WO2022000007A1

WO2022000007A1 - Lattice structure, method for stretching out a lattice structure, and computer-implemented method for creating a lattice model of the lattice structure

Info

Publication number: WO2022000007A1
Application number: PCT/AT2021/060218
Authority: WO
Inventors: Stefan PILLWEIN; Przemyslaw MUSIALSKI
Original assignee: Technische Universität Wien
Priority date: 2020-06-30
Filing date: 2021-06-23
Publication date: 2022-01-06
Also published as: AT524007A1; AT524007B1

Abstract

Lattice structure (1) for forming a three-dimensional structure, comprising border elements (2, 2') which span a surface, and lattice elements (3, 3') which are arranged between the border elements (2, 2') with the formation of crossing points (4), wherein the border elements (2, 2') and the lattice elements (3, 3') are rectilinear, elastic and have high tensile and compressive strength, wherein the lattice elements (3, 3') are fastened to the border elements (2, 2') in such a way that the lattice structure (1) can be brought from a planar position, in which it forms a planar surface section PO, into a stretched-out position, in which it forms a free-form surface section P1 approximating the structure, wherein the lattice elements (3, 3') follow geodetic lines between the border elements (2, 2') in both positions, and method for determining lattice elements (3, 3') of such a lattice structure (1).

Description

Lattice structure, method for spanning a lattice structure and computer-implemented method for creating a lattice model of the lattice structure

The invention relates to a lattice structure, a method for stretching a lattice structure and a computer-implemented method for creating a lattice model of the lattice structure.

In modern construction, so-called free-form structures are known, which realize complex three-dimensional, non-standardized curved and mostly flowing or organic-looking forms (e.g. "blobs") as architectural structures. The implementation of such a free-form architecture requires the creation of a complex lattice structure as a carrier for the facade or membrane to be applied and is associated with a high expenditure of material and time as well as considerable costs.

The elements of such a lattice structure are preferably straight lattice bars or lattice lamellas made of wood, steel, concrete or other materials. For ecological reasons in particular, the use of timber elements makes it possible to realize self-supporting, wide-span free-form structures which, due to their low weight, do not collapse under their own weight. The use of curved supports or plates or the adaptation on site by milling away material should generally be avoided for reasons of cost, instead standardized, straight support elements should be used as far as possible.

When creating free-form structures with architecturally relevant dimensions, the elements of the support grid, i.e. the bars, usually have to be connected in place. The necessary welding work has to be carried out by highly specialized workers, which means that the construction of the grating causes additional costs. The object of the invention is, inter alia, to solve these and other problems in the creation of free-form structures and to provide a lattice structure which is simple and inexpensive to manufacture and does not require complex assembly work on site. Furthermore, a method is to be provided for quick and easy mounting of the lattice structure on site. Finally, a fast and robust computer-aided method for creating a lattice model of the lattice structure is to be provided in order to facilitate the computer-aided development of the lattice structure.

These and other objects are achieved according to the invention by a lattice structure according to claim 1, a method for stretching the lattice structure according to claim 6, and a computer-implemented method for creating a lattice model of the lattice structure according to claim 7.

A lattice structure according to the invention for forming a three-dimensional free-form structure comprises edge elements which delimit an area, as well as lattice elements which are arranged to form intersection points between the edge elements. The edge elements and the grid elements are straight, elastic and resistant to tension and compression. The lattice elements are attached to the edge elements in such a way that the lattice structure can be brought from a planar position in which it forms a planar surface piece to a stretched position in which it forms a freeform surface piece approximating the freeform structure.

For this purpose, the grid elements can be rotatably arranged at the crossing points and on the edge elements. In addition, the grid elements at the crossing points can also be displaceable for a short distance; this enables the grid elements to rotate and move to a certain extent during the clamping process.

According to the invention, the edge elements enclose an angle α in the planar position and an angle α different therefrom in the open position. In both positions, the grid elements follow geodetic lines between points on the edge elements. The grid elements thus run in the planar position in one plane, and in the open position on a free-form surface piece that is to be approximated.

According to the invention it can be provided that, in the planar position, the edge elements and the grid elements are not arranged parallel to one another. This geometrically prevents any change in the planar area.

According to the invention it can be provided that the edge elements and the lattice elements are formed as straight lattice lamellas made of wood, aluminum, steel or glass fiber reinforced plastic.

According to the invention, it can in particular be provided that in the planar position a square surface is spanned by two edge elements each, with at least two grid elements not arranged in parallel being provided. In other words, at least four edge elements can be provided which define a square, between which two intersecting grid elements are provided, but which are not arranged parallel to the edge elements. Due to the fact that the grid elements and the edge elements are resistant to tension and compression, it is not possible to change the shape of the quadrilateral in the plane; Due to the elasticity of the grid elements and the edge elements, however, the grid structure can deform in the third dimension by elastic deformation of the grid elements and the edge elements, that is to say move up or down from the planar position.

