Gauss–Lobatto polygon of Pythagorean hodograph curves

Abstract

The Gauss–Legendre polygon of Pythagorean hodograph (PH) curves can be used as the rectifying control polygon, which has (i) the end point interpolation property, (ii) the rectifying property, and (iii) the same degree of freedom as the PH curve. These properties make the Gauss–Legendre polygon a nice tool to control the shape of the PH curve. A drawback of the Gauss–Legendre polygon is that it does not determine the end tangent vectors. In this paper, we introduce the Gauss–Lobatto polygon as an alternative to the Gauss–Legendre polygon. Since the Gauss–Lobatto quadrature has the end points as the predetermined nodes, the Gauss–Lobatto polygon naturally determines the end tangent vectors of the PH curve. We analyze the rectifying property of the Gauss–Lobatto polygon for both planar and spatial PH curves. Concerning the degree of freedom, we show that the Gauss–Lobatto polygons of planar PH curves are not the rectifying control polygons. For spatial PH curves, we identify the relation between the degree of a PH curve and the number of edges in the Gauss–Lobatto polygon that makes the Gauss–Lobatto polygon a rectifying control polygon. We also provide the method to compute the spatial septic PH curve with the given Gauss–Lobatto polygon, and the algorithm for the deformation of the spatial septic PH curves.