G1 Hermite interpolation by Minkowski Pythagorean hodograph cubics

As observed by [Choi, H.I., Han, Ch.Y., Moon, H.P., Roh, K.H., Wee, N.S., 1999. Medial axis transform and offset curves by Minkowski Pythagorean hodograph curves. Computer-Aided Design 31, 59–72], curves in Minkowski space R2,1 are very well suited to describe the medial axis transform (MAT) of a planar domain, and Minkowski Pythagorean hodograph (MPH) curves correspond to domains, where both the boundaries and their offsets are rational curves [Moon, H.P., 1999. Minkowski Pythagorean hodographs. Computer Aided Geometric Design 16, 739–753]. Based on these earlier results, we give a thorough discussion of G1 Hermite interpolation by MPH cubics, focusing on solvability and approximation order. Among other results, it is shown that any analytic space-like curve without isolated inflections can be approximately converted into a G1 spline curve composed of MPH cubics with the approximation order being equal to four. The theoretical results are illustrated by several examples. In addition, we show how the curvature of a curve in Minkowski space is related to the boundaries of the associated planar domain.