According to the invention it can be provided that the lattice elements at the crossing points have elongated holes with in particular different lengths, two crossing lattice elements at the crossing points being connected rotatably and mutually displaceable by connecting means, for example screws or bolts. These elongated holes and the fastening elements located therein make it possible for the grid elements to rotate and shift to a certain extent during the clamping process, so that the clamping process on site is comparatively easy. A method according to the invention for stretching a lattice structure according to the invention from a planar position to a stretched position on site comprises the following steps:

In a first step, the edge elements are rotated relative to one another in such a way that the spanned angle of the edge elements increases or decreases from - in the planar position to a in the open position. This rotation can be carried out using suitable mechanical aids, the stability of the edge elements being taken into account, so that several points of attack may be selected to be provided on the edge elements. The rotation can be done directly by setting the angle, or by pulling the grid apart at two or more points of application until the desired angle is reached. It is advantageous to design the lattice structure with lattice lamellas that are as light as possible.

In a next step, the lattice structure is bent and the end points of selected lattice elements are fixed to supports. This step is used to approximate the free-form surface piece that is to be approximated as well as possible.

In this step, in particular those end points which are closest to the extreme values of the curvature of the associated edge element in the free-form surface to be approximated are fixed on external supports. The free-form surface to be approximated is therefore first analyzed and those points of the edge elements are determined which would have the greatest curvature in the open position. Then the end points of those grid elements that are closest to these points are fixed. Preferably, all of the end points of the grid elements can also be fixed to external supports.

The end points are fixed at those points that they must occupy in the open position. Optionally, selected or all crossing points of the grid elements can be fixed to external supports.

The lattice structure can then be covered with a membrane or with facade elements in order to achieve the desired free-form structure. The invention also relates to a computer-implemented method for determining a lattice model, which depicts a lattice structure according to the invention with edge elements and lattice elements as an electronic data model for further processing.

In practice, the desired three-dimensional free-form structure is already available as an electronic data model in a computer, where it was designed, for example, by an architect or digitized based on an analog model. In order to design a lattice structure that approximates this free-form structure as well as possible in the expanded state, the method comprises the following steps.

In a first step, a planar surface piece P and a free-form surface piece P to be approximated are received on an electronic device, in particular a computer equipped for this purpose with a central data processing unit (CPU), a volatile semiconductor memory (RAM) and non-volatile semiconductor or magnetic memory ( ROM), electronic interfaces such as USB, LAN or WLAN as well as data input and data output devices. For example, the planar surface piece P can be stored in the non-volatile memory as a data structure, and the free-form surface piece P to be approximated can be entered by the user via a data input device or received by an electronic interface. The patches P and P can be stored as a data structure in any coordinates.

The next step is the discrete display of the patches of area. The planar patch is initially expressed simply by the 2-dimensional Cartesian coordinates of its four corner points. Then the 4 connections between the corner points are subdivided into a certain number (typically 100 each) of equidistant network nodes.

The geometric shape of the planar surface piece P is determined by the freely selected angle, which can be, for example, 30 °, 45 °, 60 °, 75 ° or the like. The angle a of the free-form surface piece P is usually different and not freely selectable, but rather determined mathematically by the shape of the surface.

The free-form surface piece P is first stored in the form of a NURBS surface (Non-Uniform Rational B-Spline), then it is converted into a discrete triangular network with a resolution of typically 100x100 network nodes by tessellation. The network node of the discrete triangular network, which approximates the free-form surface piece, contains its position in 3-dimensional Cartesian coordinates. Geodetic distances between any two network nodes can be calculated efficiently on this discrete triangular network.

In this step, the freeform surface piece to be approximated can already be analyzed with regard to its properties. If, for example, it is determined that the freeform surface piece has a local curvature that exceeds a predetermined maximum value (so that the grid lines do not reach certain areas of the freeform surface piece), then the freeform surface piece P is already smoothed here. This smoothing can take place iteratively until the local curvature complies with the predetermined threshold value.

The tessellation is also designed in such a way that opposing edge elements on the planar surface piece P and on the freeform surface piece P receive the same number of equidistant network nodes. Each network node can be assigned a ui, u ₂ , vi, or v ₂ coordinate, depending on which edge element it is located on.

