Many algorithms for shape analysis and shape processing rely on accurate estimates of differential information such as normals and curvature. In most settings, however, care must be taken around non-smooth areas of the shape where these quantities are not easily defined. This problem is particularly prominent with point-cloud data, which are discontinuous everywhere. In this paper we present an efficient and robust method for extracting principal curvatures, sharp features and normal directions of a piecewise smooth surface from its point cloud sampling, with theoretical guarantees. Our method is integral in nature and uses convolved covariance matrices of Voronoi cells of the point cloud which makes it provably robust in the presence of noise. We show analytically that our method recovers correct principal curvatures and principal curvature directions in smooth parts of the shape, and correct feature directions and feature angles at the sharp edges of a piecewise smooth surface, with the error bounded by the Hausdorff distance between the point cloud and the underlying surface. Using the same analysis we provide theoretical guarantees for a modification of a previously proposed normal estimation technique. We illustrate the correctness of both principal curvature information and feature extraction in the presence of varying levels of noise and sampling density on a variety of models.
{"title":"Robust Voronoi-based curvature and feature estimation","authors":"Q. Mérigot, M. Ovsjanikov, L. Guibas","doi":"10.1145/1629255.1629257","DOIUrl":"https://doi.org/10.1145/1629255.1629257","url":null,"abstract":"Many algorithms for shape analysis and shape processing rely on accurate estimates of differential information such as normals and curvature. In most settings, however, care must be taken around non-smooth areas of the shape where these quantities are not easily defined. This problem is particularly prominent with point-cloud data, which are discontinuous everywhere. In this paper we present an efficient and robust method for extracting principal curvatures, sharp features and normal directions of a piecewise smooth surface from its point cloud sampling, with theoretical guarantees. Our method is integral in nature and uses convolved covariance matrices of Voronoi cells of the point cloud which makes it provably robust in the presence of noise. We show analytically that our method recovers correct principal curvatures and principal curvature directions in smooth parts of the shape, and correct feature directions and feature angles at the sharp edges of a piecewise smooth surface, with the error bounded by the Hausdorff distance between the point cloud and the underlying surface. Using the same analysis we provide theoretical guarantees for a modification of a previously proposed normal estimation technique. We illustrate the correctness of both principal curvature information and feature extraction in the presence of varying levels of noise and sampling density on a variety of models.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115838663","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We present a new parallel algorithm for interactive and continuous collision detection between deformable models. Our algorithm performs incremental hierarchical computations between successive frames and parallelizes the computation among multiple cores on current CPUs. The main computations include front building and updating and performing the elementary tests between the triangle primitives. The overall algorithm can perform inter- and intra-object collisions at interactive rates on current commodity processors on models with many tens of thousands of triangles. In practice, the performance of our algorithm almost scales linearly with the number of cores.
{"title":"Multi-core collision detection between deformable models","authors":"Min Tang, Dinesh Manocha, Ruofeng Tong","doi":"10.1145/1629255.1629303","DOIUrl":"https://doi.org/10.1145/1629255.1629303","url":null,"abstract":"We present a new parallel algorithm for interactive and continuous collision detection between deformable models. Our algorithm performs incremental hierarchical computations between successive frames and parallelizes the computation among multiple cores on current CPUs. The main computations include front building and updating and performing the elementary tests between the triangle primitives. The overall algorithm can perform inter- and intra-object collisions at interactive rates on current commodity processors on models with many tens of thousands of triangles. In practice, the performance of our algorithm almost scales linearly with the number of cores.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"66 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130838607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jing Bai, Shuming Gao, Weihua Tang, Yusheng Liu, Song Guo
In this paper, we present a semantic-based partial retrieval approach of CAD models for design reuse. Based on the observation of reusable regions for design of 3D CAD models, for each model in the model library, we propose an algorithm that automatically extracts its reusable regions for partial retrieval. To further effectively support the partial retrieval of these reusable regions through simple queries, we represent each reusable region by all its local matching regions and describe each local matching region in a hierarchical way. Based on the hierarchical descriptor, a partial retrieval method for reusable regions is put forward. The approach proposed is implemented and tested by hundreds of mechanical parts. Preliminary results show that the method can effectively support partial retrieval for design reuse.
