Rational Motions of Minimal Quaternionic Degree with Prescribed Line Trajectories

TL;DR

This paper develops a method to find minimal-degree rational motions using dual quaternions that move lines along prescribed rational ruled surfaces, extending understanding from point to line trajectories.

Contribution

It introduces explicit formulas and conditions for constructing minimal-degree rational motions with line trajectories, a problem not previously studied.

Findings

Explicit formulas for line-moving rational motions

Conditions for existence and uniqueness of solutions

Applications to mechanism design

Abstract

In this paper, we study how to find rational motions that move a line along a given rational ruled surface. Our goal is to find motions with the lowest possible degree using dual quaternions. While similar problems for point trajectories are well known, the case of line trajectories is more complicated and has not been studied. We explain when such motions exist and how to compute them. Our method gives explicit formulas for constructing these motions and shows that, in many cases, the solution is unique. We also show examples and explain how to use these results to design simple mechanisms that move a line in the desired way. This work helps to better understand the relationship between rational motions and ruled surfaces and may be useful for future research in mechanism design.