Controlling extremal Pythagorean hodograph curves by Gauss–Legendre polygons

The problem of constructing spatial Pythagorean hodograph (PH) curves with a given Gauss–Legendre polygon is addressed. For planar/spatial PH curves of degree 2n+1, the Gauss–Legendre polygon, which consists of n+1 edges, obtained by evaluating the hodograph at the nodes of the Gauss–Legendre quadrature, is the rectifying polygon, which has the same length as the PH curve. On the other hand, if a planar polygon with n+1 edges is given, there are 2ⁿ planar PH curves whose Gauss–Legendre polygon is the given polygon. We here generalize this result to the spatial PH curves. For a given spatial polygon with n+1 edges, we construct n parameter family of PH curves of degree 2n+1. Among those PH curves, we identify 2ⁿ extremal solutions by choosing the quaternion preimages of the hodograph to have the maximal or the minimal distances from the adjacent quaternion solutions. We show that the extremal PH curves are one possible generalization of 2ⁿ planar PH curves with the planar Gauss–Legendre polygon by proving the planarity condition: the extremal PH curves are planar if the provided polygon is planar.