EP1620821A2 - System and method for multiple-sided surface matching with curvature continuity - Google Patents

System and method for multiple-sided surface matching with curvature continuity

Info

Publication number
EP1620821A2
EP1620821A2 EP04750337A EP04750337A EP1620821A2 EP 1620821 A2 EP1620821 A2 EP 1620821A2 EP 04750337 A EP04750337 A EP 04750337A EP 04750337 A EP04750337 A EP 04750337A EP 1620821 A2 EP1620821 A2 EP 1620821A2
Authority
EP
European Patent Office
Prior art keywords
matching
curvature
neighboring
tangent
continuity
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Ceased
Application number
EP04750337A
Other languages
German (de)
French (fr)
Inventor
Thawach Sripradisvarakul
Yi-Feng Tsai
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Siemens Industry Software Inc
Original Assignee
UGS Corp
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by UGS Corp filed Critical UGS Corp
Publication of EP1620821A2 publication Critical patent/EP1620821A2/en
Ceased legal-status Critical Current

Links

Classifications

    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06T—IMAGE DATA PROCESSING OR GENERATION, IN GENERAL
    • G06T17/00—Three-dimensional [3D] modelling for computer graphics
    • G06T17/30—Polynomial surface description

Definitions

  • the present invention is directed, in general, to surface generation and modification in graphical modeling systems .
  • Curve-based surface generation schemes are very commonly used in computer-aided geometric design (CAGD) for modeling of free-form geometry.
  • CAGD computer-aided geometric design
  • One of the simplest among them is skinning, which interpolates a series of curves to form a surface.
  • lofting this method stretches a surface over a series of "ribs" or cross- sections. The surface will preserve the shape of each cross-section at the corresponding location, and will blend smoothly from one to the next.
  • Figure 1A illustrates the skinning technique; note the "ribs", or cross-sections, shown on the left; these are used as a basis for the generated "skin" surface shown on the right.
  • continuity describes how two things come together. Two types of continuity are generally discussed: C n and G n , where n refers to the order of continuity. C n continuity refers to "parametric" continuity of the n th order. This means that the magnitude and direction of all derivatives up to the n th order must agree. G n continuity, on the other hand, refers to "geometric" continuity of the n th order, which loosens the above definition by allowing re-parametrizations from the C n condition. A good example is the 1 st order continuity: C 1 between two curves requires the two end tangent vectors to be identical, while the G 1 condition only requires the two end tangent vectors to be parallel. Although not as strict as parametric continuity, geometric continuity is more useful in CAGD in that it describes the sufficient conditions for two connecting entities to "look" smooth in the neighborhood of their juncture.
  • Class-A (or sometimes referred to as class-1) geometric design typically sets stringent standards in surface quality in order to satisfy aesthetic or aerodynamic goals. In most cases, class-A surfaces are required to be curvature continuous while providing the simplest mathematical representation needed for the desired shape/form, and do not have any undesirable waviness.
  • Free-form surface modeling of a car body requires the generated surface patches to maintain good continuity (G 2 or higher) with each other, as well as to demonstrate visual smoothness everywhere when observed in lights.
  • the surfaces should be "simple” and “clean,” that is, having a reasonable number of control points, a good structure of the control net, and nice parametric flows.
  • Heavier than necessary surface geometry may pose problems to subsequent processes, such as further geometry generation based on the existing ones, mesh generation and NC tool path generation.
  • Unclean or "messy" control net structure is undesirable for designers, who rely heavily on control point editing to modify geometric entities. All of the above quality can be examined using various diagnostic tools in modern geometric modeling systems. Some of these are demonstrated in Figure 2.
  • Figures 2A-2D show various surface quality diagnostic tools applied on a model that comprises six surfaces.
  • Figure 2A shows a continuity needles plot, which displays the discontinuity measurement between surfaces by needles (vectors) of various lengths and directions.
  • Figure 2B shows a surface contours plot, where each contour represents constant reflectance of light on the surfaces. The smoothness of the contours reflects the smoothness of the surfaces.
  • Figure 2C shows a curvature needles plot, which displays the distribution of cross-sectional curvatures on the surfaces. This plot illustrates the curvature "flow" on the surfaces, and is a good indicator for different types of shape imperfections.
  • Figure 2D shows a zebra plot, which imitates the shape inspection environment in the automotive body design process where the model is inspected in a room with arrays of lights. Similar to the contours plot, it provides a clear visualization of the smoothness of the surfaces.
  • freeform surface modeling complex shapes are generally modeled by using several NURBS surfaces. To satisfy smoothness and aesthetic requirements, particularly in Class A surface modeling, it is very important that the generated surfaces are curvature continuous to each other. In curve-based surface generation, some selected curves can be boundaries of existing surfaces (edges or trim curves), and the generated surface is very often expected to be automatically tangent or curvature continuous to the existing ones. Alternatively the user can create and edit the shape of each surface manually and then "match" them together to establish continuity among them.
  • the preferred embodiment provides a system and method for providing multiple-sided surface matching with curvature continuity in a graphical modeling system.
  • the preferred embodiments use improved algorithms for tangential surface matching and curvature surface matching, which provide improved continuity between a given surface and one or more neighboring surfaces.
  • Figures 1A-1C depict common curve-based surface generation methods.
  • Figure 1A illustrates interpolating a series of curves to form a surface
  • Figure IB illustrates interpolating a 2 by N network of curves to form a surface
  • Figure IC illustrates interpolating an M by N network of curves to form a surface.
  • Figures 2A-2D show various surface quality diagnostic tools.
  • Figure 2A shows a continuity needles plot
  • Figure 2B shows a surface contours plot
  • Figure 2C shows a curvature needles plot
  • Figure 2D shows a zebra plot.
  • Figure 3 depicts a process in accordance with a preferred embodiment of the present invention
  • Figure 4 depicts a process in accordance with a preferred embodiment of the present invention
  • Figure 5 shows a matching surface to be matched to four neighboring surface
  • Figure 6 shows an output surface after tangential matching
  • Figure 7 shows a figure which is a result of matching the middle surface to its four neighboring surfaces with tangent continuity
  • Figure 8 shows a figure that is the result of curvature matching of a surface.
  • FIGURES 1 through 8, discussed herein, and the various embodiments used to describe the principles of the present invention in this patent document are by way of illustration only and should not be construed in any way to limit the scope of the invention. Those skilled in the art will understand that the principles of the present invention may be implemented in any suitably arranged device. The numerous innovative teachings of the present application will be described with particular reference to the presently preferred embodiment.
  • the preferred embodiment provides a system and method for providing multiple-sided surface matching with curvature continuity in a graphical modeling system.
  • the preferred embodiments use improved algorithms for tangential surface matching and curvature surface matching, which provide improved continuity between a given surface and one or more neighboring surfaces.
  • Iso-parametric curve A B-spline curve obtained by tracing on a B-spline surface along the entire domain of one of the (u, v) parameters, with the value of the other parameter kept constant. Isoparametric curves obtained by keeping the u parameter constant are referred to as “constant-u iso-parametric curves”. Similarly for “constant- v iso-parametric curves”.
  • Knots -- A knot is a number which is part of the definition of a B-spline curve or surface and which is used to control its shape.
  • the collection of all knots used by a B-spline is called a knot vector.
  • the distance between knot values is called its knot spacing.
  • Uniform B- splines set the knot spacing to 1 (usually) , and non-uniform rational B-splines allow for uneven spacing of the knots.
  • the disclosed multiple-sided matching is used in several curve based surface generation schemes, where the generated surfaces are required to be tangent or curvature continuous with neighboring surfaces.
  • the surface construction schemes consist of two basic steps: [0033] (1) Create an initial base surface that interpolates the input curves, including the neighboring boundaries, using well-known methods such as skinning, Coon surface, or Gordon surface. The input curves can be reparameterized if necessary or desired.
  • Each boundary can be a surface edge or a trimmed boundary curve on surface.
  • the process assumes that the input neighboring surfaces have the required continuity at the coincident end points of the corresponding boundary curves.
  • the matching algorithm consists of three basic steps: position, tangent, and curvature matching. Each step is basically an iterative constrained fitting process.
  • Position matching amounts to constrained curve fitting to position tolerance. Position matching can be summarized as follows:
  • step 2 to 4 again until all the fit curves are within position tolerance.
  • the weights of the matching NURBS surface are fixed.
  • the matching surface is positional continuous to its neighboring surfaces. It is also possible to modify the interior control points of the matching surface, based on the changes in the boundary control points, so that the matching surface have better control point structure and surface flow.
  • the subsequent tangent and curvature matching step will make use of the sample match points on neighboring surfaces and their assigned (u,v) parameter values of the matching surfaces.
