Shape-preserving interpolation by G^1 and G^2 PH quintic splines

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.