adding a scaled version of the respective vectors v0 to v3 to the coordinates of vertices VS to VS and V7 (which also form vertices VO' to VS' and V6' of the second/subsequent layer of components in the containers).

The scaling factors used for this vector addition are determined by inheritance rules illustrated in Figure 31. As will be appreciated from this figure, the length of component edges along those edges of the first container defined by vectors v0 and v2 for of second components in the first container have to be the same as the length of corresponding edges of a preceding component (along the edges defined by vectors v1 and v3) if a step free surface is to be generated. Under this assumption the scalling factor used for the vector addition for the second component in the first container is thus unambiguously determined by the geometry of the first component in the second container. The edge of the second component in the first container extending parallel to vector v1 is, in this particular embodiment, also defined by the length of the edge of the preceding component in the second container, in that the further component in the first container inherits the length of the edge along vector v5, as shown in Figure 31. As also illustrated in Figure 31, this "inherited" edge length, will in turn be handed further to the edge v0 of yet another component, so that this yet further component matches the preceding components created in the two containers.

As mentioned above, the first row of components shown in Figure 27 assumes that the radius of the curvature changes from convex to concave or vice versa. At the point of the surface where this change occurs the first and second containers described above become unsuitable for generating further components for this row of components and third and fourth container need to be defined. The definition of these containers is discussed in the following with reference to Figure 32.

As a first step of defining the geometry of the third and fourth containers the lowermost components positioned in these containers are generated/defined in the same manner as any other of the components located in a container on top of a previously defined component. Put in other words, the first components in the third and fourth container respectively are generated in the above described manner, as if they were simply another component in the first and second container respectively. This is shown in Figure 32, where the last/topmost component of the first container of the row is referred to using reference numeral 900 and the last/topmost component of the second container of the row is referred to using reference numeral 910. The firstflowermost components of the third container is also shown and referred to by reference numeral 920. The first/lowermost component of the fourth container (as far as define at this point of the process) is shown in Figure 32 in dashed lines and identified by reference numeral 930.

The first/lowermost component of the fourth container is then modified. As shown in Figure 32 a vector v is defined for this purpose. This vector lies in the plane defined by the three vertices of the lower surface of the lowermost component of the third container and bisects the internal angle of this lowermost triangular surface starting at vertex VI. Further defined is a rotation axis perpendicular to both vectors v, and v1 by calculating the cross-product between these two vectors. Vertices V4, V6 and V7 are rotated about this rotation axis by an angle C. The angle C can be user defined or automatically/randomly selected. The lowermost component of the fourth container is fully defined by vertices Vi, V2 and V4 to V7. Once the first components of the third and fourth container have been defined in the above described manner further components are generated, if required, in the same manner as described above with reference to the first and second containers.

If further changes in the inclination angle in the U-direction are required, then further container can be defined in the same manner as described above.

With the above any containers required for defining the components 810-ito 810-4 of the first row of components in the U-direction can be created. Turning now to defining the containers required for creating the second row of components, it will be appreciated, for a surface to be continuous/step free on both the upper and the lower faces, the conditions: coordinates (VU) = coordinates (V2), coordinates (VI) = coordinates(V6'), coordinates (V3) = coordinates (V5') and coordinates (V4) = coordinates (V7'), need to be fulfilled, wherein the shorthand "coordinates(..)" refers to the coordinates of the vertex indicated in the brackets and wherein vertices referred to with a prime are vertices of the first component in the second container of the first row and vertices not referred to with a prime are the vertices of the first component in the first container of the second row.

Once the new component/first component in the second row has inherited these coordinates of the first component in the second container in the first row only vertices V2 and VS of the first component in the second row need to be defined. These vertices can be defined to replicate a change in the curvature of the surface in the V-direction in the following manner.

lnititial/temporary coordinates of vertices V2 and VS are defined by rotating the coordinates of vertices VO and V4 respectively by 60° about vector v1 of the first container of the second row (corresponding to vector v3 described above with reference to the second container used for the first row). The temporary vertices V2 and V5 define the direction of a vector v2. This vector has unit length in the described embodiment. It wift be appreciated that, as the coordinates of V2 and VS are not constrained the ability to freely choose their coordinates can be used to shape the components to meet aesthetic goals, for example so that a surface created using the assembled components comprises similarly sized components. Equally though, should the surface that is to be assembled from the components be subject to severe constraints, such as considerable undulations, for example, then the freedom of choosing V2 and V5 can be utilised to accommodate these constraints.

A rotation axis is defined as the axis perpendicular to vector v2 and the vertical direction (as defined with reference to the input surface as it existed prior to segmentation/triangularisation) by forming the cross-product of the vertical direction and v2. Temporary vertex V5 is then rotated about this axis by an angle A (which can again be user defined or randomly chosen)to provide the final coordinates of the last vertex V5" of the first component of the first container of the second row.