In a next step, distance fields are calculated for the planar surface piece P and for the freeform surface piece P, the distance fields each having the coordinates of point pairs ui, u ₂ and vi, v ₂ on opposing edge elements depending on the included angle relate ä or a. While the angle a can be varied, the angle a of the free-form surface piece is fixed. For this purpose, the u and v coordinates can be normalized in equidistant steps from 1 to 50, 1 to 100 or the like, and the values of ui, u ₂ and vi, v ₂ are increased linearly accordingly. In the next step, the distance fields become distance maps D _u (ui, u ₂ , ä), D _v (vi, v ₂ , ä) for the planar patch P and D _u (ui, u ₂ ), D _v (vi, v ₂ ) combined for the free-form surface piece P, the distance images each representing the geodetic distances of all possible point pair combinations on mutually opposite edge elements. In other words, the function D _u (ui, u ₂ ), for example, delivers exactly the geodetic distance on the free-form surface between two opposite edge points with the given ui, u ₂ coordinates. In the case of the planar patch, this distance also depends on the angle selected. While the distance images of the free-form _{surface piece depend only on ui and u 2} or vi and v ₂ due to the fixed angle a, the distance images of the planar surface piece are also functions of the freely selectable angle Grids of 100 x 100 points are evaluated.

In a next step, the distance maps D _u (ui, u ₂ , ä) and D _u (ui, u ₂ ) as well as D _v (vi, v ₂ , ä) and D _v (vi, v ₂ ) are intersected for the formation of covering functions F _u (ui, ä) and F _v (vi, ä). The point combinations ui, u ₂ and vi, v _{2 are} determined which are identical for the planar surface piece P and the freeform surface piece P (to enable these end points to be fixed during the clamping process) and which, on the other hand, also have identical geodetic distances on the surfaces (since the grid elements cannot stretch or compress).

In order to create the covering functions, this step can provide that a numerical optimization problem is solved in ä, whereby the value of ä is minimized and the boundary conditions are observed that on the one hand there is exactly one value of u ₂ for each value of ui, and on the other hand, there is exactly one value of v ₂ for every value of vi.

The condition that a diagonal of the planar surface piece P is shorter than the corresponding diagonal of the freeform surface piece P and vice versa can be introduced as a further boundary condition of the optimization problem. This ensures that the distance mappings intersect in the space spanned _{by ui, u 2} or vi, v _2.

As a further boundary condition of the optimization problem, a minimum slope k _min and a maximum slope k _{max of} the segments of the clothing functions F _u (ui, ä) and F _v (vi, ä) can be specified.

As a result, clothing functions F _u (ui, ä) and F _v (vi, ä) are found which supply _{exactly one valid value of u 2} and v _{2 for each value of ui and vi.}

In a next step, coordinates are selected for grid elements, the end point coordinates of which are part of the covering function F _u (ui, a) and F _v (vi, ä). The number of grid elements depends on the shape of the free-form surface piece; Sufficient grid elements must be selected to approximate the freeform surface with sufficient density.

For example, a certain density of intersection points can be provided per unit area, which must be met. This depends on the curvature of the free-form surface to be approximated, whereby surfaces with many changes in curvature (positively curved, negatively curved) require a higher density of intersection points than surfaces without a change in curvature. The mentioned selection of coordinates for grid elements relates here to a purely electronic determination of coordinates or lengths of the grid elements.

On the basis of the results of the computer-implemented method, the physical grid elements (grid lamellas) at Fland can then be manufactured and provided using these electronically determined parameters. The material and thickness of the grid elements depend on the respective field of application.

After selecting the coordinates of suitable grid elements, by comparing the arc lengths s and s of these grid elements found, on the one hand at the intersections in the planar area P and on the other hand in the freeform area P, the required length of elongated holes of the intersecting grid elements at these intersections can be calculated. In other words, by comparing the coordinates of the intersection points in P and P, the length of the elongated holes required in the intersecting grid elements can be determined. In order to determine the direction and length of the elongated holes at the intersection of intersecting grid elements, the partial lengths s and s in the planar area P and in the free-form area P can be compared.

According to the invention, it can also be provided that when selecting the grid elements, a minimal set of grid elements is initially defined by choosing _{those grid elements in the intersection curves C u} (s) and C _v (s) of the distance maps which are at the extreme values of C _u (s) and C _v (s) and at the extreme values of the curvature of C _u (s) and C _v (s). These grid elements cover the area of greatest curvature and should be selected in any case.

As a result, the remaining gaps can be compacted by inserting additional grid elements, wherein the additional grid elements can be positioned such that the sum of the square distances to the existing grid elements is minimal.

The invention further relates to a computer-readable storage medium, comprising instructions which, when executed by a computer, cause the computer to carry out a method according to the invention.

