Deformation of spatial septic Pythagorean hodograph curves using Gauss–Legendre polygon

The notion of the rectifying control polygon of planar Pythagorean hodograph (PH) curve is extended to the spatial PH curves. We construct the Gauss–Legendre polygon by applying the Gauss–Legendre quadrature to the Pythagorean hodograph. Since the parametric speed of a PH curve is a polynomial, the Gauss–Legendre polygon with enough edges has the rectifying property, that is, the length of the polygon is the same as the length of the spatial PH curve. We also present the method to compute septic PH curves with the given Gauss–Legendre polygon. This method can be used to develop the deformation algorithm for the spatial septic PH curves.