Anti-commutative Dual Complex Numbers and 2D Rigid Transformation

TL;DR

This paper introduces anti-commutative dual complex numbers as a concise and efficient way to represent 2D rigid transformations, enabling easy interpolation and blending with low computational cost.

Contribution

It presents a novel algebraic structure for 2D rigid transformations that simplifies computation and interpolation compared to traditional matrix methods.

Findings

Provides a new mathematical framework for 2D transformations

Develops a C++ library implementing the approach

Demonstrates practical application in an interactive deformation tool

Abstract

We introduce a new presentation of the two dimensional rigid transformation which is more concise and efficient than the standard matrix presentation. By modifying the ordinary dual number construction for the complex numbers, we define the ring of the anti-commutative dual complex numbers, which parametrizes two dimensional rotation and translation all together. With this presentation, one can easily interpolate or blend two or more rigid transformations at a low computational cost. We developed a library for C++ with the MIT-licensed source code and demonstrate its facility by an interactive deformation tool developed for iPad.