The coordinates of the vertices of the first component of the second container of the second row are fully constrained by the coordinates of the respective adjacent vertices of the first container of the second row and the second container of the first row. This is shown in Figure 34. The definition of these vertices is thus not described in this context in more detail.

Further components in the first and second containers of the second row can be defined in the same way as described above for further components in the first and second containers of the first row. Equally, should it be desired that third and fourth containers (or indeed further pairs of containers) be required to accommodate a change in incline of the row, then these containers can be defined in the same way as described above for the definition of the third and fourth containers of the first row. This is illustrated in Figures 35 and 36.

ASSEMBLY

CLADDING

In terms of assembly, there are different parameters that will affect the manner in which individual components are connected. The application to which the assembled component may be put (cladding for facades or roofs, structure, pavement, landscape, etc), the material used in the fabrication process (cements, plaster, ceramics or polymers) and scale of the component/tile and the overall shape will have an influence on the type and strength of fixing means used for connecting the components to each other or to supporting structures.

It the assembled surface is to be used as a claddhig system the connections between components have join the components to form a system that is able to resist lateral loads directly imparted on it, for example by wind or earthquakes, as well as vertical loads resulting from the cladding system's own weight. These loads must be transmitted through the wall system and through secondary structural elements to the building's structure. For roofs or façades simple and generic substructures for the assembly may be useable. Particular benefits may be achievable in this case from the above discussed assembly advantages of the cast on cast process.

For a cladding system the components could be made using, for example, cements, plaster, ceramics or polymers. For cements and plaster, a suitable connection between components is a type of solid pin joint or anchor, the dimensions of which can vary depending on the scale of the component. Joints between components may be sealed with sealant to prevent water penetration. Ceramics or polymer components can be connected with a mechanical joint between components.

STRUCTURAL COMPONENTS

If assembled surface is to be used as a structural component the surface may either form a permanent formwork for the structure or may form the structure itself, possibly without the need for any other layerslsupporting structures. If the surface is to form a permanent formwork for the structure it may be reinforced. A fluid casting material may be poured on to of the surface to generate a monolithic structure. The surface produced using the cast on cast technique may thus provide the monolithic structure with its shape. The reinforcement, together with the additional casting material may improve the structural properties of the structure. No specific scaffolding is required for either of these two uses of the assembled surface, thus saving construction time and funds.

Cement may be used for creating the components for either of the above mentioned structural applications and the above described solid pin joint or anchor connections between components are again suitable.

If it is desired that not further material be cast on the surface once it has been assembled from the cast components, then the surface may be post tensioned, for example by embracing all the components with metallic bars and/or by passing reinforcing members through recesses in all of the components of the surface, thereby forming a monolithic system.

PAVEMENT/TiLES If the components are to find use as tiles or pavement ceramics or cements may be used when creating the components. The components may be interconnected with solid pin joints or anchors.

SCALE

The scale of the components can depend on the material used for the fabrication process maximum toad for transportation and assembly, the scale of the overall shape that is to be created etc. It will be appreciated that for very large structures it may become necessary that scaffolding is used to support the components during assembly, while permanent interconnections between the components are being put in place. As such scaffolding is, however, not required on a permanent basis it can be re-used for the assembly of further components once the components that the scaffolding has originally supported have been connected sufficiently firmly. The scaffolding may, for example be used for assembly of individual rows of components.

The discussion provided above is based on the assumption that the top and bottom surfaces created by the cast on cast process are continuous, that is gap free as well as step free. If it is not required for both surfaces to be gap free and step free, then further degrees of freedom will be available for creating the cast components in the containers. In such situations further optimisation to minimise or optimise material use may become available. It can, for example, be envisaged that a small or thin form part can be introduce on top of an already cast component (or at least on top of a centre part thereof). This form component may then help rendering the next component thinner or leave a recess in the central part of this next component. In either case material costs may be saved without the need for creating large form pieces.

It is further noted that the above discussion focuses on converting XYZ coordinates of vertices of components formed by projecting basic geometric shapes onto the surface that is to be manufactured. It will be appreciated that, if these vertices are the only points defining the surface, then the surface may be substantially planar (if the basic geometric shape used is a triangle) or roughly approximated by planar surfaces (for example if polygons with a larger number of corners are used). Situations can be envisaged where the surface that is to be generated comprises detail that cannot adequately be approximated by such surfaces. This may, for example, be the case if the components that are to be created are large in size and/or if very fine detail is to be included in the surfaces of the components. The present invention is particularly useful in situations like these. The above described algorithm in particular provides a first approximation of the position each component has to take within a container for the cast on cast technique to generate components that can be assembled in the desired manner to form the desired surface. A survey of the topology of those part of the input surface that are represented by the individual components can reveal whether or not the components are a sufficiently accurate approximation of the input surface. Should this not be the case additional points can be defined on the surface, either automatically or manually by a user. The coordinates of these points in the original XYZ coordinate system comprising the input surface can then the translated easily into corresponding XYZ coordinates of points within the container.

