Abstract
Many algorithms rely on mesh parametrization. In particular, the mapping from a mesh to a 2D domain (and vice versa) is essential to rendering 3D models and an essential component of e.g. remeshing. In this chapter, we study several algorithms for such flattening of a patch of disk topology. The basic algorithm—quite similar to mesh smoothing—is introduced, and various vertex weights are covered, ranging from simple uniform weights to mean value and harmonic weights which result in less distortion. Finally, we discuss the so called natural boundary conditions which allow us to flatten the mesh with minimum angle distortion in the least squares sense.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Lévy, B., Petitjean, S., Ray, N., Maillot, J.: Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. 21(3), 362–371 (2002)
Praun, E., Hoppe, H.: Spherical parametrization and remeshing. ACM Trans. Graph. 22(3), 340–349 (2003)
Kälberer, F., Nieser, M., Polthier, K.: QuadCover-surface parameterization using Branched Coverings. Comput. Graph. Forum 26(3), 375–384 (2007). Wiley Online Library
Alliez, P., Meyer, M., Desbrun, M.: Interactive geometry remeshing. ACM Trans. Graph. 21(3), 347–354 (2002)
Farkas, H.M., Kra, I.: Riemann Surfaces. Graduate Texts in Mathematics, vol. 71, p. 337. Springer, New York (1980)
Forster, O.: Lectures on Riemann Surfaces. Graduate Texts in Mathematics, vol. 81, p. 254. Springer, New York (1991). Translated from the 1977 German original by Bruce Gilligan, Reprint of the 1981 English translation
Radó, T.: Aufgabe 41. Dtsch. Math.-Ver. 35, 49 (1926)
Kneser, H.: Lösung der Aufgabe 41. Dtsch. Math.-Ver. 35, 123–124 (1926)
Choquet, G.: Sur un type de transformation analytique généralisant la représentation conforme et définie au moyen de fonctions harmoniques. Bull. Sci. Math. 69, 156–165 (1945)
Floater, M.S., Hormann, K.: Surface parameterization: a tutorial and survey. In: Advances in Multiresolution for Geometric Modelling, pp. 157–186 (2005)
Botsch, M., Kobbelt, L., Pauly, M., Alliez, P., Levy, B.: Polygon Mesh Processing. AK Peters, Wellesley (2010)
Floater, M.S.: Mean value coordinates. Comput. Aided Geom. Des. 20(1), 19–27 (2003)
Pinkall, U., Polthier, K.: Computing discrete minimal surfaces and their conjugates. Exp. Math. 2(1), 15–36 (1993)
Desbrun, M., Meyer, M., Alliez, P.: Intrinsic parameterizations of surface meshes. Comput. Graph. Forum 21(3), 209–218 (2002)
Mullen, P., Tong, Y., Alliez, P., Desbrun, M.: Spectral conformal parameterization. Comput. Graph. Forum 27(5), 1487–1494 (2008)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer-Verlag London
About this chapter
Cite this chapter
Bærentzen, J.A., Gravesen, J., Anton, F., Aanæs, H. (2012). Parametrization of Meshes. In: Guide to Computational Geometry Processing. Springer, London. https://doi.org/10.1007/978-1-4471-4075-7_10
Download citation
DOI: https://doi.org/10.1007/978-1-4471-4075-7_10
Publisher Name: Springer, London
Print ISBN: 978-1-4471-4074-0
Online ISBN: 978-1-4471-4075-7
eBook Packages: Computer ScienceComputer Science (R0)