{"title":"Semantic-based partial retrieval of CAD models for design reuse","authors":"Jing Bai, Shuming Gao, Weihua Tang, Yusheng Liu, Song Guo","doi":"10.1145/1629255.1629289","DOIUrl":"https://doi.org/10.1145/1629255.1629289","url":null,"abstract":"In this paper, we present a semantic-based partial retrieval approach of CAD models for design reuse. Based on the observation of reusable regions for design of 3D CAD models, for each model in the model library, we propose an algorithm that automatically extracts its reusable regions for partial retrieval. To further effectively support the partial retrieval of these reusable regions through simple queries, we represent each reusable region by all its local matching regions and describe each local matching region in a hierarchical way. Based on the hierarchical descriptor, a partial retrieval method for reusable regions is put forward. The approach proposed is implemented and tested by hundreds of mechanical parts. Preliminary results show that the method can effectively support partial retrieval for design reuse.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133156271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Model resizing is a useful technique for model reuse. Introduced by Kraevoy et al., non-homogeneous model resizing is able to preserve important geometric features during anisotropic scaling. However, many practical objects contain various geometric textures. Different from similar objects without geometric textures, such objects seem as if extremely difficult to resize even if the underlying surfaces are relatively simple. In this paper, we present an automatic model resizing method based on geometric texture transfer. Geometric textures are separated from the underlying surfaces and reproduced on the non-homogenously scaled surfaces using geometric texture synthesis. By utilizing the natural correspondence between the surfaces before and after resizing, surfaces with multiple geometric textures can be resized and geometric texture recovered automatically. Experimental results show that our method effectively and automatically preserves geometric textures during the model resizing process.
{"title":"Anisotropic resizing of model with geometric textures","authors":"Lin Chen, Xiangxu Meng","doi":"10.1145/1629255.1629292","DOIUrl":"https://doi.org/10.1145/1629255.1629292","url":null,"abstract":"Model resizing is a useful technique for model reuse. Introduced by Kraevoy et al., non-homogeneous model resizing is able to preserve important geometric features during anisotropic scaling. However, many practical objects contain various geometric textures. Different from similar objects without geometric textures, such objects seem as if extremely difficult to resize even if the underlying surfaces are relatively simple. In this paper, we present an automatic model resizing method based on geometric texture transfer. Geometric textures are separated from the underlying surfaces and reproduced on the non-homogenously scaled surfaces using geometric texture synthesis. By utilizing the natural correspondence between the surfaces before and after resizing, surfaces with multiple geometric textures can be resized and geometric texture recovered automatically. Experimental results show that our method effectively and automatically preserves geometric textures during the model resizing process.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122228677","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The progress of nanotechnology has made it possible to make miniature electromechanical devices of sub-micrometer scale. This means that we will be in need of design packages that can model the physical properties of objects and their interactions involved down in nanometer scale. Toward this goal, our aim in this paper is to develop a computing procedure for determining molecular interaction forces, i.e. van der Waals force, between objects of arbitrary geometry. Currently there are two types of approaches for calculating van der Waals force. The first type is analytical where analytical force equations are derived for interactions between simple geometries such as spheres and half-spaces. The second type is numerical where volume integrals or surface integrals are conducted over discretized object domains where the object boundaries are approximated by simple mesh geometries. This paper presents a numerical approach that uses non-uniform rational B-spline (NURBS) based surface integrals. The integrals are done on the parametric domains of the NURBS surfaces and Gaussian quadrature points lie exactly on the object surfaces. Salient features of this approach include: 1) Orders of magnitude in accuracy improvement is achieved over other numerical approaches; The fundamental reason for such accuracy improvement is that molecular interaction force is very sensitive to surface geometry since it falls off at the rate of inverse power of 6 ~ 7. Any geometric approximation in object discretization would lead to significant bias in the calculation result. 2) Molecular interactions between arbitrary-shaped objects can be represented and evaluated since the NURBS model can represent exactly common analytical geometries such as spheres in nano-particles and cylinders in nano-rods, and complex geometries such as corrugated sample surfaces. We demonstrate its general shape applicability by calculating van der Waals force between complex geometries such as micro-gears. Further, we give error bounds for NURBS based numerical simulation and develop an adaptive subdivision scheme to improve both calculation accuracy and efficiency.
