Research On The Transition Curve Of Bézier-like Curve

Transition curve is the middle curves between two curves,although it looks like be simple,actually it is useful,such as it have a wide range of applications in highway and railway design,the design of the profile of gear and other industrial fields.Some traditional spiral,such as clothoid and Bloss spiral,is principal tools as transition curve,but their parameter expression mostly contain Fresnel integral and it is very inconvenient that we only get their numerical solution in practical applications.At the earliest,in order to solve this problem,some scholars utilizes the Bézier and PH transition curve to replace them,in recent years,with the development of the modeling theory of Bézier-like curve,using Bézier-like curve instead of traditional spiral as transition curve in the practical application of research become a hot issue.Basing on the modeling theory cubic T-Bézier curve and cubic H-Bézier curve,This paper construct a cubic T-Bezier transition curve and H-Bézier transition curve between the two circular arcs,which their shape is S type and C type.For cubic T-Bézier transition curve between the two circular arc,firstly we construct a cubic T-Bézier spiral whose has zero curvature in the starting point,and then using a pair of the spiral to transit between two circular arc.Because the curvature derivative about arc length is zero in the starting and ending point,it can replace the Bloss spiral as the transition curve in the high-speed way design.For the cubic H-bezier transition curve between two arcs,as it is a transition between two circular arc with a cubic H-Bézier curve,it is simpler than using a pair of curve to be the transition.In addition,for requirement of curve smoothness,when it is the S-shaped transition curve between the two circular arc,it's curvature monotone decreasing and when it is C type transition curve,it's contains only one curvature extremum points.The paper gives their sufficient conditions and the detail derivation process to meet the above requirements respectively.In conclusion,we found that S type H-Bézier transition curve between the two circular can satisfy the bigger restriction of two circle radius for its curvature monotone decreasing than other the transition curve in documents before,so it has higher application value.