Computational Circle Packing: Geometry and Discrete Analytic Function Theory

Geometric Circle Packings are of interest not only for their aesthetic appeal but also their relation to discrete analytic function theory. This thesis presents new computational methods which enable additional practical applications for circle packing geometry along with providing a new discrete analytic interpretation of the classical Schwarzian derivative and traditional univalence criterion of classical analytic function theory. To this end I present a new method of computing the maximal packing and solving the circle packing layout problem for a simplicial 2-complex along with additional geometric variants and applications. This thesis also presents a geometric discrete Schwarzian quantity whose value is associated with the classical Schwarzian derivative. Following Hille, I present a characterization of circle packings as the ratio of two linearly independent solutions of a discrete difference equation taking the discrete Schwarzian as a parameter. This characterization then gives a discrete interpretation of the classical univalence criterion of Nehari in the circle packing setting.