G2 Surface Interpolation Over General Topology Curve Networks

The basic idea of curve network‐based design is to construct smoothly connected surface patches, that interpolate boundaries and cross‐derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G1) continuity between the adjacent patches, curvature c...

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Veröffentlicht in:	Computer graphics forum Jg. 33; H. 7; S. 151 - 160
Hauptverfasser:	Salvi, Péter, Várady, Tamás
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	Oxford Blackwell Publishing Ltd 01.10.2014
Schlagworte:	Analysis Categories and Subject Descriptors (according to ACM CCS) Computer graphics Computer Graphics [I.3.5]: Computational Geometry and Object Modeling-curvenet-based design Computer Graphics [I.3.5]: Computational Geometry and Object Modeling—curvenet‐based design, transfinite surfaces, Gregory patches, G2 continuity G2 continuity Gregory patches Network topologies Optimization Studies Topological manifolds transfinite surfaces
ISSN:	0167-7055, 1467-8659
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Abstract	The basic idea of curve network‐based design is to construct smoothly connected surface patches, that interpolate boundaries and cross‐derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G1) continuity between the adjacent patches, curvature continuous connections (G2) may also be required. Examples include special curve network configurations with supplemented internal edges, “master‐slave” curvature constraints, and general topology surface approximations over meshes. The first step is to assign optimal surface curvatures to the nodes of the curve network; we discuss different optimization procedures for various types of nodes. Then interpolant surfaces called parabolic ribbons are created along the patch boundaries, which carry first and second derivative constraints. Our construction guarantees that the neighboring ribbons, and thus the respective transfinite patches, will be G2 continuous. We extend Gregory's multi‐sided surface scheme in order to handle parabolic ribbons, involving the blending functions, and a new sweepline parameterization. A few simple examples conclude the paper.
AbstractList	The basic idea of curve network-based design is to construct smoothly connected surface patches, that interpolate boundaries and cross-derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G1) continuity between the adjacent patches, curvature continuous connections (G2) may also be required. Examples include special curve network configurations with supplemented internal edges, "master-slave" curvature constraints, and general topology surface approximations over meshes. The first step is to assign optimal surface curvatures to the nodes of the curve network; we discuss different optimization procedures for various types of nodes. Then interpolant surfaces called parabolic ribbons are created along the patch boundaries, which carry first and second derivative constraints. Our construction guarantees that the neighboring ribbons, and thus the respective transfinite patches, will be G2 continuous. We extend Gregory's multi-sided surface scheme in order to handle parabolic ribbons, involving the blending functions, and a new sweepline parameterization. A few simple examples conclude the paper. The basic idea of curve network‐based design is to construct smoothly connected surface patches, that interpolate boundaries and cross‐derivatives extracted from the curve network. While the majority of applications demands only tangent plane (G1) continuity between the adjacent patches, curvature continuous connections (G2) may also be required. Examples include special curve network configurations with supplemented internal edges, “master‐slave” curvature constraints, and general topology surface approximations over meshes. The first step is to assign optimal surface curvatures to the nodes of the curve network; we discuss different optimization procedures for various types of nodes. Then interpolant surfaces called parabolic ribbons are created along the patch boundaries, which carry first and second derivative constraints. Our construction guarantees that the neighboring ribbons, and thus the respective transfinite patches, will be G2 continuous. We extend Gregory's multi‐sided surface scheme in order to handle parabolic ribbons, involving the blending functions, and a new sweepline parameterization. A few simple examples conclude the paper.
Author	Salvi, Péter Várady, Tamás
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Copyright	2014 The Author(s) Computer Graphics Forum © 2014 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd. 2014 The Eurographics Association and John Wiley & Sons Ltd.
