Quadratic Forms for Measuring Geometric Trees in 3-dimensional Space

Abstract

Tree-like structures appear in many areas of science, and their shapes can help understand the underlying processes they drive or that give rise to them. By thinking of these structures as geometric graphs in $\mathbb{R}^3$, we gain access to tools from computational geometry and topology to study them. In this paper, we adopt the theory of quadratic forms to measure the directional spread of geometric graphs, and we introduce the hexplot model -- equipped with a metric derived from the Fisher metric on the standard triangle -- to visualize, measure, and collect statistics.