Orientations of Simplices Determined by Orderings on the Coordinates of their Vertices

TL;DR

This paper investigates how coordinate orderings of points determine the affine independence and orientation of simplices, providing solutions in dimensions 2 and 3 with potential for generalization.

Contribution

It offers a complete combinatorial characterization and a formal calculus method for determining simplex orientation from coordinate orderings in low dimensions.

Findings

Complete solutions in dimensions 2 and 3

A decision algorithm based on cofactor expansion

Conjecture on generalization to higher dimensions

Abstract

Provided n points in an (n-1)-dimensional affine space, and one ordering of the points for each coordinate, we address the problem of testing whether these orderings determine if the points are the vertices of a simplex (i.e. are affinely independent), regardless of the real values of the coordinates. We also attempt to determine the orientation of this simplex. In other words, given a matrix whose columns correspond to affine points, we want to know when the sign (or the non-nullity) of its determinant is implied by orderings given to each row for the values of the row. We completely solve the problem in dimensions 2 and 3. We provide a direct combinatorial characterization, along with a formal calculus method. It can also be viewed as a decision algorithm, and is based on testing the existence of a suitable inductive cofactor expansion of the determinant. We conjecture that our method generalizes in higher dimensions. This work aims to be part of a study on how oriented matroids encode shapes of 3-dimensional landmark-based objects. Specifically, applications include the analysis of anatomical data for physical anthropology and clinical research.