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References	Várady T., Rockwood A., Salvi P.: Transfinite surface interpolation over irregular n-sided domains. Computer Aided Design 43, 11 (2011), 1330-1340. Hermann T.: G2 interpolation of free form curve networks by biquintic Gregory patches. Computer Aided Geometric Design 13, 9 (1996), 873-893. Gregory J.A., Hahn J.M.: A C2 polygonal surface patch. Computer Aided Geometric Design 6, 1 (1989), 69-75. Selesnick S.A.: Local invariants and twist vectors in computer-aided geometric design. Computer Graphics and Image Processing 17, 2 (1981), 145-160. Wang W., Jüttler B., Zheng D., Liu Y.: Computation of rotation minimizing frames. Transactions on Graphics 27, 1 (2008), 2. Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical recipes in C (2nd ed.): the art of scientific computing. Cambridge University Press, 1992. Worsey A.: A modified C2 Coons' patch. Computer Aided Geometric Design 1, 4 (1984), 357-360. Pegna J., Wolter F.-E.: Geometrical criteria to guarantee curvature continuity of blend surfaces. Journal of Mechanical Design 114, 1 (1992), 201-210. Peters J.: Smooth interpolation of a mesh of curves. Constructive Approximation 7, 1 (1991), 221-246. Charrot P., Gregory J.A.: A pentagonal surface patch for computer aided geometric design. Computer Aided Geometric Design 1, 1 (1984), 87-94. Hall R., Mullineux G.: Continuity between Gregory-like patches. Computer Aided Geometric Design 16, 3 (1999), 197-216. Hermann T., Peters J., Strotman T.: Curve networks compatible with G2 surfacing. Computer Aided Geometric Design 29, 5 (2012), 219-230. 1984; 1 1989; 6 1999; 16 1992; 114 2008; 27 1996 2011; 43 2012; 29 1994 1981; 17 1992 2014 1996; 13 1991; 7
References_xml	– reference: Hall R., Mullineux G.: Continuity between Gregory-like patches. Computer Aided Geometric Design 16, 3 (1999), 197-216. – reference: Selesnick S.A.: Local invariants and twist vectors in computer-aided geometric design. Computer Graphics and Image Processing 17, 2 (1981), 145-160. – reference: Wang W., Jüttler B., Zheng D., Liu Y.: Computation of rotation minimizing frames. Transactions on Graphics 27, 1 (2008), 2. – reference: Charrot P., Gregory J.A.: A pentagonal surface patch for computer aided geometric design. Computer Aided Geometric Design 1, 1 (1984), 87-94. – reference: Peters J.: Smooth interpolation of a mesh of curves. Constructive Approximation 7, 1 (1991), 221-246. – reference: Worsey A.: A modified C2 Coons' patch. Computer Aided Geometric Design 1, 4 (1984), 357-360. – reference: Hermann T.: G2 interpolation of free form curve networks by biquintic Gregory patches. Computer Aided Geometric Design 13, 9 (1996), 873-893. – reference: Gregory J.A., Hahn J.M.: A C2 polygonal surface patch. Computer Aided Geometric Design 6, 1 (1989), 69-75. – reference: Pegna J., Wolter F.-E.: Geometrical criteria to guarantee curvature continuity of blend surfaces. Journal of Mechanical Design 114, 1 (1992), 201-210. – reference: Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P.: Numerical recipes in C (2nd ed.): the art of scientific computing. Cambridge University Press, 1992. – reference: Várady T., Rockwood A., Salvi P.: Transfinite surface interpolation over irregular n-sided domains. Computer Aided Design 43, 11 (2011), 1330-1340. – reference: Hermann T., Peters J., Strotman T.: Curve networks compatible with G2 surfacing. Computer Aided Geometric Design 29, 5 (2012), 219-230. – volume: 13 start-page: 873 year: 1996 end-page: 893 article-title: G publication-title: Computer Aided Geometric Design – volume: 27 start-page: 2 year: 2008 article-title: Computation of rotation minimizing frames publication-title: Transactions on Graphics – volume: 1 start-page: 357 year: 1984 end-page: 360 article-title: A modified C2 Coons' patch publication-title: Computer Aided Geometric Design – volume: 43 start-page: 1330 year: 2011 end-page: 1340 article-title: Transfinite surface interpolation over irregular n‐sided domains publication-title: Computer Aided Design – volume: 1 start-page: 87 year: 1984 end-page: 94 article-title: A pentagonal surface patch for computer aided geometric design publication-title: Computer Aided Geometric Design – volume: 16 start-page: 197 year: 1999 end-page: 216 article-title: Continuity between Gregory‐like patches publication-title: Computer Aided Geometric Design – volume: 6 start-page: 69 year: 1989 end-page: 75 article-title: A C polygonal surface patch publication-title: Computer Aided Geometric Design – volume: 17 start-page: 145 year: 1981 end-page: 160 article-title: Local invariants and twist vectors in computer‐aided geometric design publication-title: Computer Graphics and Image Processing – start-page: 411 year: 1996 end-page: 427 – volume: 114 start-page: 201 year: 1992 end-page: 210 article-title: Geometrical criteria to guarantee curvature continuity of blend surfaces publication-title: Journal of Mechanical Design – volume: 7 start-page: 221 year: 1991 end-page: 246 article-title: Smooth interpolation of a mesh of curves publication-title: Constructive Approximation – start-page: 123 year: 1994 end-page: 159 – year: 1992 – year: 2014 – volume: 29 start-page: 219 year: 2012 end-page: 230 article-title: Curve networks compatible with G surfacing publication-title: Computer Aided Geometric Design
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SubjectTerms	Analysis Categories and Subject Descriptors (according to ACM CCS) Computer graphics Computer Graphics [I.3.5]: Computational Geometry and Object Modeling-curvenet-based design Computer Graphics [I.3.5]: Computational Geometry and Object Modeling—curvenet‐based design, transfinite surfaces, Gregory patches, G2 continuity G2 continuity Gregory patches Network topologies Optimization Studies Topological manifolds transfinite surfaces
Title	G2 Surface Interpolation Over General Topology Curve Networks
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