Rational Motions with Generic Trajectories of Low Degree
This work addresses a specific theoretical problem in kinematics and geometry, providing incremental insights into rational motion properties.
The paper tackled the problem of rational motions with lower-than-expected trajectory degrees by identifying algebraic and geometric criteria for degree reduction, enabling systematic construction of such motions and explaining differences in trajectory degrees between a motion and its inverse.
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.