In the doctoral dissertation we consider rational parametrizations of surfaces, given by non-negative basis functions that form a convex partition of unity and have local supports. In the first part of the dissertation we review polynomial and rational parametrizations of Bézier curves, surfaces and multisided Bézier surfaces, called S-patches. Then we focus on rational representations of the sphere. By using the stereographic projection and platonic solids inscribed into the sphere we derive a method that enables exact representations of sphere sections in terms of rational S-patches. The method is based on parameters that are determined by the basic properties of the platonic solids.
In the second part of the dissertation we consider rational splines on triangulations. We first review polynomial cubic continuously differentiable Powell–Sabin splines and, following a standard procedure, extend them to rational splines of the same type by introducing positive weights. Then we discuss properties of the defined splines and illustrate their practical applications. We study representations of rational Bézier surfaces, quadratic version of rational Powell–Sabin splines and some representatives of ruled surfaces in rational cubic Powell–Sabin form. By using the derived methods we also parametrize some curved domains which we later use in the context of the isoparametric finite element method to numerically solve certain boundary value problems. We determine numerical solutions to various examples of Poisson and biharmonic problem in rational cubic Powell–Sabin form.
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