Further features according to the invention emerge from the claims, the figures and the description of the exemplary embodiments. The invention is explained in more detail below on the basis of non-exclusive exemplary embodiments. Show it:

Figs. 1a-1b are schematic representations of an exemplary embodiment of a lattice structure according to the invention in an open position and in a planar position;

Figs. 1 c-1 d are schematic representations of a mathematical lattice model for creating the lattice structure;

Figs. 2a-2b schematic representations of a planar surface piece P and a freeform surface piece P to be approximated; Figs. 3a-3b are schematic representations of a planar grid element g (u1, u2) on P and a non-planar grid element g (u1, u2) on P;

Figs. 4a-4c are schematic representations of a covering function and a planar grid element g (u1, u2) and a non-planar grid element g (u1, u2); Figs. 5a-5c are schematic representations to illustrate the formation of distance fields in the case of a planar surface piece P and a freeform surface piece P; Figs. 6a-6c are schematic representations of the distance mapping D _u for the planar surface piece from FIGS. 5a-5c in different views;

Figs. 7a-7d: schematic representation of two intersections of the distance maps D _u for the planar surface piece with the distance map D _u for the freeform surface piece and the resulting covering functions;

Figs. 8a-8c: schematic representations of a free-form surface P to be approximated, a valid covering function F _u (ui, a) and the resulting grid model; Figs. 9a-9b: schematic representations of a free-form surface to be approximated and a smoothed free-form surface;

Figs. 10a-10b: schematic representations of grid models and intersection curves C _u (s) before and after compression;

Figs. 11 a-11 b: schematic representations of a planar surface piece and the corresponding free-form surface piece as well as the points of intersection of the grid lines.

Figs. 1a-1b show schematic representations of an exemplary embodiment of a physical lattice structure 1 according to the invention in an open position (FIG. 1a) and in a planar position (FIG. 1b). The physical lattice structure 1 comprises edge elements 2, 2 ', which span an area, and lattice elements 3, 3', which are arranged between the edge elements 2, 2 'with the formation of crossing points 4. In each case two edge elements 2, 2 'lie opposite one another, and in each case a family of grid elements 3, 3' is arranged between the opposite edge elements 2, 2 '. A family denotes those grid elements 3, 3 'which extend between two specific edge elements 2, 2'. The edge elements 2, 2 'and the grid elements 3, 3' are designed as grid slats made of wood, aluminum, steel or glass fiber reinforced plastic. The lattice slats are straight, elastic and resistant to tension and compression. The grid elements 3, 3 'are not parallel within a family, so that the grid structure 1 can only deform upwards or downwards when the grid elements 3, 3' bend. Adjacent edge elements 2, 2 'are arranged at an angle α to one another in the planar position (FIG. 1b) and at an angle α in the open position (FIG. 1a), where a ä. Swept sizes always relate to the planar grid .

The lattice elements 3, 3 'are attached to the edge elements 2, 2' in such a way that the lattice structure 1 has two compatible geometric states: a planar position (FIG. 1b) and an open position (FIG. 1a), the lattice structure 1 in approximates a given three-dimensional free-form surface in the clamped position. The grid elements 3, 3 ‘follow geodetic lines between the edge elements 2, 2‘ in both positions. Geodetic lines provide the shortest and most straight line connection between two points on a surface.

The two positions can be continuously converted into one another by means of a clamping process. In order to approximate the desired free-form surface, all grid elements must therefore lie on the specified free-form surface in the expanded state.

The grid elements 3, 3 ‘are provided with (schematically shown) elongated holes at the crossing points 4, which have different lengths in the intersecting grid elements 3, 3‘ so that a translational movement of the grid elements 3, 3 ‘is possible to a certain extent. The grid elements 3, 3 'are rotatably and movably connected at the crossing points 4 by connecting means, for example screws or bolts. The connecting means protrude through both elongated holes of the intersecting grid elements 3, 3 ‘.

Figs. 1c-1d show an abstract schematic representation of the lattice structure 1 in the form of a computer-generated lattice model with lattice lines which correspond to the edge elements 2, 2 'and lattice elements 3, 3'. This grid model is implemented as a data structure in a computer. The computer is a conventional PC with an electronic data processing unit, one or more storage units (for example in the form of a magnetic memory and a semiconductor memory), and interfaces. However, the specific design of the computer is not relevant to the present invention.

To generate the grid model, provision is made for a free-form surface P to be approximated as a target surface and, if necessary, also a planar surface to be input into the computer, for example by reading them out from a memory unit. The surfaces can be entered into the computer in the form of coordinates or also in the form of vectors or splines. The planar surface is preferably a square surface. For efficient calculation, the areas are discretized by a triangular network. The resolution of the triangular network can be freely chosen; the minimum resolution should be around 50x50 to 100x100 network nodes. The surfaces are represented by the computer for further processing in the normalized coordinates u [0 .. 1] and v [0 .. 1]. However, the specific procedure for discretization and conversion into normalized coordinates is not relevant; It is only important that identical coordinate systems are used for the planar surface and the free-form surface.

To ensure that the desired freeform surface piece P can be sufficiently approximated by the grid, P must be checked. If P cannot be sufficiently approximated by a grid, P is smoothed. This check takes place at the beginning of the method, since distances on the free-form surface piece P can change during the smoothing process.

