 
Summary: Fitting Scattered Data on SphereLike
Surfaces using Spherical Splines
by
Peter Alfeld 1) , Marian Neamtu 2) , and Larry L. Schumaker 3)
Abstract. Spaces of polynomial splines defined on planar triangulations are
very useful tools for fitting scattered data in the plane. Recently, [4, 5], using
homogeneous polynomials, we have developed analogous spline spaces defined on
triangulations on the sphere and on spherelike surfaces. Using these spaces, it
is possible to construct analogs of many of the classical interpolation and fitting
methods. Here we examine some of the more interesting ones in detail. For
interpolation, we discuss macroelement methods and minimal energy splines,
and for fitting, we consider discrete least squares and penalized least squares.
1. Introduction
Let S be the unit sphere or a spherelike surface (see Sect. 2 below) in IR 3 . In
addition, suppose that we are given a set of scattered points located on S, along with
real numbers associated with each of these points. The problem of interest in this
paper is to find a function defined on S which either interpolates or approximates
these data.
This problem arises in a variety of settings. For example, in geodesy, geo
physics, and metereology, S is chosen to be some model of the earth. But it also