{"title":"NURBS based molecular force calculation","authors":"Pinghai Yang, Xiaoping Qian","doi":"10.1145/1629255.1629304","DOIUrl":"https://doi.org/10.1145/1629255.1629304","url":null,"abstract":"The progress of nanotechnology has made it possible to make miniature electromechanical devices of sub-micrometer scale. This means that we will be in need of design packages that can model the physical properties of objects and their interactions involved down in nanometer scale. Toward this goal, our aim in this paper is to develop a computing procedure for determining molecular interaction forces, i.e. van der Waals force, between objects of arbitrary geometry.\u0000 Currently there are two types of approaches for calculating van der Waals force. The first type is analytical where analytical force equations are derived for interactions between simple geometries such as spheres and half-spaces. The second type is numerical where volume integrals or surface integrals are conducted over discretized object domains where the object boundaries are approximated by simple mesh geometries.\u0000 This paper presents a numerical approach that uses non-uniform rational B-spline (NURBS) based surface integrals. The integrals are done on the parametric domains of the NURBS surfaces and Gaussian quadrature points lie exactly on the object surfaces. Salient features of this approach include: 1) Orders of magnitude in accuracy improvement is achieved over other numerical approaches; The fundamental reason for such accuracy improvement is that molecular interaction force is very sensitive to surface geometry since it falls off at the rate of inverse power of 6 ~ 7. Any geometric approximation in object discretization would lead to significant bias in the calculation result. 2) Molecular interactions between arbitrary-shaped objects can be represented and evaluated since the NURBS model can represent exactly common analytical geometries such as spheres in nano-particles and cylinders in nano-rods, and complex geometries such as corrugated sample surfaces.\u0000 We demonstrate its general shape applicability by calculating van der Waals force between complex geometries such as micro-gears. Further, we give error bounds for NURBS based numerical simulation and develop an adaptive subdivision scheme to improve both calculation accuracy and efficiency.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123924640","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to Riemann uniformization theorem, every surface with negative Euler number has a unique conformal Riemannian metric, which induces Gaussian curvature of --1 everywhere. Hence, the surface admits hyperbolic geometry. Such uniformization metric can be computed using the discrete curvature flow method: hyperbolic Ricci flow. Consequently, the basis function for each control point can be naturally defined over a hyperbolic disk, and through the use of partition-of-unity, we build a freeform surface directly over hyperbolic domains while having C∞ property. The use of radial, exponential basis functions gives rise to a true meshless method for modeling freeform surfaces with greatest flexibilities, without worrying about control point connectivity. Our algorithm is general for arbitrary surfaces with negative Euler characteristic. Furthermore, it is C∞ continuous everywhere across the entire hyperbolic domain without singularities. Our experimental results demonstrate the efficiency and efficacy of the proposed new approach for shape and solid modeling.
{"title":"C∞ smooth freeform surfaces over hyperbolic domains","authors":"W. Zeng, Ying He, Jiazhi Xia, X. Gu, Hong Qin","doi":"10.1145/1629255.1629305","DOIUrl":"https://doi.org/10.1145/1629255.1629305","url":null,"abstract":"Constructing smooth freeform surfaces of arbitrary topology with higher order continuity is one of the most fundamental problems in shape and solid modeling. This paper articulates a novel method to construct C∞ smooth surfaces with negative Euler numbers based on hyperbolic geometry and discrete curvature flow. According to Riemann uniformization theorem, every surface with negative Euler number has a unique conformal Riemannian metric, which induces Gaussian curvature of --1 everywhere. Hence, the surface admits hyperbolic geometry. Such uniformization metric can be computed using the discrete curvature flow method: hyperbolic Ricci flow. Consequently, the basis function for each control point can be naturally defined over a hyperbolic disk, and through the use of partition-of-unity, we build a freeform surface directly over hyperbolic domains while having C∞ property. The use of radial, exponential basis functions gives rise to a true meshless method for modeling freeform surfaces with greatest flexibilities, without worrying about control point connectivity. Our algorithm is general for arbitrary surfaces with negative Euler characteristic. Furthermore, it is C∞ continuous everywhere across the entire hyperbolic domain without singularities. Our experimental results demonstrate the efficiency and efficacy of the proposed new approach for shape and solid modeling.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127497837","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A-patches are a form of representation of an algebraic curve or surface over a simplex. The A-patch conditions can be used as the basis for an adaptive subdivision style marching tetrahedra algorithm whose advantage is that it guarantees that we do not miss features of the algebraic: singularities are localized, and in regions around nearby multiple sheets, the subdivision process continues until the sheets are separated. Unfortunately, the A-patch single sheet conditions are too strict: for some algebraics, the subdivision process converges slowly or fails to converge. In this paper, I give an additional single sheet condition for curves that allows for convergence of this process. I also give additional conditions for surfaces that trades off some of the single sheet guarantees for improved convergence.
