# Rational Motions with Generic Trajectories of Low Degree

## TL;DR

This paper investigates special rational motions with lower-than-expected trajectory degrees, providing algebraic and geometric criteria for their characterization and construction, and explaining degree differences between motions and their inverses.

## Contribution

It introduces algebraic and geometric criteria for identifying and constructing rational motions with reduced trajectory degrees, advancing understanding of their properties.

## Key findings

- Criteria for degree drop involve right factors and rulings on an invariant quadric.
- Systematic methods for constructing such motions are developed.
- It explains why a motion and its inverse can have different trajectory degrees.

## Abstract

The trajectories of a rational motion given by a polynomial of degree n in the dual quaternion model of rigid body displacements are generically of degree 2n. In this article we study those exceptional motions whose trajectory degree is lower. An algebraic criterion for this drop of degree is existence of certain right factors, a geometric criterion involves one of two families of rulings on an invariant quadric. Our characterizations allow the systematic construction of rational motions with exceptional degree reduction and explain why the trajectory degrees of a rational motion and its inverse motion can be different.