Conformal Spherical Splines for the Laplace--Beltrami Operator on Genus-Zero Surfaces: Construction, Algorithm, and Experiments

Abstract

Every smooth closed Riemannian surface of genus zero is isometric to the unit sphere carrying a conformal metric, the round metric multiplied by a positive factor. This paper solves the Poisson equation and the eigenvalue problem of the Laplace--Beltrami operator of such a metric with smooth spherical splines, taking the conformal factor as the only description of the geometry. Because the Dirichlet energy is conformally invariant in two dimensions, the stiffness matrix is the round-sphere matrix for every conformal metric, and the factor enters only the load vector, the mean-value constraint and the weighted mass matrix. A surface embedded in space with a conformal parametrization, a surface of revolution, and a surface given only as a triangle mesh are instances of the same problem that differ in how the factor is obtained; the first two are treated here, and the factor is checked against an area identity before any equation is discretized. The unknown is a spline on a fixed spherical triangulation in broken Bernstein--Bézier form, with $C^r$ smoothness imposed algebraically through edge functionals and the conforming space realized by a null-space matrix computed once and reused for every metric. Each step of the computation is stated with the formulas it needs. Experiments on the round metric, a prolate spheroid, a dumbbell of revolution, an Evans--Fung red-blood-cell profile and a prescribed factor with no embedding show the rates $d+1$ in $L^2$ and $d$ in energy for spline degree $d$, the rate $2d$ for eigenvalues with reproduction of the even-order eigenvalues of the round metric up to degree $d$ to roundoff, and, against parametric surface finite elements on the same triangulations, smaller errors at comparable numbers of unknowns for degrees four and six.