$G^{0}$ Pythagorean-Hodograph Curves Closest to Prescribed Planar B\'{e}zier Curves

Abstract:

      The task of identifying the quintic PH curve $G^{0}$ ``closest'' to a given planar B\'{e}zier curve with or without prescribed arc length is discussed here using Gauss-Legendre polygon and Gauss-Lobatto polygon respectively. By expressing the sum of squared differences between the vertices of Gauss-Legendre or Gauss-Lobatto polygon of a given B\'{e}zier and those of a PH curve, it is shown that this problem can be formulated as a constrained polynomial optimization problem in certain real variables, subject to two or three quadratic constraints, which can be efficiently solved by Lagrange multiplier method and Newton-Raphson iteration. Several computed examples are used to illustrate implementations of the optimization methodology. The results demonstrate that compared with B\'{e}zier control polygon, the method with Gauss-Legendre and Gauss-Lobatto polygon can produce the $G^{0}$ PH curve closer to the given B\'{e}zier curve with close arc length. Moreover, good approximations with prescribed arc length can also be achieved.