Minkowski Pythagorean Hodographs

The Minkowski Pythagorean hodograph curves — or MPH curves for brevity — were first introduced by H. P. Moon [328–330]. Their distinctive feature is that the hodographs satisfy a Pythagorean condition under the metric of the Minkowski space ℝ^2,1 with two “space-like” and one “time-like” coordinates, rather than the Euclidean metric of ℝ³ with three space-like coordinates. The motivation for the definition of MPH curves is that they allow the medial axis transform of a planar domain to be specified in such a way that the domain boundary is exactly describable by rational curves. The medial axis transform (MAT) represents a planar domain as the union of a one-parameter family of variable-radius disks, and the problem of constructing the domain boundary from the MAT is a generalization of the problem of constructing offset curves, which correspond to envelopes of families of fixed-radius disks. Rationality of the domain boundary is ensured by requiring the components of a polynomial MAT to satisfy a Pythagorean condition under the metric of ℝ^2,1.