The surfaces within the container can therefore have intricate shapes, despite the use of the above described robust and computationally inexpensive way of determining the coordinates of vertices of the components within the container.

The above description focussed on the assembly of components where abutting surfaces are substantially in contact over their entire height. It will be appreciated that this is not a necessary condition for rendering one of the top and bottom surfaces of the assembled surface gap and step free. To render only one of the top and bottom surfaces step free it is merely required that the edges lying in this surface are in continuous contact. If these edges are straight, then the side faces of the components may be arranged such that gaps are formed between the components. Such gaps may be useful to save material or providing space in which fixing means for supporting the surface components can be located.

The top surface of assembled components can be rendered gap free without a need for the entire side faces of the components to touch even if the abutting edges of the components are not straight. Such non-straight edges will be located in a single plane. A curved edge may, for example define a planar circular segment. For the top or bottom surface of the structure that is to be assembled to be gap free it is sufficient if the surfaces defined by the edges on the two contacting components are in the same plane and in contact with each other.

While the above description refers to casting within a container it will be appreciated that this container does not have to be physically present and that any such container (for example if the manufacturing method used is the above discussed container free cast on cast manufacturing method), if not required for casting can merely be used as a concept that allows the calculation of the coordinates of the vertices of the components for later manufacture.

OVERLAPSO far the description has assumed that all of the components contact each other along the side faces of the components. The following part of the description relates to an embodiment in which adjacent rows or adjacent columns of components can overlap over each other, for example in the manner known from roof-tiles. Figures 37A to 37C each show two overlapping components. The components are part two different rows of component. A can be seen the two rows of component do not only overlap but are also laterally offset against each other. It will be appreciated that it may not be important whether these components are considered to overlap in the row direction and are offset against each other in the column direction or vice versa. In Figure 37A the topmost components covers 25% (1/4) of the depths of the lowermost component and is offset against the lowermost component by either 25% (114) of the width of the lowermost component (if it is assumed that the two components were in different rows and in different columns before they were offset against each other) or by 75% (3/4) of the width of the lowermost component (if it is assumed that the two components were in the same rows (if the components are offset against each other in the column direction) or in the same column (if the components are offset against each other in the row direction) before they were offset against each other). Figure 37B shows a 50% (112) overlap and 50% (1/2) offset between the components and Figure 37C shows a 75% (314) overlap and 75% (3(4) offset between the components. It will be appreciated that, because the overlap between the components shown in Figures 37A to 37C progresses from picture to picture in steps that have the size of a quarter of the component width/depths, this is not essential and the overlap can be any desirable fraction of the width/depth of the component.

To allow expressing the degree of overlap between adjacent rows/columns (and/or to quantify the degree of offset) the components are no longer solely described by eight corner vertices but are additionally subdivided into two matrices of vertices on the top and bottom surfaces. In Figures 37A to 37C the components are indicated as being subdivided into imaginary 4x4 matrices. lt will, however, be appreciated that other appropriate subdivisions, as required by the desired overlap/offset, are equally possible. The components could for example be subdivided into a matrix that has a length and/or depth of 10 subdivisions or any number of other subdivisions. The resolution of the matrix may (and in the example provided in the following does) form a parameter of determining the parameters of the components, as does the percentage of overlap between adjacent rows/columns.

it will also be appreciated that it is by no means necessary that the same number of subdivisions be used in both the length and depth directions. The number of vertices in the top and bottom matrices is the same in the following embodiments.

Figure 38 illustrates other ways of subdividing components, into a 2 x 2 matrix in Figure 38A, a 3 x 3 matrix in Figure 388 and an i x i matrix in Figure 38C, wherein i may be any desired integer. Figure 38 at the same time introduced a terminology for referring to the vertices of the components of one (the upper or tower) surface in this matrix.

In the following a description of a method of a way of determining containers for casting overlapping components is described. Each of the side walls of the components can be considered as lying in a plane defined by two orthogonal vectors, a first vector extending in the plane occupied by the component as seen in a top plan view and a second vector extending perpendicular to the first vector. The following description is based on the assumption that the first vectors of adjacent side walls of the components are perpendicular to each other, that opposing side walls are parallel to each other and that all of the second vectors are parallel to each other.

