Fractions, Projective Representation, Duality, Linear Algebra and Geometry

TL;DR

This paper explores the connections between fractions, projective geometry, duality, and linear algebra, demonstrating that division is unnecessary for solving linear systems when using projective and homogeneous coordinates, with efficient CPU and GPU implementations.

Contribution

It introduces a novel approach to solving linear equations using projective space and homogeneous coordinates, eliminating the need for division operations.

Findings

Division operation is unnecessary in solving linear systems with projective representation.

Efficient CPU and GPU algorithms for solving linear equations are developed.

Application to barycentric coordinate computation demonstrates practical utility.

Abstract

This contribution describes relationship between fractions, projective representation, duality, linear algebra and geometry. Many problems lead to a system of linear equations. This paper presents equivalence of the Cross-product operation and solution of a system of linear equations Ax=0 or Ax=b using projective space representation and homogeneous coordinates. It leads to conclusion that division operation is not required for a solution of a system of linear equations, if the projective representation and homogeneous coordinates are used. An efficient solution on CPU and GPU based architectures is presented with an application to barycentric coordinates computation as well.