Geometric similarity measures and their applications

Similarity measures are used in many applications, from comparing and searching for images to procedural content generation. They let us rank objects in order of similarity to a query, and they let us guide many search-based algorithms by telling us how good a chess move is, or how much our generated content resembles a given example. In this thesis, we specifically explore geometric similarity measures, which are used to quantify the resemblance between two geometric objects. We expand on the theoretical understanding of similarity measures, showing how the earth mover's distance can be approximated for a variety of simple geometric objects. We also find new applications for well-studied similarity measures by showing how the Hausdorff distance can be used to interpolate between two shapes, creating a morph. Additionally, we take a detour to study a new model of indeterminacy in graphs.