FIG. 1c shows the computer-generated grid model in the planar position and FIG. 1d in the expanded position. A first grid element family consisting of three grid elements 3, which are not parallel to one another, is arranged between the two edge elements 2. A second grid element family consisting of two grid elements 3 'which are not parallel to one another is arranged between the two edge elements 2'. By changing the angle of the edge elements 2, 2 ' the lattice structure 1 can be transferred from the planar position to the expanded position. In both positions, the grid elements 3, 3 'run as geodetic lines on the planar surface or on the free-form surface.

Figs. 2a-2b show the computer-generated representations of the edge elements of the planar grid and the spatial grid on the free-form surface in the coordinates u and v. The edge square of the planar grid delimits a planar area P, the edge quadrangle on the free-form surface delimits a free-form area P. The four edge elements 2, 2 'that make up the edge are of equal length in the planar and spatial grid, since the edge elements should be tensile and compressive, and have the coordinates ui, u _2, vi, v ₂ . In order to be able to generate both the planar and the expanded state, each grid element 3, 3 'lying between the edge elements 2, 2' must meet the following requirements:

(i) The distances g (IM, U ₂ ) and g (u, u ₂ ) are of equal length, and

(ii) The end point coordinates m and u ₂ are the same for g and g.

These conditions are shown in Figs. 3a - 3b shown schematically: The lattice elements must fit between the edge elements in the planar and in the stretched state and begin and end at the same edge coordinates.

In addition to these central conditions, further restrictions must be met, which ensure the kinematic functionality of the grid:

(iii) Each point with coordinate ui has one and only one corresponding point on the opposite edge element with coordinate u ₂ , whereby a grid element is uniquely defined.

(iv) The edges of the freeform patch P and the planar patch P must be convex.

These conditions apply to both families of the grid elements, i.e. also to those grid elements that run along the coordinates vi, v ₂ . To the above To meet conditions, the distances between opposite edge elements are systematically recorded and used to find the functions it2 = F _M (U _I , ä) and v ₂ = F _v (vi, ä), the conditions (i), (ii) and (iii) satisfy. These functions link the planar patch P and the freeform patch P.

Each point of the functions F _u or F _v corresponds to a geodetic line along the coordinates u or v on P and P. Since the edge of P has the angle as a degree of freedom which influences the distances between the edge elements, «is a variable of these functions . The functions F _u and F _v are therefore used to cover P and P with geodetic lines; they control the position and alignment of the geodetic lines and are referred to as covering functions.

4a shows a valid covering function for a family of grid elements along the coordinates u1, u2. The function U2 = F _u (ui, ä) delivers exactly one grid element with a point on U2 for each point on ui. The function v ₂ = F _v (vi, ä) delivers exactly one grid element with a point on v ₂ for each point on vi.

Here ui, U2 and vi, v _{2 take on} values in the range from [0 .. 1]. Figs. 4b and 4c show exemplary images of a possible grid element in the planar surface and in the free-form surface.

To determine F _u and F _v , distances are measured using numerous distance fields and combined to form distance maps D _u , D _M , D _v , D _v . The distance fields have a starting point on ui or vi and provide a geodetic distance value to the starting point for each opposite point. In the planar plane, the geodetic distance coincides with the Euclidean distance. All distances are recorded separately for the u and v coordinates. D _u (ui, u ₂ ), D _u (ui, u ₂ , ä) summarize, for example, the distances for the family of geodetic lines between the opposite edge elements with the coordinates ui, u ₂ for P and P. Figs. 5a -5c show the acquisition of distances from a distance field between the edge elements with the coordinates ui, u ₂ . Since the grid elements for the family shown run from the lower edge element with coordinate ui to the upper _{edge element with coordinate u 2} , the distances from all points p (ui) to all points q {u ₂ ) are measured. In practice, the number of points p and q is limited and is related to the resolution of the triangular network.

For the representation of the geodetic distances d between two edge elements, the lengths of many distance fields are combined to form distance maps. A distance mapping is a collection of geodetic distances in a three-dimensional space with the coordinate axes ui, u ₂ and d {ui, ₂ ). The distance mappings are created for both the free-form patch P and the planar patch P. The distance images D from the planar grid are dependent on the angle ä, since the distances in the area depend on the shape of the planar edge. The distance maps are shown in Figs. 6a-6c for the planar surface piece D are shown schematically.

Conditions (i) and (ii) require that grid elements with the same coordinates ui, u ₂ and vi, v _2, respectively, are of equal length. These grid elements can be found as common points in the distance mappings of the planar surface piece and the freeform surface piece.