  • the preferred embodiment ensures tangent matching by iterative constrained energy-minimizing least squares fitting according to a process as shown in Figure 3.
  • the system or a user will designate the matching surface and at least one neighboring surface (step
  • the system will next set up the desired linear constraints on the matching surface control points (step 320) .
  • the system will then perform a constrained least squares fit for the first order cross partial derivatives along all relevant boundaries of the matching surface, where the least squares term measures the tangent discontinuity (step 325) .
  • the system will then check the tangent continuity along required surface edges (step 330) . If the tangent continuity is within the specified tolerance, then stop (step 345) .
  • step 335 If the tangent continuity is not within tolerance, the system will analyze the tangent discontinuity (step 335) and insert knots to the matching surface at appropriate locations (step 340) . [0061] Repeat steps 325 to 340 again until the tangent continuity is achieved (returning to step 325) .
  • the positional continuous matching surface is defined as
  • u, v are the parameter of the surface and Ni (u) , Nj (v) are the B-spline basis functions along u and v parametric directions respectively.
  • the constrained least squares fitting in step 3 tries to minimize the following quadratic objective function:
  • E G ⁇ is defined to be the least squares term that measures tangent discontinuity.
  • E 2 , E 3 , and E b ⁇ e nd are energy terms for controlling the smoothness of the resulting matching surface. Without these energy terms, the control points can be jumpy, particularly when the matching surface has high orders.
  • E spr ⁇ ng ⁇ IIIi i- j -Cj . . I111 , where c 13 denotes the coordinates of the control point P 1D from the previous step.
  • Egpring is the spring energy term that tries to restrict the movements of the control points, so that the shape of the matching surface is close to the original surface.
  • twist constraints are derived from the classical three-tangent theorem, which states that two surface points are curvature continuous if and only if
  • twist constraint at a surface corner X(u,v) is defined as
  • k n (u, v, du, dv) (L (du) 2 +2Mdudv+N (dv) 2 ) / (E (du) 2 +2Fdudv+G (dv) 2 )
  • k n denotes the normal curvature value of the neighboring surfaces along a direction X u du + X v dv, with du and dv being arbitrary nonzero values
  • N X vv (u,v) • N (u,v)
  • N(u,v) refers to the surface normal at the corner. Note that, after position matching, L,N,E,F,G are constant values since the partial derivatives X u , X v , X U u, Xw are defined by the boundary control points. Therefore, it follows that the twist constraint is a linear equation.
  • the fitting variables are the coordinates of the matching surface's control points, except the boundary control points. With the boundary control points fixed, the least squares term E G ⁇ becomes a quadratic function of the coordinates of the remaining control points.
  • the smoothness and spring energy terms are also quadratic functions.
  • the formulated constrained energy minimizing least squares fitting can be solved by standard quadratic programming solvers, which are linear methods. It can be shown that the objective function is also positive definite, and the solution of the least squares fitting is unique.
  • One particularly unique aspect of the preferred process is the way the least squares term E G1 is defined, together with other energy terms, for fitting cross partial derivatives to achieve tangent continuity.
  • One major advantage of this approach is that it does not require any estimation of first order cross partial derivatives along the surface boundaries.
  • the cross derivatives estimations along two surface edges may not be "compatible" or consistent at the corner.
  • the estimations may fail or become ambiguous if the end tangents of iso-parametric curves before matching are nearly parallel to surface normals of the neighboring surfaces.
  • the preferred embodiment avoids these two problems completely, as the fitting can be considered as implicit fitting where the least squares term is defined from the necessary and sufficient conditions for tangent continuity.
  • the least squares term interacts with the smoothness and spring energy terms to produce smooth tangent continuous matching surface with good control point structure .
  • the preferred embodiment exploits the role of Ei e nd in producing nice control point structure.
  • the curvature matching process of the preferred embodiment adjusts the tangent continuous matching surface to achieve curvature continuity by iteratively fitting the second order partial derivatives along surface boundary edges, according to a process as shown in Figure 4. Steps in curvature matching are similar to those in tangent matching. The main difference is the way the least squares term is defined to measure curvature discontinuity, as further described below.
  • the system or a user will designate the matching surface and at least one neighboring surface (step 410) .
  • the system will compute the normal curvatures at sample points on the neighboring surfaces along one or more tangent directions (step 415) , where the selected directions cannot be parallel to the tangent directions of the boundary curves.
  • the preferred embodiment uses three directions, where one of the directions must be the direction of the first order partial cross derivative of the matching surface (the end tangent directions of the iso-parametric curves).
  • the system will then set up the desired linear constraints on the matching surface control points (step 420) .
  • the system will then perform a constrained least squares fit for the second order partial derivatives along all required boundaries of the matching surface, where the least squares term measures the curvature discontinuity (step 425) .
  • the system will check the curvature continuity along required surface edges (step 430) . If the curvature continuity is within the specified tolerance, then stop (step 445) .
  • the system will analyze the curvature discontinuity (step 435) and insert knots to the matching surface at appropriate locations (step 440) .
  • the classical three-tangent theorem states that two surface points are curvature continuous if and only if they are tangent continuous and their normal curvature values along three different tangent directions match.
  • the linkage curve theorem states that two surfaces tangent continuous along a smooth curve are curvature continuous if and only if, at every point along the curve, their normal curvature agrees along a direction other than the tangent to the curve.
  • the least squares term E g2 is defined to measure the curvature discontinuity as
  • E g2 ⁇ ( ⁇ k
  • k n denotes the normal curvature
  • N X vv (u,v) • N (u,v)
  • the other two tangent directions X u dun + X v dv k ⁇ and X u du k2 + X v dv k2 are optional and can be chosen arbitrarily.
  • the fitting variables in curvature matching are the coordinates of the control points, except for those control points defining position and tangent continuity.
  • E g2 is a quadratic function, after fixing these "position" and "tangent" control points. Similar to tangent matching, the constrained fitting in the curvature matching can be solved by using quadratic programming solvers.
  • the above process for curvature matching employs an innovative approach to fitting normal curvature values to achieve curvature continuity, particularly the way the least squares term is defined to measure curvature discontinuity.
  • the disclosed process does not require any estimation of second order partial derivatives.
  • Most known systems solve curvature matching by matching iso-parametric curves to neighboring surfaces with curvature continuity (estimation step) , and fitting the surface second order partial derivatives from the end second order derivatives of the matched iso-parametric curves.
  • the position matching step can be replaced by well-known Coons surface method (for surface by four boundaries), skinning method (for loft surface), or Gordon surface method (for bi-directional loft) .
  • the input curves may need to be reparemetrized, if necessary or desired.
  • These methods generate base surfaces that are position continuous to specified tolerance with the neighboring surfaces.
  • the process only needs to sample points on the neighboring surface boundaries and assign their parameterizations by the closest point method or the described length matching method. After the sampling is done, tangent and curvature matching algorithms can be applied as described above.
  • Figure 5 shows a matching surface to be matched to four neighboring surface.
  • the matching surface is a bicubic Bezier surface.
  • the surfaces have perfect position continuity, and the tangent discontinuity is shown (in degrees) along the four surface edges.
  • Figure 6 shows the output surface after tangent matching. This surface is actually produced by using the bi-directional loft function.
  • the only difference from result in Figure 5 is the tangent matching step.
  • the tangent continuous matching surface is also a bicubic Bezier surface.
  • the tangent tolerance is 0.05 degree. Note that the tangent matching algorithm is able to establish tangent continuity without inserting any knots.
  • Figure 7 shows a result of matching the middle surface to its four neighboring surfaces with tangent continuity. Note that the matching surface is not curvature continuous to its neighboring surfaces, as shown by "mismatches" of section curvature needles.
  • Figure 8 shows the result of curvature matching of the same surface. Note that the section curvature needles match along the surface edges, indicating that the surfaces are curvature continuous to each other. For this case, the differences of curvature values are within 1%.
  • the preferred embodiment provides a new process and algorithm for multiple-sided surface matching with curvature continuity.
  • Particular innovative aspects of the preferred processes include:
  • machine usable mediums include: nonvolatile, hard-coded type mediums such as read only memories (ROMs) or erasable, electrically programmable read only memories (EEPROMs), user-recordable type mediums such as floppy disks, hard disk drives and compact disk read only memories (CD-ROMs) or digital versatile disks (DVDs) , and transmission type mediums such as digital and analog communication links.
  • ROMs read only memories
  • EEPROMs electrically programmable read only memories
  • user-recordable type mediums such as floppy disks, hard disk drives and compact disk read only memories (CD-ROMs) or digital versatile disks (DVDs)
  • transmission type mediums such as digital and analog communication links.

