$G^{2}$ Blending of Cubic Pythagorean Hodograph Curves with Prescribed Total Arc Length

Abstract:

      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.