Common points of the distance mapping D _{u of} the freeform surface piece and the distance mapping D _{u of} the planar surface piece thus correspond to geodetic lines between the edge elements of the coordinates ui, u ₂ , which meet the conditions (i) and (ii). The common points are found technically by intersecting the two distance maps D and D. The points of intersection form the covering function F _u and F _v . These link the

and u ₂ coordinates or the v and v ₂ coordinates such that conditions (i), (ii) and (iii) are met. Figs. 7a-7b show a satisfactory covering function for the example from Figs. 5a - 5c. In contrast, Figs. 7c-7d a covering function which does not fulfill condition (iii): condition (iii) requires that there may only be exactly one grid element _{for each u or u 2 coordinate.} This condition is in the case of Figs. 7c - 7d for the coordinates ui = 0 and u ₂ = 1 not fulfilled, because the distance maps Du and Du always have at least two points in common, namely the points with the coordinates u- _\ = u ₂ = 0 and m = u ₂ = 1. In order to achieve an optimal intersection of the distance maps, an optimization problem is solved under two constraints:

Constraint 1: So that the distance maps D _u and D _{u have} more than two points in common, the following first constraint is formulated for the existence of an intersection curve:

(e - e) (/ - /) <0

This condition relates to the diagonals e and / of the planar surface piece P and the diagonals e and f of the freeform surface piece P (these are indicated schematically in FIG. 7d). In other words, a diagonal of P must be shorter than the corresponding diagonal of P and vice versa for the other diagonal.

This manifests itself in the distance pictures such that D _u D _u is above one of the two corners ((u = 0, u ₂ = 1) and (u = 1, u ₂ = 0)) and at the remaining corner under D _u . This ensures at least a partial overlap.

Secondary condition 2: To ensure a complete intersection, the intersection in the ui, u ₂ space is considered and a piece-wise parametric representation F _{u is} introduced. F _u connects all points in ui, ii ₂ -space that were found at the intersection, is piecewise linear and throughout

and u ₂ - area defined (see Fig. 4a). Due to the dependence D _u on ä, F _{u also depends on c7.} In order to obtain a covering function F _M which fulfills (i), (ii) and (iii), limits for the slope of the segments of F _{u are used} as a second constraint in the optimization problem.

The optimization problem can thus be formulated: min a

where n is the number of segments and / c _min and k _{msK are} limits for the slope, which can be defined, _{for example, as k m \ n} = 0.1 and k _{m3X = 10.}

When performing the optimization, a minimum permissible angle _{min is} found in a first step. The minimization problem is then reformulated into a maximization problem and a maximum permissible angle _{max is} found. For angles in between, the common points of the distance mappings in the ΐί-i, ii ₂ -space form a piecewise linear, bijective tensioning function F _m . The angles in this range can thus be used, the angle can be set as _{min in} order to obtain a compact planar grating.

In addition to the conditions (i), (ii) and (iii), the covering functions should also ensure that the entire free-form surface area P can be covered with geodetic lines.

The measured distances, which are summarized in the distance figures, are the shortest distances between two points on opposite edge elements. A geodetic line can be drawn for each of these pairs of points, as shown in FIG. 5b. The paths of these geodetic lines are the shortest connections between these points. However, even for a valid covering function according to criteria (i), (ii) and (iii), these shortest geodetic lines cannot provide unambiguous solutions for certain pairs of points. Figs. 8a - 8c illustrate the problem: Despite a valid covering function, a central area of the free-form surface piece remains uncovered. For a pair of points on the two boundary elements there are two shortest geodetic connections.

For the quality of the approximation of the surface by the grid, it is therefore necessary to avoid ambiguous solutions for geodetic lines between the edges. Because this behavior of the geodetic lines is influenced by the curvature of the surface it can be ensured by gradually smoothing the surface (reducing the curvature) that only unique solutions for geodetic lines occur. During the smoothing process, a check is made in each iteration to determine whether the area that cannot be covered has already been reduced sufficiently. If there is no problem during the initial check, the free-form surface piece can be used as it is. If the freeform surface piece has to be smoothed, there is a change in shape, as shown in Figs. 9a - 9b can be seen.

In the next step, suitable grid elements are selected from the geodetic lines defined by the covering functions. The shape of the free-form surface piece P is accompanied, for example, by a certain minimum number and positioning of grid elements. In addition, the grid should allow an even distribution of the grid elements in order to meet aesthetic requirements.

The definition of the grid elements is therefore divided into two parts. In the first step the minimum set of grid elements is defined, in the second step the grid can be condensed. In order to find the minimum set of grid elements, the intersection of the distance maps is used again.

The intersection curve C _{u is} considered (cf. FIGS. 7a-7d.) If the arc length s of the intersection curve C _u and the lengths d of the associated geodetic lines are plotted, as shown in FIGS. 10a-10b, one can identify longest and shortest geodetic lines _{in C u (s).}

The relationship between the length of the geodetic lines and the shape of P is established by the curvature of P. Figuratively speaking, geodetic lines get longer as the curvature of a surface increases. This observation inspires the definition of the grid elements. The grid elements are first defined at the extreme values of C _u (s) and then at the extreme values of the curvature of C _u (s). The first pass ensures that the main features that make up the shape of P (eg large elevations) are mapped in the grid, since these grid elements correspond to the locally longest and shortest geodetic lines. The second step ensures that finer features (e.g. smaller elevations) are recorded through compression. In Fig. 10a the grid elements are entered in the C _u (s) function. The third element is an element of the first run, the remaining elements were found in the second run. If one of these grid elements is omitted, the shape of the free-form surface piece (three elevations and two constrictions) cannot be reproduced in the grid.