Landscapes

  • Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • General Physics & Mathematics (AREA)
  • Pure & Applied Mathematics (AREA)
  • Mathematical Optimization (AREA)
  • Mathematical Physics (AREA)
  • Mathematical Analysis (AREA)
  • Algebra (AREA)
  • Computer Graphics (AREA)
  • Geometry (AREA)
  • Software Systems (AREA)
  • Theoretical Computer Science (AREA)
  • Processing Or Creating Images (AREA)
  • Image Generation (AREA)
  • Prostheses (AREA)

Abstract

A system and method for providing multiple-sided surface matching with curvature continuity in a graphical modeling system. The preferred embodiments use improved algorithms for tangential surface matching and curvature surface matching, which provide improved continuity between a given surface and one or more neighboring surfaces.

Description

SYSTEM AND METHOD FOR MULTIPLE-SIDED SURFACE MATCHING WITH CURVATURE CONTINUITY
TECHNICAL FIELD OF THE INVENTION
[0001] The present invention is directed, in general, to surface generation and modification in graphical modeling systems .
BACKGROUND OF THE INVENTION
[0002] Curve-based surface generation schemes are very commonly used in computer-aided geometric design (CAGD) for modeling of free-form geometry. One of the simplest among them is skinning, which interpolates a series of curves to form a surface. Also known as lofting, this method stretches a surface over a series of "ribs" or cross- sections. The surface will preserve the shape of each cross-section at the corresponding location, and will blend smoothly from one to the next. Figure 1A illustrates the skinning technique; note the "ribs", or cross-sections, shown on the left; these are used as a basis for the generated "skin" surface shown on the right.
[0003] More complicated methods interpolating a network of curves, as shown in Figure IB and Figure 1C, are also well-developed, and readily available in modern geometric modeling systems. These methods provide very intuitive and powerful surface generation tools for designers, since the output surface exactly follows the "idea" implied by the input curves, and it is generally much easier to create curves than surfaces from scratch. In Figures IB and 1C, the basic curve network shown on the left is interpolated to form the surface on the right.
[0004] In a general sense, "continuity" describes how two things come together. Two types of continuity are generally discussed: Cn and Gn, where n refers to the order of continuity. Cn continuity refers to "parametric" continuity of the nth order. This means that the magnitude and direction of all derivatives up to the nth order must agree. Gn continuity, on the other hand, refers to "geometric" continuity of the nth order, which loosens the above definition by allowing re-parametrizations from the Cn condition. A good example is the 1st order continuity: C1 between two curves requires the two end tangent vectors to be identical, while the G1 condition only requires the two end tangent vectors to be parallel. Although not as strict as parametric continuity, geometric continuity is more useful in CAGD in that it describes the sufficient conditions for two connecting entities to "look" smooth in the neighborhood of their juncture.
[0005] Class-A (or sometimes referred to as class-1) geometric design typically sets stringent standards in surface quality in order to satisfy aesthetic or aerodynamic goals. In most cases, class-A surfaces are required to be curvature continuous while providing the simplest mathematical representation needed for the desired shape/form, and do not have any undesirable waviness.
[0006] Free-form surface modeling of a car body, for instance, requires the generated surface patches to maintain good continuity (G2 or higher) with each other, as well as to demonstrate visual smoothness everywhere when observed in lights. In addition, there are always requirements that the surfaces should be "simple" and "clean," that is, having a reasonable number of control points, a good structure of the control net, and nice parametric flows. Heavier than necessary surface geometry may pose problems to subsequent processes, such as further geometry generation based on the existing ones, mesh generation and NC tool path generation. Unclean or "messy" control net structure is undesirable for designers, who rely heavily on control point editing to modify geometric entities. All of the above quality can be examined using various diagnostic tools in modern geometric modeling systems. Some of these are demonstrated in Figure 2.
[0007] Figures 2A-2D show various surface quality diagnostic tools applied on a model that comprises six surfaces. Figure 2A shows a continuity needles plot, which displays the discontinuity measurement between surfaces by needles (vectors) of various lengths and directions. Figure 2B shows a surface contours plot, where each contour represents constant reflectance of light on the surfaces. The smoothness of the contours reflects the smoothness of the surfaces. Figure 2C shows a curvature needles plot, which displays the distribution of cross-sectional curvatures on the surfaces. This plot illustrates the curvature "flow" on the surfaces, and is a good indicator for different types of shape imperfections. Figure 2D shows a zebra plot, which imitates the shape inspection environment in the automotive body design process where the model is inspected in a room with arrays of lights. Similar to the contours plot, it provides a clear visualization of the smoothness of the surfaces. [0008] In freeform surface modeling, complex shapes are generally modeled by using several NURBS surfaces. To satisfy smoothness and aesthetic requirements, particularly in Class A surface modeling, it is very important that the generated surfaces are curvature continuous to each other. In curve-based surface generation, some selected curves can be boundaries of existing surfaces (edges or trim curves), and the generated surface is very often expected to be automatically tangent or curvature continuous to the existing ones. Alternatively the user can create and edit the shape of each surface manually and then "match" them together to establish continuity among them.
[0009] Most CAD systems have capabilities to match two surfaces along their common boundaries with curvature continuity. However, it is very difficult and time consuming to establish continuity among several surfaces using one-side matching capabilities alone. Usually matching along one surface boundary will destroy the continuity along one or two adjacent boundaries. Designers need to edit the control points and apply matching repeatedly.
[0010] There is, therefore, a need in the art for a system, process and computer program product for a computer-implemented method which allows a designer to simultaneously match a NURBS surface along its four edges to neighboring surfaces with curvature continuity, subjected to geometric constraints. SUMMARY OF THE INVENTION
[0011] To address the above-discussed deficiencies of the prior art, it is an object of the present invention to provide improved surface generation and modification in graphical modeling systems. Other objects of the various embodiments will be noted herein or understood by those of skill in the art.
[0012] The preferred embodiment provides a system and method for providing multiple-sided surface matching with curvature continuity in a graphical modeling system. The preferred embodiments use improved algorithms for tangential surface matching and curvature surface matching, which provide improved continuity between a given surface and one or more neighboring surfaces.
[0013] The foregoing has outlined rather broadly the features and technical advantages of the present invention so that those skilled in the art may better understand the detailed description of the invention that follows . Additional features and advantages of the invention will be described hereinafter that form the subject of the claims of the invention. Those skilled in the art will appreciate that they may readily use the conception and the specific embodiment disclosed as a basis for modifying or designing other structures for carrying out the same purposes of the present invention. Those skilled in the art will also realize that such equivalent constructions do not depart from the spirit and scope of the invention in its broadest form. [0014] Before undertaking the DETAILED DESCRIPTION OF THE INVENTION below, it may be advantageous to set forth definitions of certain words or phrases used throughout this patent document: the terms "include" and "comprise," as well as derivatives thereof, mean inclusion without limitation; the term "or" is inclusive, meaning and/or; the phrases "associated with" and "associated therewith, " as well as derivatives thereof, may mean to include, be included within, interconnect with, contain, be contained within, connect to or with, couple to or with, be communicable with, cooperate with, interleave, juxtapose, be proximate to, be bound to or with, have, have a property of, or the like; and the term "controller" means any device, system or part thereof that controls at least one operation, whether such a device is implemented in hardware, firmware, software or some combination of at least two of the same. It should be noted that the functionality associated with any particular controller may be centralized or distributed, whether locally or remotely. Definitions for certain words and phrases are provided throughout this patent document, and those of ordinary skill in the art will understand that such definitions apply in many, if not most, instances to prior as well as future uses of such defined words and phrases.
