Bernstein-Be´zier polynomials on spheres and sphere-like surfaces
Computer Aided Geometric Design
Finite Element Method for Elliptic Problems
Finite Element Method for Elliptic Problems
One-to-one piecewise linear mappings over triangulations
Mathematics of Computation
Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model
Mathematical and Computer Modelling: An International Journal
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In this paper we analyse a method for triangulating the sphere originally proposed by Baumgardner and Frederickson in 1985. The method is essentially a refinement procedure for arbitrary spherical triangles that fit into a hemisphere. Refinement is carried out by dividing each triangle into four by introducing the midpoints of the edges as new vertices and connecting them in the usual 'red' way. We show that this process can be described by a sequence of piecewise smooth mappings from a reference triangle onto the spherical triangle. We then prove that the whole sequence of mappings is uniformly bi-Lipschitz and converges uniformly to a non-smooth parameterization of the spherical triangle, recovering the Baumgardner and Frederickson spherical barycentric coordinates. We also prove that the sequence of triangulations is quasi-uniform, that is, areas of triangles and lengths of the edges are roughly the same at each refinement level. Some numerical experiments confirm the theoretical results.