On [formula omitted] rational motions of degree six

Abstract

In this paper two methods for constructing a C2 rational spline motion of degree six that interpolates given positions are presented. The first method requires the solution of a global linear system of equations and each rational motion segment depends on all the input data. The matrix of the system that determines the rotational part includes some extra free parameters that influence the motion and can be chosen such that the unique solution is guaranteed. The second method that is presented splits the construction of the spline motion to individual local biarc segments which beside the given positions interpolate also some additional derivative information. For this local method it is essential to use the extra degrees of freedom coming from the equivalence relation in the 3-dimensional projective space as well as the biarc construction in order to reduce the degree of the motion to six. Remaining free parameters that arise in the rotational part are examined in detail and two exceptional cases where the solution does not exist, together with possible remedies, are identified. Different possibilities for the construction of the translational part are proposed too. The theoretical results are substantiated with numerical examples.