a -patch是单纯形上代数曲线或曲面的一种表示形式。A-patch条件可以作为自适应细分风格的前进四面体算法的基础，其优点是它保证我们不会错过代数的特征:奇点是局部化的，并且在附近的多个片周围的区域中，细分过程继续直到片分离。遗憾的是，A-patch单片条件过于严格:对于某些代数，细分过程收敛缓慢或不收敛。在本文中，我给出了一个额外的单页条件，允许这个过程的收敛。我还给出了一些额外的曲面条件，这些曲面可以用一些单片保证来改善收敛性。
{"title":"Extending the A-patch single sheet conditions to enable the tessellation of algebraics","authors":"Stephen Mann","doi":"10.1145/1629255.1629300","DOIUrl":"https://doi.org/10.1145/1629255.1629300","url":null,"abstract":"A-patches are a form of representation of an algebraic curve or surface over a simplex. The A-patch conditions can be used as the basis for an adaptive subdivision style marching tetrahedra algorithm whose advantage is that it guarantees that we do not miss features of the algebraic: singularities are localized, and in regions around nearby multiple sheets, the subdivision process continues until the sheets are separated.\u0000 Unfortunately, the A-patch single sheet conditions are too strict: for some algebraics, the subdivision process converges slowly or fails to converge. In this paper, I give an additional single sheet condition for curves that allows for convergence of this process. I also give additional conditions for surfaces that trades off some of the single sheet guarantees for improved convergence.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123230415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Delaunay graph is constructed incrementally. Our first contribution is to propose robust end efficient algorithms for all required predicates, thus generalizing our earlier algorithms for ellipses, and we analyze their algebraic complexity, under the exact computation paradigm. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5 X 5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of ellipses, which is the first exact implementation for the problem. Our code spends about 98 sec to construct the Delaunay graph of 128 non-intersecting ellipses, when few degeneracies occur. It is faster than the cgal segment Delaunay graph, when ellipses are approximated by k-gons for k > 15.
