Abstract
Given a smooth surface patch we construct an approximating piecewise linear structure. More precisely, we produce a mesh for which virtually all vertices have valency three. We present two methods for the construction of meshes whose facets are tangent to the original surface. These two methods can deal with elliptic and hyperbolic surfaces, respectively. In order to describe and to derive the construction, which is based on a projective duality, we use the so–called support function representation of the surface and of the mesh, where the latter one has a piecewise linear support function.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Almegaard, H.: The Stringer System—a Truss Model of Membrane Shells for Analysis and Design of Boundary Conditions. Int. J. Space Structures 19, 1–10 (2004)
Bonnesen, T., Fenchel, W.: Theory of Convex Bodies. BCS Associates, Moscow, Idaho (1987)
Brückner, M.: Vielecke und Vielflache—Theorie und Geschichte. Teubner, Leipzig (1900)
Cutler, B., Whiting, E.: Constrained Planar Remeshing for Architecture. In: Symposium on Geometry Processing 2006, Poster proceedings (electronic), p. 5 (2006), http://sgp2006.sc.unica.it/program/PosterProceedings.pdf
Groemer, H.: Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press, Cambridge (1996)
Gruber, P.M., Wills, J.M. (eds.): Handbook of Convex Geometry. North-Holland, Amsterdam (1993)
Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. AK Peters, Wellesley, Mass (1996)
Hoschek, J.: Dual Bézier Curves and Surfaces. In: Barnhill, R.E., Boehm, W. (eds.) Surfaces in Computer Aided Geometric Design, pp. 147–156. North-Holland, Amsterdam (1983)
Kawarahada, H., Sugihara, K.: Dual Subdivision: A New Class of Subdivision Schemes using Projective Duality. In: Jorge, J., Skala, V. (eds.) Proc. WSCG 2006, pp. 9–16. University of West Bohemia, Plzen (2006)
Liu, Y., Pottmann, H., Wallner, J., Yang, Y., Wang, W.: Geometric Modeling with Conical Meshes and Developable Surfaces. ACM Trans. Graphics 25, 681–689 (2006)
Patanè, G., Spagnuolo, M.: Triangle Mesh Duality: Reconstruction and Smoothing. In: Wilson, M.J., Martin, R.R. (eds.) Mathematics of Surfaces. LNCS, vol. 2768, pp. 111–128. Springer, Heidelberg (2003)
Pottmann, H., Wallner, J.: The Focal Geometry of Circular and Conical Meshes. Adv. Comput. Math. (to appear)
Ros, L., Sugihara, K., Thomas, F.: Towards Shape Representation using Trihedral Mesh Projections. The Visual Computer 19, 139–150 (2003)
Sabin, M.: A Class of Surfaces Closed under Five Important Geometric Operations. Technical Report VTO/MS/207, British Aircraft Corporation (1974), Available at http://www.damtp.cam.ac.uk/user/na/people/Malcolm/vtoms/vtos.html
Šír, Z., Gravesen, J., Jüttler B.: Curves and surfaces represented by polynomial support functions. SFB report no. 2006-36 (2006), Available at http://www.sfb013.uni-linz.ac.at
Weisstein, E.W.: Dual Polyhedron. From MathWorld — A Wolfram Web Resource. http://mathworld.wolfram.com/DualPolyhedron.html
Wenninger, M.J.: Dual Models. Cambridge University Press, Cambridge (1983)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Almegaard, H., Bagger, A., Gravesen, J., Jüttler, B., Šír, Z. (2007). Surfaces with Piecewise Linear Support Functions over Spherical Triangulations. In: Martin, R., Sabin, M., Winkler, J. (eds) Mathematics of Surfaces XII. Mathematics of Surfaces 2007. Lecture Notes in Computer Science, vol 4647. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73843-5_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-73843-5_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73842-8
Online ISBN: 978-3-540-73843-5
eBook Packages: Computer ScienceComputer Science (R0)