In this chapter and the next we discuss spline spaces defined on triangulations of the unit sphere S in ℝ3. The spaces are natural analogs of the bivariate spline spaces discussed earlier in this book, and are made up of pieces of trivariate homogeneous polynomials restricted to S. Thus, they are piecewise spherical harmonics. As we shall see, virtually the entire theory of bivariate polynomial splines on planar triangulations carries over, although there are several significant differences. This chapter is devoted to the basic theory of spherical splines. Approximation properties of spherical splines are treated in the following chapter.

Spherical Polynomials

In this section we introduce the key building blocks for spherical splines. Throughout the chapter we write ν for a point on the unit sphere S in ℝ3. When there is no chance of confusion, at times we will also write v for the corresponding unit vector. Before introducing spherical polynomials, we need to discuss spherical triangles and spherical barycentric coordinates.

Spherical Triangles

Suppose ν1, ν2 are two points on the sphere which are not antipodal, i.e., they do not lie on a line through the origin. Then the points ν1, ν2 divide the great circle passing through ν1, ν2 into two circular arcs. We write 〈ν1, ν2〉 for the shorter of the arcs. Its length is just the geodesic distance between ν1 and ν2.

Definition 13.1.Suppose ν1, ν2, ν3are three points on the unit sphere S which lie strictly in one hemisphere.