On control polygons of Pythagorean hodograph septic curves
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摘要
For a given septic Bézier curve with a distinct ordered sequence of control points, how to determine whether it is a PH curve via exact symbolic computation in theory. This problem motivated the study of a necessary and sufficient condition for a planar septic Bézier curve to possess a Pythagorean hodograph (PH). Based on the definition of a PH curve and the complex representation of a planar curve, we develop geometric conditions in terms of the leg-lengths and angles of a control polygon that must be separated to guarantee the PH property. The relation between the compatibility of solutions with respect to the complex coefficients of PH equations and geometric constraints is analyzed. Moreover, PH septic curves with inflections are extended to construct S-shaped transition curves.
论文关键词:Septic,Pythagorean hodograph,Bézier curves,Control polygon,Compatibility,Transition curves
论文评审过程:Received 1 August 2014, Revised 2 September 2015, Available online 25 September 2015, Version of Record 11 November 2015.
论文官网地址:https://doi.org/10.1016/j.cam.2015.09.006