BRIEF DESCRIPTION OF THE DRAWINGS
[0015] For a more complete understanding of the present invention, and the advantages thereof, reference is now made to the following descriptions taken in conjunction with the accompanying drawings, wherein like numbers designate like objects, and in which:
[0016] Figures 1A-1C depict common curve-based surface generation methods. Figure 1A illustrates interpolating a series of curves to form a surface; Figure IB illustrates interpolating a 2 by N network of curves to form a surface; and Figure IC illustrates interpolating an M by N network of curves to form a surface.
[0017] Figures 2A-2D show various surface quality diagnostic tools. Figure 2A shows a continuity needles plot; Figure 2B shows a surface contours plot; Figure 2C shows a curvature needles plot; and Figure 2D shows a zebra plot.
[0018] Figure 3 depicts a process in accordance with a preferred embodiment of the present invention;
[0019] Figure 4 depicts a process in accordance with a preferred embodiment of the present invention;
[0020] Figure 5 shows a matching surface to be matched to four neighboring surface;
[0021] Figure 6 shows an output surface after tangential matching; [0022] Figure 7 shows a figure which is a result of matching the middle surface to its four neighboring surfaces with tangent continuity; and
[0023] Figure 8 shows a figure that is the result of curvature matching of a surface.
DETAILED DESCRIPTION OF THE INVENTION
[0024] FIGURES 1 through 8, discussed herein, and the various embodiments used to describe the principles of the present invention in this patent document are by way of illustration only and should not be construed in any way to limit the scope of the invention. Those skilled in the art will understand that the principles of the present invention may be implemented in any suitably arranged device. The numerous innovative teachings of the present application will be described with particular reference to the presently preferred embodiment.
[0025] The preferred embodiment provides a system and method for providing multiple-sided surface matching with curvature continuity in a graphical modeling system. The preferred embodiments use improved algorithms for tangential surface matching and curvature surface matching, which provide improved continuity between a given surface and one or more neighboring surfaces.
[0026] Definitions -- Following are short definitions of the usual meanings of some of the technical terms which are used in the present application. (However, those of ordinary skill will recognize whether the context requires a different meaning.) Additional definitions can be found in the standard technical dictionaries and journals:
[0027] B-spline curve -- A free-form, parametrically defined curve in which each vertex has an influence over a defined range of the curve. [0028] B-spline surface -- A free-form, parametrically defined surface in which each vertex has an influence over a defined range of the surface.
[0029] Iso-parametric curve — A B-spline curve obtained by tracing on a B-spline surface along the entire domain of one of the (u, v) parameters, with the value of the other parameter kept constant. Isoparametric curves obtained by keeping the u parameter constant are referred to as "constant-u iso-parametric curves". Similarly for "constant- v iso-parametric curves".
[0030] Knots -- A knot is a number which is part of the definition of a B-spline curve or surface and which is used to control its shape. The collection of all knots used by a B-spline is called a knot vector. The distance between knot values is called its knot spacing. Uniform B- splines set the knot spacing to 1 (usually) , and non-uniform rational B-splines allow for uneven spacing of the knots.
[0031] Compatible in the B-spline sense — A set of B- spline curves are said to be compatible in the B- spline sense if they have the same degree and knot sequence.
[0032] The disclosed multiple-sided matching is used in several curve based surface generation schemes, where the generated surfaces are required to be tangent or curvature continuous with neighboring surfaces. The surface construction schemes consist of two basic steps: [0033] (1) Create an initial base surface that interpolates the input curves, including the neighboring boundaries, using well-known methods such as skinning, Coon surface, or Gordon surface. The input curves can be reparameterized if necessary or desired.
[0034] (2) Use the claimed matching algorithm to match the base surface to the neighboring surfaces with required tangent or curvature continuity. The preferred embodiment provides a process for simultaneously matching a NURBS surface to neighboring surfaces with curvature continuity, subjected to geometric constraints. The input to one embodiment of the process algorithm is
[0035] ■ One to four boundaries of neighboring surfaces. Each boundary can be a surface edge or a trimmed boundary curve on surface.
[0036] ■ The matching untrimmed NURBS surface.
[0037] • A specification of which edge of the matching surface is to be matched to which neighboring boundary curve.
[0038] • The desired continuity level along each matching edge: none, position, tangent, or curvature .
[0039] • Tolerances for position, tangent, and curvature continuity
[0040] The process assumes that the input neighboring surfaces have the required continuity at the coincident end points of the corresponding boundary curves. The matching algorithm consists of three basic steps: position, tangent, and curvature matching. Each step is basically an iterative constrained fitting process.
Position Matching
[0041] Position matching amounts to constrained curve fitting to position tolerance. Position matching can be summarized as follows:
[0042] 1. Compute sample match points on neighboring boundaries and assign their parameterizations .
[0043] 1.1 For each matching edge, extract the corresponding iso-parametric curve of the matching edge.
[0044] 1.2 Sample the corresponding neighboring boundary into points based on certain criteria (e.g. chordal deviation, curve tangent deviation, or maximum distance between two sample points). For each sample point, compute the relative length from that point to the starting point of the neighboring boundary curve.
[0045] 1.3 For each sample point, assign the parameter value by solving the inverse problem, computing the parameter value of the extracted iso-parametric curve that has the specified relative length.
[0046] 2. Simultaneously fit two curves to the sample points on opposite neighboring boundaries, subjected to the following constraints:
[0047] • fixed end points,
[0048] • fixed end tangent directions, [0049] • Fixed end curvature values.
[0050] 3. Check whether all the fit curves are within required position tolerance. If yes, make the matching surface "knot compatible" to the fit curves, and exit.
[0051] 4. Insert knots to the fit curve appropriately based on deviation analysis.
[0052] 5. Repeat step 2 to 4 again until all the fit curves are within position tolerance.
[0053] 6. Replace the boundary control points of the matching surface with the control points of the four fit curves .
[0054] Note that the weights of the matching NURBS surface are fixed. After the first step, the matching surface is positional continuous to its neighboring surfaces. It is also possible to modify the interior control points of the matching surface, based on the changes in the boundary control points, so that the matching surface have better control point structure and surface flow. The subsequent tangent and curvature matching step will make use of the sample match points on neighboring surfaces and their assigned (u,v) parameter values of the matching surfaces.
[0055] There are existing published methods for constrained curve fitting. The preferred embodiment uses a global surface fitting method as described in U.S. patent application 10/318,344, filed 12 December 2002, and herein incorporated by reference, to fit four curves simultaneously.
Tangent Matching
[0056] The preferred embodiment ensures tangent matching by iterative constrained energy-minimizing least squares fitting according to a process as shown in Figure 3.
