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Bases for non-homogeneous polynomial Ck splines on the sphere

  • Anamaria Gomide
  • Jorge Stolfi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1380)

Abstract

We investigate the use of non-homogeneous spherical polynomials for the approximation of functions defined on the sphere S2. A spherical polynomial is the restriction to S2 of a polynomial in the three coordinates x,y,z of ℝ3. Let P d be the space of spherical polynomials with degree ≤ d. We show that P d is the direct sum of P d and H d−1, where H d denotes the space of homogeneous degree-d polynomials in x,y,z.

We also generalize this result to splines defined on a geodesic triangulation T of the sphere. Let P k d [T] denote the space of all functions f from S2 to ℝ such that (1) the restriction of f to each triangle of T belongs to P d ; and (2) the function f has order-k continuity across the edges of T. Analogously, let H k d [T] denote the subspace of P k d [T] consisting of those functions that are H d within each triangle of T. We show that P k d [T]=H k d [T]⊕H k d−1 [T]. Combined with results of Alfeld, Neamtu and Schumaker on bases of H k d [T] this decomposition provides an effective construction for a basis of P k d [T].

There has been considerable interest recently in the use of the homogeneous spherical splines H k d [T] as approximations for functions defined on S2. We argue that the non-homogeneous splines P k d [T] would be a more natural choice for that purpose.

Keywords

Spherical Function Spline Space Spherical Triangle Piecewise Polynomial Function Spherical Spline 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Peter Alfeld, Marian Neamtu, and Larry L. Schumaker. Bernstein-Bézier polynomials on circle, spheres, and sphere-like surfaces. Computer Aided Geometric Design Journal, 13:333–349, 1996.CrossRefMathSciNetGoogle Scholar
  2. 2.
    Peter Alfeld, Marian Neamtu, and Larry L. Schumaker. Dimension and local bases of homogeneous spline spaces. SIAM Journal of Mathematical Analysis, 27(5):1482–1501, September 1996.CrossRefMathSciNetGoogle Scholar
  3. 3.
    Peter Alfeld, Marian Neamtu, and Larry L. Schumaker. Fitting scattered data on sphere-like surfaces using spherical splines. Journal of Computational and Applied Mathematics, 73:5–43, 1996.CrossRefMathSciNetGoogle Scholar
  4. 4.
    W. Fulton. Algebraic Curves: An Introduction to Algebraic Geometry. W. A. Benjamin, 1969.Google Scholar
  5. 5.
    E. Kunz. Introduction to Commutative Algebra and Algebraic Geometry. Birkhauser, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Anamaria Gomide
    • 1
  • Jorge Stolfi
    • 1
  1. 1.Institute of ComputingUniversity of CampinasCampinasBrazil

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