Quadratic Forms for Measuring Geometric Trees in 3-dimensional Space

2026-06-18

AI Summary

This study addresses the challenge of effectively quantifying the directional distribution and morphological characteristics of three-dimensional tree-like geometric structures. To this end, it proposes a novel framework that integrates quadratic form theory with the Fisher metric from information geometry. By modeling tree-like structures as 3D geometric graphs, the method leverages quadratic forms to capture their directional spread and introduces, for the first time, a hexplot model to enable visualization and statistical analysis of directional distributions. This approach delivers the first analytical tool for 3D arboreal structures that simultaneously ensures geometric rigor and statistical interpretability, significantly enhancing the quantification of morphological features and the efficiency of cross-structure comparisons in fields such as neuroscience and botany.

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.

Problem

• quadratic forms
• geometric trees
• directional spread
• 3-dimensional space
• hexplot model

Innovation

• quadratic forms
• geometric trees
• hexplot model
• Fisher metric
• directional spread