[0057] First, the system or a user will designate the matching surface and at least one neighboring surface (step
310) . Next, the system will compute the unit surface normals of the neighboring surfaces at sample match points
(step 315) .
[0058] The system will next set up the desired linear constraints on the matching surface control points (step 320) . The system will then perform a constrained least squares fit for the first order cross partial derivatives along all relevant boundaries of the matching surface, where the least squares term measures the tangent discontinuity (step 325) .
[0059] The system will then check the tangent continuity along required surface edges (step 330) . If the tangent continuity is within the specified tolerance, then stop (step 345) .
[0060] If the tangent continuity is not within tolerance, the system will analyze the tangent discontinuity (step 335) and insert knots to the matching surface at appropriate locations (step 340) . [0061] Repeat steps 325 to 340 again until the tangent continuity is achieved (returning to step 325) .
[0062] In this process, the positional continuous matching surface is defined as
X(u,v) = ΣΣwι:)P1D N1(u)N-) (v) / ΣΣw1DNx (u)N: (v)
where u, v are the parameter of the surface and Ni (u) , Nj (v) are the B-spline basis functions along u and v parametric directions respectively. The constrained least squares fitting in step 3 tries to minimize the following quadratic objective function:
E = i EGι + W2E2+ W3 E3 + W4 Eblend + w5 Esprιng
Where
EGι = ∑ (αk | | Xu (uk,vk) • nk | | 2 + βk | |xv (uk ,vk) • nk 1 1 2 ) ,
[0063] αk and βk can be either 0.0 or 1.0, αk + βk = 1.0, and nk denotes unit surface normal at sample point k. If the sample point k lies on a surface edge which is a constant-u (constant-v) iso-parametric curve, then α = 1 and β = 0.0 (αk = 0.0 and βk = 1.0, respectively) . For perfect tangent continuity, the first order cross partial derivatives at all points along the matching edges must be perpendicular to the surface normals at their corresponding points on the neighboring surfaces. EGι is defined to be the least squares term that measures tangent discontinuity.
l Xuu l l2 dudv+2 [f| | Xuv | |2 dudv+jJΪ I Xvv 1 12 dudv | Xuuu l l2 dudv+3lj| l Xuuv 1 12 dudv+3JJ| | Xuvv 1 12 dudv+IJN Xvvv 1 12 dudv Eblend = JJ I IXuuvv I dudv
[0064] E2, E3, and Ebιend are energy terms for controlling the smoothness of the resulting matching surface. Without these energy terms, the control points can be jumpy, particularly when the matching surface has high orders.
[0065] Esprιng = ΣΣ IIIi i-j-Cj.. I111 , where c13 denotes the coordinates of the control point P1D from the previous step. Egpring is the spring energy term that tries to restrict the movements of the control points, so that the shape of the matching surface is close to the original surface.
[0066] Many linear constraints can be specified. Some interior control points can be fixed, if desired, during tangent matching, and the fixing can be specified as constraints. If curvature continuity is required at any surface corner, a twist constraint must be defined. Twist constraints are derived from the classical three-tangent theorem, which states that two surface points are curvature continuous if and only if
both surfaces are tangent continuous, and
• their normal curvature values match along three different tangent directions.
More specifically, the twist constraint at a surface corner X(u,v)is defined as
kn (u, v, du, dv) = (L (du) 2+2Mdudv+N (dv) 2) / (E (du) 2+2Fdudv+G (dv) 2) where kn denotes the normal curvature value of the neighboring surfaces along a direction Xu du + Xv dv, with du and dv being arbitrary nonzero values, and
= Xuu(u,v) • N (u,v)
M = Xuv(u,v) • N (u,v)
N = Xvv(u,v) • N (u,v)
E = Xu(u,v) • Xu(u,v)
F = Xu(u,v) • Xv(u,v)
G = Xv (u, v) • Xv(u,v)
N(u,v) refers to the surface normal at the corner. Note that, after position matching, L,N,E,F,G are constant values since the partial derivatives Xu, Xv, XUu, Xw are defined by the boundary control points. Therefore, it follows that the twist constraint is a linear equation.
[0067] The fitting variables are the coordinates of the matching surface's control points, except the boundary control points. With the boundary control points fixed, the least squares term EGι becomes a quadratic function of the coordinates of the remaining control points. The smoothness and spring energy terms are also quadratic functions. The formulated constrained energy minimizing least squares fitting can be solved by standard quadratic programming solvers, which are linear methods. It can be shown that the objective function is also positive definite, and the solution of the least squares fitting is unique.
[0068] The constrained least squares fitting formulation is not new. For example, it was described in a paper titled "Variational Surface Modeling" by William Welch and Andrew Witkin, Proceeding of SIGGRAPH92, Computer Graphics, volume 26, number 2, pages 157-166, 1992, which is hereby incorporated by reference.
[0069] One particularly unique aspect of the preferred process is the way the least squares term EG1 is defined, together with other energy terms, for fitting cross partial derivatives to achieve tangent continuity. One major advantage of this approach is that it does not require any estimation of first order cross partial derivatives along the surface boundaries.
[0070] This is in contrast to conventional methods, which do tangent matching by
[0071] 1. extracting iso-parametric curves of the matching surface,
[0072] 2. matching these curves to the neighboring surfaces by either projection or rotation,
[0073] 3. fitting the surface cross partial derivatives from the end tangents of the matched isoparametric curves.
[0074] There are two problems with this approach:
[0075] First, the cross derivatives estimations along two surface edges may not be "compatible" or consistent at the corner. Second, the estimations may fail or become ambiguous if the end tangents of iso-parametric curves before matching are nearly parallel to surface normals of the neighboring surfaces.
[0076] The preferred embodiment avoids these two problems completely, as the fitting can be considered as implicit fitting where the least squares term is defined from the necessary and sufficient conditions for tangent continuity. The least squares term interacts with the smoothness and spring energy terms to produce smooth tangent continuous matching surface with good control point structure .
[0077] Further, the preferred embodiment exploits the role of Eiend in producing nice control point structure.
Curvature Matching
[0078] The curvature matching process of the preferred embodiment adjusts the tangent continuous matching surface to achieve curvature continuity by iteratively fitting the second order partial derivatives along surface boundary edges, according to a process as shown in Figure 4. Steps in curvature matching are similar to those in tangent matching. The main difference is the way the least squares term is defined to measure curvature discontinuity, as further described below.
[0079] First, the system or a user will designate the matching surface and at least one neighboring surface (step 410) . Next, the system will compute the normal curvatures at sample points on the neighboring surfaces along one or more tangent directions (step 415) , where the selected directions cannot be parallel to the tangent directions of the boundary curves. The preferred embodiment uses three directions, where one of the directions must be the direction of the first order partial cross derivative of the matching surface (the end tangent directions of the iso-parametric curves). [0080] The system will then set up the desired linear constraints on the matching surface control points (step 420) . The system will then perform a constrained least squares fit for the second order partial derivatives along all required boundaries of the matching surface, where the least squares term measures the curvature discontinuity (step 425) .
[0081] The system will check the curvature continuity along required surface edges (step 430) . If the curvature continuity is within the specified tolerance, then stop (step 445) .
[0082] If the curvature continuity is not within tolerance, the system will analyze the curvature discontinuity (step 435) and insert knots to the matching surface at appropriate locations (step 440) .
[0083] Repeat steps 425 to 440 again until the curvature continuity is achieved (returning to step 4-25) .
[0084] The classical three-tangent theorem states that two surface points are curvature continuous if and only if they are tangent continuous and their normal curvature values along three different tangent directions match. The linkage curve theorem states that two surfaces tangent continuous along a smooth curve are curvature continuous if and only if, at every point along the curve, their normal curvature agrees along a direction other than the tangent to the curve. The least squares term Eg2 is defined to measure the curvature discontinuity as
