Abstract
This paper proposes a new subdivision scheme based on line geometry. We name the scheme ‘line subdivision’. Line subdivision scheme acts on the line space, and generates two-dimensional manifolds contained in the Klein quadric. The two-dimensional manifolds are the Klein images of line congruences in P 3. So, this new subdivision scheme generates line congruences at the limit. Here, we define the line subdivision surface as an envelope surface which is made by the line congruence. Then, this paper derives basic properties of the surface. Moreover, we show that line subdivision contains ordinary subdivision and dual subdivision.
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References
Lee, A., Moreton, H., Hoppe, H.: Displaced subdivision surface. In: SIGGRAPH 2000 Proceedings, pp. 85–94. ACM, New York (2000)
Loop, C.T.: Smooth subdivision surfaces based on triangles. Master’s thesis, University of Utah, Department of Mathematics (1987)
Derose, T., Kass, M., Truong, T.: Subdivision surfaces in character animation. In: SIGGRAPH 1998 Proceedings, pp. 85–94 (1998)
Warren, J., Weimer, H.: Subdivision Methods for Geometric Design: A Constructive Approach. Morgan Kaufmann, San Francisco (1995)
Stam, J.: Evaluation of loop subdivision surfaces. In: SIGGRAPH 1998 Conference Proceedings on CDROM, ACM, New York (1998)
Zorin, D.: Smoothness of stationary subdivision on irregular meshes. Constructive Approximation 16, 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 c k-subdivision surfaces at extraordinary points. Presented at Oberwolfach (preprint, 1995)
Cavaretta, A.S., Dahmen, W., Micchell, C.A.: Stationary subdivision. Memoirs Amer. Math. Soc. 93(453) (1991)
Goodman, T.N.T., Micchell, C.A., Derose, T., Warren, J.: 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)
Lounsbery, M., Derose, T., Warren, J.: Multiresolution analysis for surfaces of arbitrary topological type. ACM Transactions on Graphics 16, 34–73 (1997)
Eck, M., DeRose, T., Duchamp, T.: Multiresolution analysis of arbitrary meshes. In: SIGGRAPH 1995 Proceedings, pp. 173–182. ACM, New York (1995)
Stollnitz, E.J., DeRose, T.D., Salesin, D.H.: Wavelets for Computer Graphics: Theory and Applications. Morgan Kaufmann Publishers, San Francisco (1996)
Guskov, I., Sweldens, W., Schroder, P.: Multiresolution signal processing for meshes. In: SIGGRAPH 1999 Proceedings, pp. 325–334. ACM, New York (1999)
Guskov, I., Khodakovsky, A., Schroder, P., Sweldens, W.: Hybrid meshes: multiresolution using regular and irregular refinement. In: Proceedings of the eighteenth annual symposium on Computational geometry, pp. 264–272. ACM Press, New York (2002)
Gu, X., Gortler, S.J., Hoppe, H.: Geometry images. In: Proceedings of the 29th annual conference on Computer graphics and interactive techniques, pp. 355–361. ACM, New York (2002)
Claes, J., Beets, K., Reeth, F.V.: A corner-cutting scheme for hexagonal subdivision surfaces. In: Proceedings of Sape Modeling International 2002, pp. 13–24. IEEE, Los Alamitos (2002)
Farin, G., Hoschek, J., Kim, M.: Handbook of Computer Aided Geometric Design. Elsevier Science Publishers, Amsterdam (2002)
Kawaharada, H., Sugihara, K.: Dual subdivision a new class of subdivision schemes using projective duality. In: METR 2005-01, The University of Tokyo (2005)
Pottman, H., Wallner, J.: Computational Line Geometry. Springer, Heidelberg (2001)
Dietz, R., Hoschek, J., Juttler, B.: An algebraic approach to curves and surfaces on the sphere and on other quadrics. Computer Aided Geometric Design 10, 211–229 (1993)
Dietz, R., Hoschek, J., Juttler, B.: Rational patches on quadric surfaces. Computer Aided Geometric Design 27, 27–40 (1995)
Sun, K.A., Juttler, B., Kim, M.S., Wang, W.: Computing the distance between two surfaces via line geometry. In: The Tenth Pacific Conference on Computer Graphics and Applications, Los Alamitos (2002)
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Kawaharada, H., Sugihara, K. (2005). Line Subdivision. In: Martin, R., Bez, H., Sabin, M. (eds) Mathematics of Surfaces XI. Lecture Notes in Computer Science, vol 3604. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537908_15
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DOI: https://doi.org/10.1007/11537908_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28225-9
Online ISBN: 978-3-540-31835-4
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