In order to condense the grid after the important grid elements have been found, additional grid elements are systematically inserted into the gaps so that all gaps are approximately as large as the smallest gap. After insertion, the grid elements are positioned in such a way that the sum of the square distances to the existing grid elements is minimal. In this way, a visually appealing, more uniform distribution of the grating elements can be achieved, as is shown by way of example in FIG. 10b. However, grid elements can also be inserted completely freely. If the width of the physical lattice louvres is known in advance, this can be included in the compression steps. If, for example, overlaps occur, the grille slats should be narrowed.

A direct stretching of the planar lattice structure is generally not possible with pure rotational connections between the lattice elements, since the partial lengths between the connections do not match and the lattice elements must not be compressed or stretched.

The problem can be solved by introducing two additional degrees of freedom in translation for each connection in addition to the existing degree of freedom in rotation. This enables the grid elements to slide off one another during the transformation and solves the geometric problem. In the physical grids, a connection of two grille slats can be implemented using two elongated holes and a screw or bolt.

In order to determine the direction and length of the elongated holes, the partial lengths between connections in the planar and the spatial grid are compared. Compares If you put a grid element gt (s) on P and the corresponding grid element gi (s) on P and measure the arc length from the beginning of the grid element to the connections with other grid elements, the necessary length of the elongated hole i _gi can be determined as follows: lgi = S - S.

The sign of this equation gives the direction of the slot from the center of the hole. This is shown in Figs. 11 a - 11 b shown schematically.

After a grid layout has been found and the lengths of the elongated holes have been determined, it can be made. The physical lattice structure consists of flexible lattice blades. The stretching of the lattice structure consists of the following two processes:

In a first step, the angle of the edge elements is changed. By changing this angle ä - a, the kinematic mechanism of the grid is activated and the grid deforms into space. The lattice elements rotate around the connections and the connections slide in intersections along the elongated holes. The shape that the grid will take is not clearly determined in advance. The grid can deform upwards or downwards. It is a branching problem in the shape of the lattice. This branching problem can be eliminated by pushing individual points of the grid in the desired direction at the beginning of the deformation.

In general, the change from ä - a is not sufficient for the grid to approximate the desired freeform surface piece P well. Therefore, the lattice structure is bent in a second step. Since the grids are flexible, they can be bent into different shapes. The grid can be bent so that it comes to rest on P.

For a successful clamping process, it must be ensured at the beginning that the grid takes on the correct shape, and after ä - a it must be bent in such a way that the shape of the grid structure matches the shape of the freeform surface piece matches. For a good match of the shape of the lattice with the shape of P, it is enough to fix a few connections on the edge. These connections are selected depending on the curvature of the edge elements. In particular, the connections can be fixed which are closest to the extreme values of the curvature of the edge elements. The best correspondence between the shape of the grid and the free-form surface piece P is achieved when all connections are fixed at the edge.

A connection of two lattice elements is fixed by being assigned a fixed position and orientation in space. The position is the point on the free-form surface at which the corresponding grid elements meet; the alignment is determined by the surface normal at this point. This can be ensured by means of supports with inclined mating threads for screwing in the connecting screws.

However, the invention is not restricted to the exemplary embodiments shown, but rather comprises all devices and methods within the scope of the following patent claims.

Claims

1. Lattice structure (1) for forming a three-dimensional free-form structure, comprising a. Edge elements (2, 2 ‘) that delimit an area, b. Lattice elements (3, 3 ‘) which are arranged between the edge elements (2, 2‘) to form intersection points (4), c. wherein the edge elements (2, 2 ‘) and the grid elements (3, 3‘) are straight, elastic and resistant to tension and compression, characterized in that

- The lattice elements (3, 3 ') are attached to the edge elements (2, 2') in such a way that the lattice structure (1) o from a planar position in which it forms a planar surface piece, o into an open position in which it forms a free-form surface piece approximating the free-form structure, can be brought,

- The edge elements (2, 2 ‘) in the planar position enclose the angle a and in the open position enclose the angle a, where a a, and

- The grid elements (3, 3 ‘) in both positions follow geodetic lines between the edge elements (2, 2‘).

2. Lattice structure (1) according to claim 1, characterized in that in the planar position the edge elements (2, 2 ‘) and the lattice elements (3, 3‘) are not arranged parallel to one another.

3. Lattice structure (1) according to claim 1 or 2, characterized in that the

Edge elements (2, 2 ‘) and the grid elements (3, 3‘) are formed as straight lattice slats made of wood, aluminum, steel or glass fiber reinforced plastic.

