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Geometric modeling with conical meshes and developable surfaces

Authors:

Helmut Pottmann,

Johannes Wallner,

Yong-Liang Yang,

Wenping WangAuthors Info & Claims

ACM Transactions on Graphics (TOG), Volume 25, Issue 3

Pages 681 - 689

https://doi.org/10.1145/1141911.1141941

Published: 01 July 2006 Publication History

Abstract

In architectural freeform design, the relation between shape and fabrication poses new challenges and requires more sophistication from the underlying geometry. The new concept of conical meshes satisfies central requirements for this application: They are quadrilateral meshes with planar faces, and therefore particularly suitable for the design of freeform glass structures. Moreover, they possess a natural offsetting operation and provide a support structure orthogonal to the mesh. Being a discrete analogue of the network of principal curvature lines, they represent fundamental shape characteristics. We show how to optimize a quad mesh such that its faces become planar, or the mesh becomes even conical. Combining this perturbation with subdivision yields a powerful new modeling tool for all types of quad meshes with planar faces, making subdivision attractive for architecture design and providing an elegant way of modeling developable surfaces.

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References

[1]

Alliez, P., Cohen-Steiner, D., Devillers, O., Levy, B., and Desbrun, M. 2003. Anisotropic polygonal remeshing. ACM Trans. Graphics 22, 3, 485--493.]]

Digital Library

[2]

Aumann, G. 2004. Degree elevation and developable Bézier surfaces. Comp. Aided Geom. Design 21, 661--670.]]

Digital Library

[3]

Blake, A., and Isard, M. 1998. Active Contours. Springer.]]

[4]

Bobenko, A., and Suris, Y., 2005. Discrete differential geometry. Consistency as integrability. preprint, http://arxiv.org/abs/math.DG/0504358.]]

[5]

Bobenko, A., Matthes, D., and Suris, Y. 2003. Discrete and smooth orthogonal systems: C∞-approximation. Int. Math. Res. Not., 45, 2415--2459.]]

[6]

Bobenko, A., Hoffmann, T., and Springborn, B. A. 2006. Minimal surfaces from circle patterns: Geometry from combinatorics. Annals of Mathematics 164, 231--264.]]

[7]

Cecil, T. 1992. Lie Sphere Geometry. Springer.]]

[8]

Cerda, E., Chaieb, S., Melo, F., and Mahadevan, L. 1999. Conical dislocations in crumpling. Nature 401, 46--49.]]

[9]

Chen, Y., and Medioni, G. 1991. Object modeling by registration of multiple range images. In Proc. IEEE Conf. on Robotics and Automation.]]

[10]

Chu, C. H., and Sequin, C. 2002. Developable Bézier patches: properties and design. Computer-Aided Design 34, 511--528.]]

[11]

Cipolla, R., and Giblin, P. 2000. Visual Motion of Curves and Surfaces. Cambridge University Press.]]

Digital Library

[12]

Clarenz, U., Rumpf, M., and Telea, A. 2004. Robust feature detection and local classification for surfaces based on moment analysis. IEEE Trans. Visual. Comp. Graphics 10, 516--524.]]

Digital Library

[13]

Cohen-Steiner, D., and Morvan, J.-M. 2003. Restricted Delaunay triangulations and normal cycle. In Proc. 19th annual symposium on Computational geometry, ACM, 312--321.]]

Digital Library

[14]

Cohen-Steiner, D., Alliez, P., and Desbrun, M. Variational shape approximation. ACM Trans. Graphics 23, 3, 905--914.]]

Digital Library

[15]

Desbrun, M., Grinspun, E., and Schröder, P. 2005. Discrete Differential Geometry. SIGGRAPH Course Notes.]]

[16]

do Carmo, M. 1976. Differential Geometry of Curves and Surfaces. Prentice-Hall.]]

[17]

Dong, S., Kircher, S., and Garland, M. 2005. Harmonic functions for quadrilateral remeshing of arbitrary manifolds. Comp. Aided Geom. Design 22, 392--423.]]

Digital Library

[18]

Frey, W. 2004. Modeling buckled developable surfaces by triangulation. Computer-Aided Design 36, 4, 299--313.]]

[19]

Glymph, J., Shelden, D., Ceccato, C., Mussel, J., and Schober, H. 2002. A parametric strategy for freeform glass structures using quadrilateral planar facets. In Acadia 2002, ACM, 303--321.]]

[20]

Hildebrandt, K., Polthier, K., and Wardetzky, M. 2005. On the convergence of metric and geometric properties of polyhedral surfaces. Tech. Rep. 05--24, Zuse Institute Berlin.]]

[21]

Julius, D., Kraevoy, V., and Sheffer, A. 2005. D-charts: Quasi-developable mesh segmentation. Computer Graphics Forum (Proc. Eurographics 2005) 24, 3, 581--590.]]

[22]

Kelley, C. T. 1999. Iterative Methods for Optimization. SIAM.]]