Figure 39 shows an example of a first component to be generated in a first container. Using terminology introduced above, this first component is located in the first row (at the lowest position in the V-direction) and in the first column (at the lowest position in the U-direction). The component shown in Figure 39 is symmetrical about the vertical plane bisecting the component along its diagonal,that is about the plane defined by vertices V2, ye, Vii and Vi 5. it will be appreciated that this plane of symmetry is the vertical plane extending in the direction of overlap between components. This means the Z-coordinates of a number of vertices are identical. The following conditions therefore apply: zv3 = zv? zvc = = zv8 zv1 = zvs.

This means that the z-coordinates of only vertices V15 and V11 as well as of one vertex of the pair of vertices V12 and V15, one vertex of the group of vertices V91 V13 and V17 and one vertex of the pair of vertices V10 and "14 can be chosen freeiy, as desired to achieve a given surface architecture. The z-coordinates of the remaining vertices in the pairs/group follow as a result of the above conditions.

Once the geometry of the first component has been determined, subsequent components for casting in the same container can be determined by using the inheritance rules illustrated in 540. In particular vertices of the second component (indicated by a prime) have a-values that depend on the a-values of vertices of the first component (underlying the second component in the container) in the following manner: Z0 = 4g; 49 = Z'qg + (415 Z,1 = Zvlo; Zvio = Zvio + -Zv3) Zv2Zvll; ZvilZvll+(Zv13-Zv4 Z,3 = Z12; Zv'12 Z + a user defined growth parameter Zv4 = Zv13 Zvia = Zvis + (Zvis 4e) Zv's = Zv14; 4'14 = Zv14 + (4is -Z) = Z15; Z,15 = Zvjs + the growth parameter discussed above with reference to rectangular containers Z = Z16; Zv.18 = --the user defined growth parameter Zvs = Zv17; 4,17 4,17 + (Zv15 -Zvs) The growth parameter described above for rectangular components is also used for calculating Z,15. This parameter is used to define the overall curvature of the surface assembled from the components. The user defined grows parameter can be used to define or influence the local curvature of the components.

Put in words, the a-coordinates of the vertices of the lower surface are inherited/are the same as the a-coordinates of vertices of the upper surface of the first components. The a-coordinates of vertices V'9 to V'11, V'13, V'14 and V'17 corresponds to the sum of the a-coordinate of the relevant underlying vertex and the thickness (in the a-direction) of the first component at vertex V15 (for vertices V'9 and V'17), the thickness (in the i-direction) of the first component at vertex V12N15 (for vertices V'10 and V'14), the thickness (in the i-direction) of the first component at vertex V15 (for verticex V'13) and the thickness (in the i-direction) of the first component at vertex V13 (for verticex V'11) respectively. It will be appreciated that these inheritance rules ensure that the topology of the part of the bottom surface of the second component that overlaps with a part of the top component exactly corresponds to the part of the topology of the part of the top surface of the first component with which it overlaps. The above rules ensure that this is the case in situation where the overlap between adjacent rows corresponds to the offset between components. Put more generally, however, overlap can be achieved if the lower edge of the component overlapping another component conforms to the topology of the underlying component along the contact line. More preferably the topology of the entire overlapping lower surface of the upper component should conform to the overlapped part of the upper surface of the overlapped component.

The manner in which a first component of a second row is created corresponds to the above described manner of determining the first component of the first row.

However, for a first component in the second row that is partially overlapped by the second component in the first row, the part of the upper surface that is overlapped by the second component of the first row confirms to the overlapping part of the lower surface of the overlapping component in the embodiment. This is achieved by an inheritance rule that ensures that i-coordinates of vertices V12, V13, V15 and V16 (in the case of a component being subdivided into a 2x2 array) are inherited from the second component of the preceding row, so that the surface topologies of the overlapping components are the same in the overlapping region. It will be appreciated that this example is specific to surfaces with a 50% overlap and a 50% offset between components. This is illustrated in Figure 42. If the overlap/offset is different, then the z-coordinates of more vertices of the first component in the second row will need to be inherited from the second component in the first row. As a rule it can be said that those vertices that lie below an overlapping part of the second component of a preceding row (including those vertices at the very edge of the overlapping area) need to inherit coordinates from the second component of the first row.

If the overlap between adjacent rows/columns is larger than 50% of the size of the components, then the growth parameters useable in the container covering the row of components are completely constrained by the geometry of the preceding row of components. In this case the growth parameters can only be defined for the components in the first row. The coordinates of the vertices of components in the remaining rows are constrained by the requirement that the top surfaces of preceding rows match the bottom surfaces of the next rows.