$G^{2}$ Blending of Cubic Pythagorean Hodograph Curves with Prescribed Total Arc Length
Received:July 13, 2020  Revised:November 15, 2020
Key Word: Pythagorean-hodograph curve   $G^{2}$ blending   prescribed arc length   nonlinear equations  
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11801225), University Science Research Project of Jiangsu Province (Grant No.18KJB110005) and the Research Foundation for Advanced Talents of Jiangsu University (Grant No.14JDG034).
Author NameAffiliation
Yongxia HAO School of Mathematical Sciences, Jiangsu University, Jiangsu 212000, P. R. China 
Lianxing LIAO School of Mathematical Sciences, Jiangsu University, Jiangsu 212000, P. R. China 
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      Pythagorean-hodograph (PH) curve is widely used in curve modeling because of its advantages in arc length and equidistant curve calculation. This paper discusses the $G^{2}$ continuous blending of cubic PH curves under total arc length constraint. Given three points including two end control points and a joint point, construct two cubic PH curves such that they interpolate the end control points and are $G^{2}$ continuous at joint point with prescribed total arc length. It can also be regarded as a curve extension problem. According to the arc length formula of cubic PH curve and the condition of $G^{2}$ blending, the problem is transformed into a constrained minimization problem. Several examples are served to illustrate our method.
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