Abstract
This paper proposes a method for computing of normals for stationary subdivision surfaces.
In [1, 2], we derived a new necessary and sufficient condition for C k-continuity of stationary subdivision schemes. First, we showed that tangent plane continuity is equivalent to the convergence of difference vectors. Thus, using “normal subdivision matrix” [3], we derived a necessary and sufficient condition of tangent plane continuity for stationary subdivision at extraordinary points (including degree 6). Moreover, we derived a necessary and sufficient condition for C 1-continuity.
Using the analysis, we show that at general points on stationary subdivision surfaces, the computation of the exact normal is an infinite sum of linear combinations of cross products of difference vectors even if the surfaces are C 1-continuous. So, it is not computable. However, we can compute the exact normal of subdivision surfaces at the limit position of a vertex of original mesh or of j-th subdivided mesh for any finite j even if the surfaces are not regular.
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
Preview
Unable to display preview. Download preview PDF.
References
Kawaharada, H., Sugihara, K.: ck-continuity of stationary subdivision schemes. METR 2006-01, The University of Tokyo (2006)
Kawaharada, H., Sugihara, K.: c k-continuity for stationary subdivisions at exraordinary points. Computer Aided Geometric Design XX (submitted, 2006)
Kawaharada, H., Sugihara, K.: Dual subdivision a new class of subdivision schemes using projective duality. METR 2005-01, The University of Tokyo (2005)
Loop, C.T.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics (1987)
Warren, J., Weimer, H.: Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann Publishers, San Francisco (1995)
Zorin, D.: Smoothness of stationary subdivision on irregular meshes. Constructive Approximation 16(3), 359–397 (2000)
Reif, U.: A unified approach to subdivision algorithms near extraordinary points. Computer Aided Geometric Design 12, 153–174 (1995)
Prautzsch, H.: Analysis of ck-subdivision surfaces at extraordinary points. Oberwolfach (preprint, 1995) (presented)
Cavaretta, A.S., Dahmen, W., Micchell, C.A.: Stationary subdivision. Memoirs Amer. Math. Soc. 93(453) (1991)
Goodman, T.N.T., Micchell, C.A.: Spectral radius formulas for subdivision operators. In: Schumaker, L.L., Webb, G. (eds.) Recent Advances in Wavelet Analysis, pp. 335–360. Academic Press, London (1994)
Zorin, D.: Subdivision and Multiresolution Surface Representations. PhD thesis, University of California Institute of Technology (1997)
Reif, U.: A degree estimate for polynomial subdivision surfaces of higher regularity. Proc. Amer. Math. Soc. 124, 2167–2174 (1996)
Doo, D., Sabin, M.A.: Behaviour of recursive subdivision surfaces near extraordinary points. Computer Aided Geometric Design 10, 356–360 (1978)
Meyer, M., Desbrum, M., Schroder, P., Barr, A.H.: Discrete differential geometry operators for triangulated 2-manifolds. In: Visualization and Mathematics III, pp. 35–57 (2003)
Stam, J.: Evaluation of loop subdivision surfaces. In: SIGGRAPH 1998 Conference Proceedings on CDROM. ACM, New York (1998)
Stam, J.: Exact evaluation of catmull-clark subdivision surfaces at arbitrary parameter values. In: SIGGRAPH 1998 Conference Proceedings, pp. 395–404. ACM, New York (1998)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kawaharada, H., Sugihara, K. (2006). Computation of Normals for Stationary Subdivision Surfaces. In: Kim, MS., Shimada, K. (eds) Geometric Modeling and Processing - GMP 2006. GMP 2006. Lecture Notes in Computer Science, vol 4077. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11802914_44
Download citation
DOI: https://doi.org/10.1007/11802914_44
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-36711-6
Online ISBN: 978-3-540-36865-6
eBook Packages: Computer ScienceComputer Science (R0)