The geometry of rational parameterized representations

Abstract

We study the projective space \(\mathbb{S}_n^d \) of univariate rational parameterized equations of degree d or less in real projective space \(\mathbb{P}^n. \) The parameterized equations of degree less than d form a special algebraic variety \(\mathcal{K}_1. \) We investigate the subspaces on \(\mathcal{K}_1 \) and their relation to rational curves in \(\mathbb{P}^n, \) give a geometric characterization of the automorphism group of \(\mathcal{K}_1 \) and outline applications of the theory to projective kinematics.