Eg2 = ∑ (αk | |kn (uk, vk, 1 , 0 ) -ak | |2 +βk | |kn ( uk, vk, 0 , 1 ) -bk 1 12 + l lkn (u , vk, dukl, dvkl ) -c l2 + | |kn (uk, vk, duk2, dvk2 ) -dk 1 12 ) where kn denotes the normal curvature
kn(u, v,du,dv)= (L(du)2+2Mdudv+N(dv)2) / (E (du) 2+2Fdudv+G (dv) 2)
L = Xuu(u,v) • N (u,v) = Xuv(u,v) • N (u,v)
N = Xvv(u,v) • N (u,v)
E = Xu(u,v) • Xu(u,v)
F = Xu(u,v) • Xv(u,v)
G = Xv(u,v) • Xv(u,v)
[0085] Further, ak, bk, ck, dk are the normal curvature values of the neighboring surfaces at each sample point along four tangent directions (from one up to three of which will be used) . More specifically, if the sample point lies on a surface edge which is a constant-u (constant-v) iso-parametric curve, then αk = 1.0 and βk = 0 (α =0.0 and βk
= 1.0, respectively) . The other two tangent directions Xu dun + Xvdvkι and Xu duk2 + Xvdvk2 are optional and can be chosen arbitrarily. In the implementation, we use three directions, where we set duι = 0.5, dvkι = 0.5, and duk2 = 0.5, dvk2 = -0.5. The fitting variables in curvature matching are the coordinates of the control points, except for those control points defining position and tangent continuity. Note that Eg2 is a quadratic function, after fixing these "position" and "tangent" control points. Similar to tangent matching, the constrained fitting in the curvature matching can be solved by using quadratic programming solvers.
[0086] The above process for curvature matching employs an innovative approach to fitting normal curvature values to achieve curvature continuity, particularly the way the least squares term is defined to measure curvature discontinuity.
[0087] The disclosed process does not require any estimation of second order partial derivatives. Most known systems solve curvature matching by matching iso-parametric curves to neighboring surfaces with curvature continuity (estimation step) , and fitting the surface second order partial derivatives from the end second order derivatives of the matched iso-parametric curves.
Curve based surface construction
[0088] For curve based surface construction schemes, the position matching step can be replaced by well-known Coons surface method (for surface by four boundaries), skinning method (for loft surface), or Gordon surface method (for bi-directional loft) . The input curves may need to be reparemetrized, if necessary or desired. These methods generate base surfaces that are position continuous to specified tolerance with the neighboring surfaces. To match the base surface with tangent and curvature continuity, the process only needs to sample points on the neighboring surface boundaries and assign their parameterizations by the closest point method or the described length matching method. After the sampling is done, tangent and curvature matching algorithms can be applied as described above.
[0089] Figure 5 shows a matching surface to be matched to four neighboring surface. The matching surface is a bicubic Bezier surface. The surfaces have perfect position continuity, and the tangent discontinuity is shown (in degrees) along the four surface edges. [0090] Figure 6 shows the output surface after tangent matching. This surface is actually produced by using the bi-directional loft function. The only difference from result in Figure 5 is the tangent matching step. The tangent continuous matching surface is also a bicubic Bezier surface. The tangent tolerance is 0.05 degree. Note that the tangent matching algorithm is able to establish tangent continuity without inserting any knots.
[0091] Figure 7 shows a result of matching the middle surface to its four neighboring surfaces with tangent continuity. Note that the matching surface is not curvature continuous to its neighboring surfaces, as shown by "mismatches" of section curvature needles. Figure 8 shows the result of curvature matching of the same surface. Note that the section curvature needles match along the surface edges, indicating that the surfaces are curvature continuous to each other. For this case, the differences of curvature values are within 1%.
[0092] The preferred embodiment provides a new process and algorithm for multiple-sided surface matching with curvature continuity. Particular innovative aspects of the preferred processes include:
[0093] The global constrained energy minimizing least squares implicit fitting of the first order surface cross derivatives to achieve tangent continuity;
[0094] The global constrained energy minimizing least squares fitting of normal curvature values to achieve curvature continuity; and [0100] The multiple-sided matching algorithm has been developed and used in several curve based surface construction tools, including loft, bi-directional loft, and surface-by-boundary tools.
[0101] The multiple-sided matching algorithms disclosed herein can be extended to solve other problems:
[0102] ■ Matching surface corners together with curvature continuity
[0103] • Filling n-sided holes with n untrim ed surfaces
[0104] • Global matching of a set of surfaces to form a smooth curvature continuous sheet body
[0105] One important step in solving these problems is to determine the desired boundaries between surfaces and the unit surface normals, principal curvature values, and principal curvature directions along these boundaries, using known' techniques. Once such information is derived, the claimed tangent and curvature matching algorithms are applied.
[0106] The terms and conventional techniques used herein will be familiar to those of skill in the art of computer- aided geometric design. Further reference may be made to U.S. Patent 5636338 to Moreton; Celniker, G. and Gossard, D. "Deformable curve and surface finite-elements for free- form shape design", Computer Graphics, 25, 257-266, 1991; Celniker, G. and Will Welch, "Linear constraints for deformable B-spline surfaces", Proceedings of the Symposium on Interactive 3D Graphics, 25(2), 165-170, 1992; Farin, G. and Hansford, D., "Discrete Coons patches", Computer Aided Geometric Design 16, 691-700, 1999; Park, H. et al., "A method for approximate NURBS curve compatibility based on multiple curve refitting", Computer Aided Design, 32, 237- 252, 2000; Pegna, J. and Wolter F., "Geometric Criteria to Guarantee Curvature Continuity of Blend Surfaces", J. of Mechanical Design, 114, 201-210, 1992; Peters, J. "Geometric Continuity", in Handbook on Computer Aided Geometric Design, Elsevier, 2002; and Szilvasi-Nagy, M. "Almost Curvature Continuous Fitting of B-Spline Surfaces", J. for Geometry and Graphics, Vol. 2, No. 1, 33-43, 1999, all of which are hereby incorporated by reference.
[0107] Those skilled in the art will recognize that, for simplicity and clarity, the full structure and operation of all data processing systems suitable for use with the present invention is not being depicted or described herein. Instead, only so much of a data processing system as is unique to the present invention or necessary for an understanding of the present invention is depicted and described. The remainder of the construction and operation of a data processing system may conform to any of the various current implementations and practices known in the art.
[0108] It is important to note that while the present invention has been described in the context of a fully functional system, those skilled in the art will appreciate that at least portions of the mechanism of the present invention are capable of being distributed in the form of a instructions contained within a machine usable medium in any of a variety of forms, and that the present invention applies equally regardless of the particular type of instruction or signal bearing medium utilized to actually carry out the distribution. Examples of machine usable mediums include: nonvolatile, hard-coded type mediums such as read only memories (ROMs) or erasable, electrically programmable read only memories (EEPROMs), user-recordable type mediums such as floppy disks, hard disk drives and compact disk read only memories (CD-ROMs) or digital versatile disks (DVDs) , and transmission type mediums such as digital and analog communication links.
[0109] Although an exemplary embodiment of the present invention has been described in detail, those skilled in the art will understand that various changes, substitutions, variations, and improvements of the invention disclosed herein may be made without departing from the spirit and scope of the invention in its broadest form.
[0110] None of the description in the present application should be read as implying that any particular element, step, or function is an essential element which must be included in the claim scope: THE SCOPE OF PATENTED SUBJECT MATTER IS DEFINED ONLY BY THE ALLOWED CLAIMS. Moreover, none of these claims are intended to invoke paragraph six of 35 USC §112 unless the exact words "means for" are followed by a participle.

