The problem of surface parametrization has a long history. The first known examples of parametrization are maps - mappings of a sphere onto a two dimensional surface. It is known that some surfaces in ${\mathbb R}^3$ cannot be mapped into two dimensions without angle or area distortions. The main focus of this master's thesis is to find parametrizations that preserve angles or that have an upper bound for the angle distortion. Such mappings are called conformal or quasiconformal. We will show that instead of searching for a conformal mapping we can search for a harmonic one. In this thesis, two algorithms for spherical parametrization of genus 0 closed surfaces and an algorithm for disk parametrization of simply connected open surfaces are presented.
In many applications, especially in computer graphics, it is common practice to approximate the surface by a set of piecewise linear surfaces in the form of a triangular mesh. This thesis presents the properties of admissible and regular triangular meshes and the data structures that can be used to represent a triangular mesh in a computer.
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