Linear computational approach to interpolations with polynomial Minkowski Pythagorean hodograph curves

Abstract

Methods using Pythagorean hodographs both in Euclidean plane and Minkowski space are often used in geometric modelling when necessary to solve the problem of rationality of offsets of planar domains. A main justification for studying and formulating approximation and interpolation algorithms based on the called Minkowski Pythagorean hodograph (MPH) curves is the fact that they make the trimming procedure of inner offsets considerably simpler. This is why one can find many existing techniques in literature. In this paper a simple computational approach to parametric/geometric Hermite interpolation problem by polynomial MPH curves in R 2 , 1 is presented and an algorithm to construct such interpolants is described. The main idea is to construct first not a tangent but a normal vector space satisfying the prescribed MPH property that matches the given first order conditions, and then to compute a curve possessing this constructed normal vector space and satisfying all the remaining interpolation conditions. Compared to other methods using special formalisms (e.g. Clifford algebra), the presented approach is based only on solving systems of linear equations. The results are confirmed by number of examples.