The Study Variety of Conformal Kinematics

TL;DR

This paper introduces the Study variety of conformal kinematics, a high-dimensional projective variety that generalizes rigid body kinematics models and is computationally accessible through conformal geometric algebra.

Contribution

It defines and explores the properties of the Study variety in conformal kinematics, extending the dual quaternion framework to conformal motions and analyzing straight lines related to conformal motions.

Findings

Study variety is a 10-dimensional, degree 12 projective variety in 15-dimensional space.

Straight lines on the Study variety correspond to conformal motions introduced by Dorst.

The framework facilitates decomposition of rational conformal motions into simpler motions.

Abstract

We introduce the Study variety of conformal kinematics and investigate some of its properties. The Study variety is a projective variety of dimension ten and degree twelve in real projective space of dimension 15, and it generalizes the well-known Study quadric model of rigid body kinematics. Despite its high dimension, co-dimension, and degree it is amenable to concrete calculations via conformal geometric algebra (CGA) associated to three-dimensional Euclidean space. Calculations are facilitated by a four quaternion representation which extends the dual quaternion description of rigid body kinematics. In particular, we study straight lines on the Study variety. It turns out that they are related to a class of one-parametric conformal motions introduced by L. Dorst in 2016. Similar to rigid body kinematics, straight lines (that is, Dorst's motions) are important for the decomposition of rational conformal motions into lower degree motions via the factorization of certain polynomials with coefficients in CGA.