© 2003 by Institute of Mathematics and its Applications
Shape-preserving interpolation by G1 and G2 PH quintic splines
1 Department of Mechanical and Aeronautical Engineering, University of California, Davis, CA 95616, USA 2 Dipartimento di Matematica, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy 3 Dipartimento di Energetica, Università di Firenze, Via Lombroso 6/17, 50134 Firenze, Italy
The interpolation of a planar sequence of points p0, ..., pN by shape-preserving G1 or G2 PH quintic splines with specified end conditions is considered. The shape-preservation property is secured by adjusting ‘tension’ parameters that arise upon relaxing parametric continuity to geometric continuity. In the G2 case, the PH spline construction is based on applying Newton–Raphson iterations to a global system of equations, commencing with a suitable initialization strategy—this generalizes the construction described previously in Numerical Algorithms 27, 35–60 (2001). As a simpler and cheaper alternative, a shape-preserving G1 PH quintic spline scheme is also introduced. Although the order of continuity is lower, this has the advantage of allowing construction through purely local equations.
Key Words: Pythagorean-hodograph curves; shape-preserving spline interpolation; geometric continuity; tension parameters; Newton–Raphson method
Received 7 December 2001. Revised 21 May 2002.