Study On The Construction Of G~2 Continuous Cubic PH Transition Curves

In CAGD,due to the need of geometric splicing,smooth splicing between plane curves is often involved,especially between straight lines,straight lines and arcs,and smooth splicing between circular arcs and arcs.The splicing has a wide application in solving practical problems and scientific research.For example,road trajectory design,robot walking route,aesthetics and satellite trajectory design.In general,we construct G~2 continuous transition curve.The curve is requested to keep parallel to the tangent vector of the original curve at the two ends and the curvature equal.And the curve should be as smooth as possible,which requires the curvature of the curve to be monotonous.If it could not be satisfied,it is also required to have as few extreme points of curvature as possible.This paper discusses and studies the transition curve between two arcs,which is mainly divided into three parts.The first part mainly introduces the construction methods of transition curves,including Clothoid line and Béizer curve and PH curve.The second part is the work of this paper.The cubic PH transition curve is constructed between two circles that are mutually exclusive.This transition curve is G~2 continuous C-type transition curve.The control polygon satisfies the certain geometric characteristic conditions.And the internal curvature of the curve contains only one extreme point of curvature.And the parameter range,existence and generation algorithm of C-type transition curve are given.Finally,two numerical examples are given to illustrate the feasibility and effectiveness of the method.The third part is another part of this paper.By using the definition of cubic PH curve,the C-type transition curve between two mutually contained circles is constructed.The generation algorithm of cubic PH curve is given.And a numerical example generated by this algorithm is represented to illustrate the effectiveness of this method.