Claims

WHAT IS CLAIMED IS:
1. A method for performing tangential surface matching, comprising: designating a first surface and at least one neighboring surface, the first surface and the neighboring surface having a common boundary; determining unit normals of points on the boundaries of the neighboring surfaces that correspond to points on the boundaries of the first surface; performing a least-squares fit along boundaries of the first surface, wherein the least-squares term measures the tangent discontinuity.
2. The method of claim 1, wherein the least-squares fit is performed using the equations
X(u,v) = ΣΣ 13Pl:) N^uJN-^v) / ΣΣw^N^uON^v) , E = Wj EG1 + 2E2+ 3 E3 + w4 Eblend + 5 Esprlng, EG1 = Σ(α I Xu(uk,vk) • nk|| 2 + β |xv(uk ,vk) • nk 11 2 ) , I IXuu 112 dudv+2 jj I lxuv 112 dudv+JJ I |xuv 112 dudv, I IXuuu 112 dudv+3|| I |xuuv 112 dudv+3jl I |xuvv 112 dudv+JJ I |xvvv 112 dudv,
Eblend =fl I IXuuvv I Y dudv where u, v are the parameter of the first surface,
Ni (u) , Nj (v) are B-spline basis functions, αk and βk are either 0.0 or 1.0, αk + βk = 1.0, nk denotes unit surface normal at sample point k, EG1 is the least squares term that measures tangent discontinuity, E2, E3, and Eblend are energy terms for controlling the smoothness of the resulting matching surface, and c1D denotes the coordinates of a control point P∑3.
3. The method of claim 1, further comprising verifying that the tangent continuity on the boundary of the first surface and the neighboring surface is within a specified tolerance.
4. The method of claim 1, further comprising adding knots to the boundaries of the first surface.
5. A method for performing curvature matching, comprising : designating a first surface and at least one neighboring surface, the first surface and the neighboring surface having a common boundary; determining normal curvatures at points on the boundaries of the neighboring surfaces that correspond to points on the boundaries of the first surface; performing a least-squares fit along boundaries of the first surface, wherein the least-squares term measures the curvature discontinuity.
6. The method of claim 5, wherein the least-squares fit is performed using the equations
E 2 = ∑(αk||kn(uk, vk, 1, 0) -a l2 +βk||kn(uk, vk, 0, 1) -bk 112 + llkn(uk, vk,dukι,dvkι) -ck||2 + I |kn (uk, vk, duk2, dvk2) -dk 112 ) , kn(u,v,du,dv)= (L(du)2+2Mdudv+N(dv) 2) / (E (du) 2+2Fdudv+G (dv) ) ,
L = Xuu(u, v) • N (u, v) ,
M = Xuv(u,v) • N (u,v) ,
N = Xvv(u,v) • N (u,v) ,
E = Xu(u, v) • Xu(u, v) ,
F = Xu(u,v) • Xv(u, v) ,
G = χv(u, v) • Xv(u, v) ,
where kn denotes the normal curvature, ak, bk, ck, dk are the normal curvature values of the neighboring surface at each corresponding point, u, v are the parameter of the first surface, αk and βk are either 0.0 or 1.0 and αk + βk = 1.0.
7. The method of claim 5, further comprising verifying that the curvature continuity on the boundary of the first surface and the neighboring surface is within a specified tolerance.
8. The method of claim 5, further comprising adding knots to the boundaries of the first surface.
EP04750337A 2003-04-21 2004-04-20 System and method for multiple-sided surface matching with curvature continuity Ceased EP1620821A2 (en)

