Construction and Application of 3-Point Tensor Product Scheme

Affiliation(s)

The Islamia University of Bahawalpur, Bahawalpur, Pakistan.

Tsinghua University, Beijing, China.

The Islamia University of Bahawalpur, Bahawalpur, Pakistan.

Tsinghua University, Beijing, China.

ABSTRACT

In this paper, we propose and analyze a tensor product subdivision scheme which is the extension of three point scheme for curve modeling. The usefulness of the scheme is illustrated by considering different examples along with its application in surface modeling.

Cite this paper

A. Ghaffar, G. Mustafa and K. Qin, "Construction and Application of 3-Point Tensor Product Scheme,"*Applied Mathematics*, Vol. 4 No. 3, 2013, pp. 477-485. doi: 10.4236/am.2013.43071.

A. Ghaffar, G. Mustafa and K. Qin, "Construction and Application of 3-Point Tensor Product Scheme,"

References

[1] E. Catmull and J. Clark, “Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes,” Computer Aided Design, Vol. 10, No. 6, 1978, pp. 350-355. doi:10.1016/0010-4485(78)90110-0

[2] D. Doo and M. A. Sabin, “Behaviour of Recursive Subdivision Surfaces Near Extraordinary Points,” Computer Aided Design, Vol. 10, No. 6, 1978, pp. 356-360. doi:10.1016/0010-4485(78)90111-2

[3] U. Reif, “A Unified Approach to Subdivision Algorithms near Extraordinary Vertices,” Computer Aided Geometric Design, Vol. 12, 1995, pp. 153-174. doi:10.1016/0167-8396(94)00007-F

[4] D. Zorin, “Subdivision and Multiresolution Surface Representations,” Ph.D. Thesis, Caltech, Pasadena, 1997.

[5] J. Stam, “Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values,” Proceedings of the Annual Conference Series of Computer Graphics, Orlando, July 1998, pp. 395-404.

[6] H. Hoppe, T. De-Rose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer and W. Stuetzle, “Piecewise Smooth Surface Reconstruction,” Proceedings of the Association for Computing Machinery’s Special Interest Group on Computer Graphics and Interactive Techniques, Orlando, 1994, pp. 295-302.

[7] C. Loop, “Smooth Subdivision Surfaces Based on Triangles,” Master’s Thesis, Department of Mathematics, University of Utah, Salt Lake City, 1987.

[8] G. Morin, J. Warren and H. Weimer, “A Subdivision Scheme for Surfaces of Revolution” Computer Aided Geometric Design, Vol. 18, No. 5, 2001, pp. 483-502. doi:10.1016/S0167-8396(01)00043-7

[9] M. J. Jena, P. Shunmugaraj and P. J. Das, “A Non-Stationary Subdivision Scheme for Generalizing Trigonometric Spline Surfaces to Arbitrary Meshes,” Computer Aided Geometric Design, Vol. 20, No. 2, 2003, pp. 61-77. doi:10.1016/S0167-8396(03)00008-6

[10] X. Li and J. Zheng, “An Alternative Method for Constructing Interpolatory Subdivision from Approximating Subdivision,” Computer Aided Geometric Design, Vol. 29, No. 7, 2012, pp. 474-484. doi:10.1016/j.cagd.2012.03.008

[11] N. Dyn, “Interpolatory Subdivision Schemes and Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials,” In: A. Iske, E. Quak and M. S Floater, Eds, Tutorials on Multiresolution in Geometric Modeling, Springer, 2002, pp. 51-68. doi:10.1007/978-3-662-04388-2_3

[12] G. Mustafa, F. Khan and A. Ghaffar, “The m-Point Approximating Subdivision Scheme,” Lobachevskii Journal of Mathematics, Vol. 30, No. 2, 2009, pp. 138-145. doi:10.1134/S1995080209020061

[1] E. Catmull and J. Clark, “Recursively Generated B-Spline Surfaces on Arbitrary Topological Meshes,” Computer Aided Design, Vol. 10, No. 6, 1978, pp. 350-355. doi:10.1016/0010-4485(78)90110-0

[2] D. Doo and M. A. Sabin, “Behaviour of Recursive Subdivision Surfaces Near Extraordinary Points,” Computer Aided Design, Vol. 10, No. 6, 1978, pp. 356-360. doi:10.1016/0010-4485(78)90111-2

[3] U. Reif, “A Unified Approach to Subdivision Algorithms near Extraordinary Vertices,” Computer Aided Geometric Design, Vol. 12, 1995, pp. 153-174. doi:10.1016/0167-8396(94)00007-F

[4] D. Zorin, “Subdivision and Multiresolution Surface Representations,” Ph.D. Thesis, Caltech, Pasadena, 1997.

[5] J. Stam, “Exact Evaluation of Catmull-Clark Subdivision Surfaces at Arbitrary Parameter Values,” Proceedings of the Annual Conference Series of Computer Graphics, Orlando, July 1998, pp. 395-404.

[6] H. Hoppe, T. De-Rose, T. Duchamp, M. Halstead, H. Jin, J. McDonald, J. Schweitzer and W. Stuetzle, “Piecewise Smooth Surface Reconstruction,” Proceedings of the Association for Computing Machinery’s Special Interest Group on Computer Graphics and Interactive Techniques, Orlando, 1994, pp. 295-302.

[7] C. Loop, “Smooth Subdivision Surfaces Based on Triangles,” Master’s Thesis, Department of Mathematics, University of Utah, Salt Lake City, 1987.

[8] G. Morin, J. Warren and H. Weimer, “A Subdivision Scheme for Surfaces of Revolution” Computer Aided Geometric Design, Vol. 18, No. 5, 2001, pp. 483-502. doi:10.1016/S0167-8396(01)00043-7

[9] M. J. Jena, P. Shunmugaraj and P. J. Das, “A Non-Stationary Subdivision Scheme for Generalizing Trigonometric Spline Surfaces to Arbitrary Meshes,” Computer Aided Geometric Design, Vol. 20, No. 2, 2003, pp. 61-77. doi:10.1016/S0167-8396(03)00008-6

[10] X. Li and J. Zheng, “An Alternative Method for Constructing Interpolatory Subdivision from Approximating Subdivision,” Computer Aided Geometric Design, Vol. 29, No. 7, 2012, pp. 474-484. doi:10.1016/j.cagd.2012.03.008

[11] N. Dyn, “Interpolatory Subdivision Schemes and Analysis of Convergence and Smoothness by the Formalism of Laurent Polynomials,” In: A. Iske, E. Quak and M. S Floater, Eds, Tutorials on Multiresolution in Geometric Modeling, Springer, 2002, pp. 51-68. doi:10.1007/978-3-662-04388-2_3

[12] G. Mustafa, F. Khan and A. Ghaffar, “The m-Point Approximating Subdivision Scheme,” Lobachevskii Journal of Mathematics, Vol. 30, No. 2, 2009, pp. 138-145. doi:10.1134/S1995080209020061