The Rational Motion of Minimal Dual Quaternion Degree With Prescribed Trajectory

TL;DR

This paper proves the existence of a unique minimal-degree rational motion in dual quaternion space that follows a prescribed rational trajectory, with the degree depending on the trajectory's properties.

Contribution

It provides a constructive proof for minimal rational motions with prescribed trajectories, linking motion degree to the trajectory's degree and circularity.

Findings

Minimal motion degree equals trajectory degree minus circularity.

Minimal degree is lower than that of trivial curvilinear translation for circular curves.

Unique rational motion exists for given rational trajectories.

Abstract

We give a constructive proof for the existence of a unique rational motion of minimal degree in the dual quaternion model of Euclidean displacements with a given rational parametric curve as trajectory. The minimal motion degree equals the trajectory's degree minus its circularity. Hence, it is lower than the degree of a trivial curvilinear translation for circular curves.