On Curvature Monotony for a PH Cubic Curve and Constructing Transition Curve

The PH curves used to construct transition curves are all quintic,a PH quintic curve not only has higher degree but also the determination of its relevant parameters involves the complicated system of non-linear equations.PH cubic curves with lower degree are introduced to avoid the above problems in CAGD.Firstly,a sufficient condition for the curvature of a PH cubic curve to be monotonous is derived by using the geometric property of the legs and the angles of PH cubic curve control polygon and combining with the expression of its curvature derivative.Based on the condition,we discuss how to construct transition curve of G² contact,between two circles with one circle inside the other by PH cubic curve.We compute an attainable range of the distance between the centers of two circles and prove that the transition curve is unique respectively,the algorithm steps for constructing transition curve are given.Compared with the existing methods in examples,the results show that our method is effective and practical.