On G2 interpolation by degree seven Minkowski Pythagorean-hodograph spline curves

It is well established that curves in Minkowski space R2,1 are particularly well suited for representing the medial axis transform (MAT) of planar domains. In particular, Minkowski Pythagorean-hodograph (MPH) curves correspond to domains whose boundaries and offset curves are all rational (Moon, 1999). Motivated by recent results on G2 Hermite interpolation by spatial PH curves (Knez et al., 2024), this paper extends that framework through the use of Clifford algebras. In more detail, we present the construction of a G2 continuous MPH spline interpolant of degree seven, equipped with a rational G1 continuous Lorentz orthonormal frame, under the additional constraint of a prescribed Minkowski length. To ensure applicability to arbitrary interpolation data, each spline segment is formed as a biarc composed of two polynomial MPH curves that match the geometric Hermite data at the endpoints. The proposed method is local and yields a closed-form solution with additional free parameters, where appropriate guidelines for their selection are given. Furthermore, two methods are proposed for determining the input frame data, where one is based on discrete point and derivative information, and the other introduces a novel approach that employs rotation-minimizing Lorentz frames, computed numerically via a generalization of the double reflection method (Wanget al., 2008) to Minkowski space. Several numerical experiments are included to demonstrate the effectiveness of the derived interpolation scheme and to show that the proposed parameter selection and frame data computation result in spline approximants with a high approximation order.