Identifying Pythagorean-Hodograph Curves Closest to Prescribed Planar Bézier Curves. (August 2022) - Farouki, Rida T.
Abstract: The problem of identifying the planar Pythagorean-hodograph curve that is "closest" to a given Bézier curve, and has the same end points (or end points and tangents), is considered. The "closeness" measure employed in this context is the root-mean-square magnitude of the differences between pairs of corresponding control points for the two curves. The methodology is developed in the context of quintic PH curves, although it readily generalizes to PH curves of higher degree. Using the complex representation for planar curves, it is shown that this problem can be reduced to the minimization of a quartic penalty function in certain real variables, subject to two quadratic constraints, which can be efficiently solved by the Lagrange multiplier method. By expressing the penalty function and constraints in terms of variables that identify a complex pre-image polynomial, the closest solution is guaranteed to be a PH curve. Several computed examples are used to illustrate implementation of the optimization methodology and typical approximation results that can be obtained. Highlights: A methodology to identify the PH curve closest to a prescribed planar Bezier curve is developed. The methodology focuses on quintic PH curves, and can accommodate G 0 or G 1 end conditions. The identification of the closest PH curve is formulated as a constrained optimization problem. Computed examples are used to illustrate the implementation and performance of the method.