我们研究了在参数形式给出的欧几里得平面上一组可能相交的光滑凸伪圆的精确计算Delaunay图(和对偶Voronoi图)的问题。伪圆是(凸)点，每一对最多有两个交点。Delaunay图是增量构造的。我们的第一个贡献是为所有必需的谓词提出鲁棒的端高效算法，从而推广了我们之前的椭圆算法，并在精确的计算范式下分析了它们的代数复杂性。其次，我们重点讨论了最难的谓词InCircle，并用一个简单的稀疏5 X 5多项式系统来表示它，该系统允许通过连续Sylvester结果和一个新的因式分解引理来有效地实现它。第三个贡献是针对省略号情况的基于cgal的c++软件，这是该问题的第一个精确实现。我们的代码花了大约98秒来构造由128个不相交椭圆组成的Delaunay图，当简并很少发生时。当k > 15时，椭圆近似为k-gons时，它比合法段Delaunay图更快。
{"title":"Exact Delaunay graph of smooth convex pseudo-circles: general predicates, and implementation for ellipses","authors":"I. Emiris, Elias P. Tsigaridas, George M. Tzoumas","doi":"10.1145/1629255.1629282","DOIUrl":"https://doi.org/10.1145/1629255.1629282","url":null,"abstract":"We examine the problem of computing exactly the Delaunay graph (and the dual Voronoi diagram) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Delaunay graph is constructed incrementally. Our first contribution is to propose robust end efficient algorithms for all required predicates, thus generalizing our earlier algorithms for ellipses, and we analyze their algebraic complexity, under the exact computation paradigm. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5 X 5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of ellipses, which is the first exact implementation for the problem. Our code spends about 98 sec to construct the Delaunay graph of 128 non-intersecting ellipses, when few degeneracies occur. It is faster than the cgal segment Delaunay graph, when ellipses are approximated by k-gons for k > 15.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123638012","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We undertake a study of the local properties of 2-Gabriel meshes: manifold triangle meshes each of whose faces has an open Euclidean diametric ball that contains no mesh vertices. We show that, under mild constraints on the dihedral angles, such meshes are Delaunay meshes: the open geodesic circumdisk of each face contains no mesh vertex. The analysis is done by means of the Delaunay edge flipping algorithm and it reveals the details of the distinction between these two mesh structures. In particular we observe that the obstructions which prohibit the existence of Gabriel meshes as homeomorphic representatives of smooth surfaces do not hinder the construction of Delaunay meshes.
{"title":"Gabriel meshes and Delaunay edge flips","authors":"R. Dyer, Hao Zhang, Torsten Möller","doi":"10.1145/1629255.1629293","DOIUrl":"https://doi.org/10.1145/1629255.1629293","url":null,"abstract":"We undertake a study of the local properties of 2-Gabriel meshes: manifold triangle meshes each of whose faces has an open Euclidean diametric ball that contains no mesh vertices. We show that, under mild constraints on the dihedral angles, such meshes are Delaunay meshes: the open geodesic circumdisk of each face contains no mesh vertex.\u0000 The analysis is done by means of the Delaunay edge flipping algorithm and it reveals the details of the distinction between these two mesh structures. In particular we observe that the obstructions which prohibit the existence of Gabriel meshes as homeomorphic representatives of smooth surfaces do not hinder the construction of Delaunay meshes.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"176 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121263128","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The behavior of Catmull-Rom curves heavily depends on the choice of parameter values at the control points. We analyze a class of parameterizations ranging from uniform to chordal parameterization and show that, within this class, curves with centripetal parameterization contain properties that no other curves in this family possess. Researchers have previously indicated that centripetal parameterization produces visually favorable curves compared to uniform and chordal parameterizations. However, the mathematical reasons behind this behavior have been ambiguous. In this paper we prove that, for cubic Catmull-Rom curves, centripetal parameterization is the only parameterization in this family that guarantees that the curves do not form cusps or self-intersections within curve segments. Furthermore, we provide a formulation that bounds the distance of the curve to the control polygon and explain how globally intersection-free Catmull-Rom curves can be generated using these properties.
{"title":"On the parameterization of Catmull-Rom curves","authors":"Cem Yuksel, S. Schaefer, J. Keyser","doi":"10.1145/1629255.1629262","DOIUrl":"https://doi.org/10.1145/1629255.1629262","url":null,"abstract":"The behavior of Catmull-Rom curves heavily depends on the choice of parameter values at the control points. We analyze a class of parameterizations ranging from uniform to chordal parameterization and show that, within this class, curves with centripetal parameterization contain properties that no other curves in this family possess. Researchers have previously indicated that centripetal parameterization produces visually favorable curves compared to uniform and chordal parameterizations. However, the mathematical reasons behind this behavior have been ambiguous. In this paper we prove that, for cubic Catmull-Rom curves, centripetal parameterization is the only parameterization in this family that guarantees that the curves do not form cusps or self-intersections within curve segments. Furthermore, we provide a formulation that bounds the distance of the curve to the control polygon and explain how globally intersection-free Catmull-Rom curves can be generated using these properties.","PeriodicalId":216067,"journal":{"name":"Symposium on Solid and Physical Modeling","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2009-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115616515","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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