Hermite interpolation of space curves using the symmetric algebra

Abstract

Conditions are presented which describe the construction of a degree 5 Bezier curve which interpolates given positional, tangent, curvature, and torsion data at end points. The result is extended to provide technique for interpolation of a given curve by a sequence of Bezier segments that interpolate positional, tangent, curvature, and torsion data associated with intermediate points on the curve. It is shown that the order of approximation is O(h8), 0 < h < 1. This corroborates a conjecture of Hollig and Koch. Techniques which simplify calculation are used. The techniques involve the consideration of a class of fundamental Bezier curves which lie in the graded symmetric algebra associated with R2.