# Bases for non-homogeneous polynomial *C*_{k} splines on the sphere

## Abstract

We investigate the use of non-homogeneous spherical polynomials for the approximation of functions defined on the sphere S^{2}. A *spherical polynomial* is the restriction to S^{2} of a polynomial in the three coordinates *x,y,z* of ℝ^{3}. 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 S^{2} 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## Preview

Unable to display preview. Download preview PDF.

## References

- 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.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.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.W. Fulton.
*Algebraic Curves: An Introduction to Algebraic Geometry*. W. A. Benjamin, 1969.Google Scholar - 5.E. Kunz.
*Introduction to Commutative Algebra and Algebraic Geometry*. Birkhauser, 1993.Google Scholar