4. Lattice structure (1) according to one of claims 1 to 3, characterized in that in the planar position a square surface is spanned by two edge elements (2, 2 '), with at least two non-parallel lattice elements (3, 3') are provided.

5. Lattice structure (1) according to any one of claims 1 to 4, characterized in that the lattice elements (3, 3 ') at the intersection points (4) have elongated holes with in particular different lengths, with two intersecting lattice elements (3, 3') are connected at the crossing points (4) by connecting means, for example screws or bolts, rotatably and displaceably relative to one another.

6. The method for stretching a lattice structure (1) according to one of claims 1 to 5 from a planar position to a stretched position, characterized in that the method comprises the following steps: a. Rotation of the edge elements (2, 2 ') relative to one another in such a way that the spanned angle of the edge elements (2, 2') increases or decreases from - in the planar position to a in the open position, b. Bending of the lattice structure (1) to approximate the free-form surface to be approximated, c. Fixing of the end points of selected grid elements (3, 3 '), in particular those end points which are closest to the extreme values of the curvature of the associated edge element (2, 2') in the free-form surface to be approximated, preferably all of the end points of the grid elements (3, 3 ') ), on external supports.

7. Computer-implemented method for determining a lattice model which depicts a lattice structure (1) with edge elements (2, 2 ') and lattice elements (3, 3') according to one of claims 1 to 5, characterized in that the method comprises the following steps : a. Receipt of a planar surface piece P and a free-form surface piece P to be approximated, b. Discretization of the patches in coordinates u, v; c. Calculation of distance fields for the planar surface piece P and for the freeform surface piece P, the distance fields each being the coordinates of point pairs ui, u ₂ and vi, v ₂ on opposite edge elements (2, 2 ') depending on the included angle relate ä and a, d. Combination of the distance fields into distance maps D _u (ui, u ₂ , ä), D _v (vi, v ₂ , ä) for the planar patch P and D _u (ui, u ₂ ), D _v (vi, v ₂ ) for the free-form surface piece P, the distance images each representing the geodetic distances of all possible point pair combinations on mutually opposite edge elements (2, 2 '), e. Intersection of the distance mappings D _u (ui, u ₂ , ä) and D _u (ui, u ₂ ) as well as D _v (vi, v ₂ , ä) and D _v (vi, v ₂ ) to form clothing functions F _u (ui, ä) and F _v (vi, ä), with covering functions denoting those point pair coordinates ui, u ₂ and vi, v ₂ , i. which are identical for the planar patch P and for the freeform patch P, and ii. the distances of which are identical for planar surface piece P and for the free-form surface piece P, f. selection of grid elements (3, 3 ') whose end point coordinates are part of the covering function F _u (ui, ä) and F _v (vi, ö).

8. The method according to claim 7, characterized in that, in order to form the covering functions in step e.), An optimization problem in is solved, the value of being minimized and the boundary conditions being observed that a. for each value of ui there is exactly one value of U2, and b. for each value of vi there is exactly one value of v ₂ .

9. The method according to claim 8, characterized in that the condition that a diagonal of the planar surface piece P is shorter than the corresponding diagonal of the freeform surface piece P and vice versa is introduced as a further boundary condition of the optimization problem.

10. The method according to claim 8 or 9, characterized in that a minimum slope k _min and a maximum slope k _{max of} the segments of the clothing functions F _u (ui, ä) and F _v (vi, ö) are specified as a further boundary condition of the optimization problem .

11. The method according to any one of claims 7 to 10, characterized in that the discretization in step b) takes place by triangulation, the resolution preferably being at least 50 x 50 points.

12. The method according to any one of claims 7 to 11, characterized in that after the selection of the grid elements (3, 3 ') by comparing the arc lengths s and s of the grid elements (3, 3') at intersections (4) in the planar area P and in the free-form surface piece P the required length of elongated holes of the intersecting grid elements (3, 3 ') at these intersection points (4) is calculated.

13. The method according to any one of claims 7 to 12, characterized in that in step b) the maximum curvature of the freeform surface piece P is determined and the freeform surface piece P is smoothed when the curvature exceeds a predetermined maximum value.

14. The method according to any one of claims 7 to 13, characterized in that in step f) to select the grid elements (3, 3 ') a. a minimal set of grid elements is defined by choosing _{those grid elements (3, 3 ') in the intersection curves C u} (s) or C _v (s) of the distance maps that are at the extreme values of C _u (s) or . C _v (s) and lie at the extreme values of the curvature of C _u (s) and C _v (s), and b. the remaining gaps are compacted by inserting additional grid elements (3, 3 '), the additional grid elements being positioned in such a way that the sum of the square distances to the existing grid elements is minimal.

15. A computer-readable storage medium, comprising instructions which, when executed by a computer, cause the computer to carry out a method according to any one of claims 7 to 14.