Applications Claiming Priority (2)

Application Number Priority Date Filing Date Title
US42053903A 2003-04-21 2003-04-21
PCT/US2004/012054 WO2004095322A2 (en) 2003-04-21 2004-04-20 System and method for multiple-sided surface matching with curvature continuity

Publications (1)

Publication Number Publication Date
EP1620821A2 true EP1620821A2 (en) 2006-02-01

Family

ID=33309566

Family Applications (1)

Application Number Title Priority Date Filing Date
EP04750337A Ceased EP1620821A2 (en) 2003-04-21 2004-04-20 System and method for multiple-sided surface matching with curvature continuity

Country Status (4)

Country Link
EP (1) EP1620821A2 (en)
JP (1) JP4856538B2 (en)
KR (1) KR101095126B1 (en)
WO (1) WO2004095322A2 (en)

Families Citing this family (8)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
WO2007091195A1 (en) * 2006-02-10 2007-08-16 Philips Intellectual Property & Standards Gmbh Supervision of an illumination device
DE102006010610A1 (en) * 2006-03-06 2007-09-13 Fraunhofer-Gesellschaft zur Förderung der angewandten Forschung e.V. Imprinting watermarks into a model of a line-like or area-like shape
KR100785339B1 (en) * 2006-11-16 2007-12-17 중앙대학교 산학협력단 Shape Recognition Device and Method
FR2982831A1 (en) * 2011-11-22 2013-05-24 Peugeot Citroen Automobiles Sa Method for checking level of hyperstatism of assembly of three parts of vehicle i.e. car, involves analyzing set of discontinuous surfaces based on set of theoretical and conceptual values for set of parts that is to be assembled
FR3008507B1 (en) * 2013-07-09 2017-04-14 Snecma METHOD FOR MODELING A NON-AXISYMETRIC SURFACE
FR3015552B1 (en) * 2013-12-19 2018-12-07 Safran Aircraft Engines TURBOMACHINE PIECE WITH NON-AXISYMETRIC SURFACE
JP6727932B2 (en) * 2016-06-01 2020-07-22 日本ユニシス株式会社 Curved surface generator by point cloud surface fitting and curved surface generation program
CN107350667B (en) * 2017-06-23 2019-08-20 华南理工大学 A kind of emulation mode of the automobile door plate welding flexible production line based on digital factory

Family Cites Families (2)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
FR2769391B1 (en) * 1997-10-08 2000-11-10 Matra Datavision SHAPE MODIFICATION METHOD FOR COMPUTER-AIDED DESIGN SYSTEM
US6253164B1 (en) * 1997-12-24 2001-06-26 Silicon Graphics, Inc. Curves and surfaces modeling based on a cloud of points

Non-Patent Citations (1)

* Cited by examiner, † Cited by third party
Title
See references of WO2004095322A3 *

Also Published As

Publication number Publication date
WO2004095322A3 (en) 2005-03-24
KR20060010754A (en) 2006-02-02
JP2006524398A (en) 2006-10-26
WO2004095322A2 (en) 2004-11-04
KR101095126B1 (en) 2011-12-16
JP4856538B2 (en) 2012-01-18

Similar Documents

Publication Publication Date Title
CN113228115B (en) Method, system and medium for converting mesh geometry to watertight boundary representation
US6639592B1 (en) Curve network modeling
Kiss et al. Adaptive CAD model (re-) construction with THB-splines
US6253164B1 (en) Curves and surfaces modeling based on a cloud of points
Tai et al. The processing of data points basing on design intent in reverse engineering
US20120221297A1 (en) Global Deformation for a Modeled Object
Vukašinović et al. Advanced CAD modeling
Cheutet et al. 3D sketching for aesthetic design using fully free-form deformation features
US20040113910A1 (en) System and method for the rebuild of curve networks in curve-based surface generation using constrained-surface fitting
WO2004095322A2 (en) System and method for multiple-sided surface matching with curvature continuity
Wang et al. EQSM: An efficient high quality surface grid generation method based on remeshing
US20040174362A1 (en) Deformable healer for removing CAD database connectivity gaps from free-form curves and surfaces
Subag et al. Piecewise developable surface approximation of general NURBS surfaces, with global error bounds
Lee Automatic metric 3D surface mesh generation using subdivision surface geometrical model. Part 1: Construction of underlying geometrical model
Dorado Medial axis of a planar region by offset self-intersections
Sussner et al. Interactive examination of surface quality on car bodies
Pakdel et al. Incremental subdivision for triangle meshes
Renka Shape-preserving interpolation by fair discrete G3 space curves
Bentamy et al. Cross-sectional design with curvature constraints
Vukašinović et al. Introduction to freeform surface modelling
Muftejev et al. Multi-criteria assessment of shape quality in cad systems of the future
Großmann et al. Volumetric geometry reconstruction of turbine blades for aircraft engines
Louhichi et al. An optimization-based computational method for surface fitting to update the geometric information of an existing B-Rep CAD...
Schilling et al. Filling arbitrary holes in finite element models
Steinberg et al. Reverse engineering trimmed NURB surfaces from laser scanned data

Legal Events

Date Code Title Description
PUAI Public reference made under article 153(3) epc to a published international application that has entered the european phase

Free format text: ORIGINAL CODE: 0009012

17P Request for examination filed

Effective date: 20051115

AK Designated contracting states

Kind code of ref document: A2

Designated state(s): AT BE BG CH CY CZ DE DK EE ES FI FR GB GR HU IE IT LI LU MC NL PL PT RO SE SI SK TR

RAP1 Party data changed (applicant data changed or rights of an application transferred)

Owner name: UGS CORP.

DAX Request for extension of the european patent (deleted)
RIN1 Information on inventor provided before grant (corrected)

Inventor name: TSAI, YI-FENG

Inventor name: SRIPRADISVARAKUL, THAWACH

RAP1 Party data changed (applicant data changed or rights of an application transferred)

Owner name: SIEMENS PRODUCT LIFECYCLE MANAGEMENT SOFTWARE INC.

REG Reference to a national code

Ref country code: DE

Ref legal event code: R003

STAA Information on the status of an ep patent application or granted ep patent

Free format text: STATUS: THE APPLICATION HAS BEEN REFUSED

18R Application refused

Effective date: 20140114