$G^{0}$ Pythagorean-Hodograph Curves Closest to Prescribed Planar B\'{e}zier Curves
Received:May 19, 2023  Revised:August 14, 2023
Key Words: Pythagorean-hodograph curves   Gauss-Legendre polygon   Gauss-Lobatto polygon   constrained optimization   Lagrange multiplier   Newton-Raphson iteration  
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.11801225).
Author NameAffiliation
Wenqing FEI School of Mathematical Sciences, Jiangsu University, Jiangsu 212000, P. R. China 
Yongxia HAO School of Mathematical Sciences, Jiangsu University, Jiangsu 212000, P. R. China 
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      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.
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