[23]

Kilian, A. 2006. Design exploration through bidirectional modeling of constraints. PhD thesis, Massachusets Inst. Technology.]]

Digital Library

[24]

Kim, S.-J., and Yang, M.-Y. 2005. Triangular mesh offset for generalized cutter. Computer-Aided Design 37, 10, 999--1014.]]

Digital Library

[25]

Madsen, K., Nielsen, H. B., and Tingleff, O., 2004. Optimization with constraints. Lecture Notes.]]

[26]

Marinov, M., and Kobbelt, L. 2004. Direct anisotropic quad-dominant remeshing. In Proc. Pacific Graphics, 207--216.]]

Digital Library

[27]

Martin, R. R., de Pont, J., and Sharrock, T. J. 1986. Cyclide surfaces in computer aided design. In The mathematics of surfaces, J. A. Gregory, Ed. Clarendon Press, Oxford, 253--268.]]

[28]

Mitani, J., and Suzuki, H. 2004. Making papercraft toys from meshes using strip-based approximate unfolding. ACM Trans. Graphics 23, 3, 259--263.]]

Digital Library

[29]

Polthier, K. 2002. Polyhedral surfaces of constant mean curvature. Habilitationsschrift TU Berlin.]]

[30]

Porteous, I. R. 1994. Geometric Differentiation for the Intelligence of Curves and Surfaces. Cambridge Univ. Press.]]

[31]

Pottmann, H., and Wallner, J. 2001. Computational Line Geometry. Springer.]]

Digital Library

[32]

Pottmann, H., Huang, Q.-X., Yang, Y.-L., and Kölpl, S., 2005. Integral invariants for robust geometry processing. Geometry Preprint 146, TU Wien.]]

[33]

Ray, N., Li, W.-C., Levy, B., Sheffer, A., and Alliez, P., 2005. Periodic global parameterization. INRIA preprint.]]

Digital Library

[34]

Sauer, R. 1970. Differenzengeometrie. Springer.]]

[35]

Sequin, C. 2004. CAD tools for aesthetic engineering. CAD & Appl. 1, 301--309.]]

[36]

Sullivan, J. 2005. The aesthetic value of optimal geometry. In In The Visual Mind II, M. Emmer, Ed. MIT Press, 547--563.]]

[37]

Wang, C., and Tang, K. 2004. Achieving developability of a polygonal surface by minimum deformation: a study of global and local optimization approaches. Vis. Computer 20, 521--539.]]

Digital Library

[38]

Wang, W., Wallner, J., and Liu, Y., 2006. An angle criterion for conical mesh vertices. Geometry Preprint 157, TU Wien. http://www.geometrie.tuwien.ac.at/ig/papers/tr157.pdf.]]

[39]

Wunderlich, W. 1951. Zur Differenzengeometrie der Flächen konstanter negativer Krümmung. Sitz. öst. Ak. Wiss. 160, 41--77.]]

[40]

Yamauchi, H., Gumhold, S., Zayer, R., and Seidel, H.-P. 2005. Mesh segmentation driven by Gaussian curvature. Visual Computer 21, 659--668.]]

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Index Terms

Geometric modeling with conical meshes and developable surfaces
1. Computing methodologies
  1. Computer graphics
    1. Shape modeling
      1. Parametric curve and surface models
      2. Volumetric models
2. Theory of computation
  1. Randomness, geometry and discrete structures

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Published In

cover image ACM Transactions on Graphics

ACM Transactions on Graphics Volume 25, Issue 3

July 2006

742 pages

ISSN:0730-0301

EISSN:1557-7368

DOI:10.1145/1141911

Issue’s Table of Contents

Seminal Graphics Papers: Pushing the Boundaries, Volume 2
August 2023
893 pages
ISBN:9798400708978
DOI:10.1145/3596711
Editor:
Mary C. Whitton
Department of Computer Science, UNC Chapel Hill, USA

Copyright © 2006 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

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Publication History

Published: 01 July 2006

Published in TOG Volume 25, Issue 3

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https://doi.org/10.1016/j.ijmecsci.2025.109939
Maekawa TScholz F(2024)All you need is rotation: Construction of developable stripsACM Transactions on Graphics10.1145/368794743:6(1-15)Online publication date: 19-Dec-2024
https://dl.acm.org/doi/10.1145/3687947
Charrondière RNeukirch SBertails-Descoubes F(2024)MERCI: Mixed Curvature-Based Elements for Computing Equilibria of Thin Elastic RibbonsACM Transactions on Graphics10.1145/367450243:5(1-26)Online publication date: 9-Aug-2024
https://dl.acm.org/doi/10.1145/3674502
Rodriguez EBonneau GHahmann SSkouras M(2024)Designing Bending-Active Freeform SurfacesProceedings of the 9th ACM Symposium on Computational Fabrication10.1145/3639473.3665793(1-11)Online publication date: 7-Jul-2024
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