The Shape Adjustable Transition Curve And The Construction Of The Triangle Bézier Surface

The property of the transition curve is closely related to the property of the potential function chosen.In order to construct a transition curve that can reach Ck(k is an arbitrary natural number)continuity at both endpoints,it is necessary to find the potential function with the corresponding property first.Considering the simplicity of calculation,the construction problem of potential function is solved in polynomial function space firstly.In order to deduce the potential function that conforms to the expectation,starting from the Ck continuity conditions at the endpoint of the transition curve,the expected properties of the potential function are obtained.According to the conditions,the potential function is represented as a linear combination of Bernstein basis functions with the combined coefficients to be determined.The value of the coefficient to be determined is calculated according to the properties of the Bernstein basis functions at 0 and 1.A general expression of the potential function that meets the expected nature is obtained,which does not contain any free parameters.Further,in order to make the shape of the resulting transition curve can be adjusted freely,the expression of the existing potential function is firstly ascended,and then the shape parameters are introduced in it,so as to obtain an expression that enables the transition curve to satisfy Ck(kis an arbitrary natural number)at the endpoint continuous and the shape can be freely adjusted.In order to further enhance the continuous order of the transition curve at the endpoint,the two potential functions obtained will be triangularized,and two new potential functions are obtained.The properties of the derived potential function and the characteristics of the corresponding transition curve are analyzed comprehensively,and the correctness of the theoretical analysis results is verified by using the legend of the potential function and transition curve.In addition to the study of transition curves,this paper also defines a new surface with the same properties and the same structure as the cubic triangle BezierBezier surface by introducing new parameters in the control points.As a result of the introduction of new parameters,it is possible to adjust the surface shape and surface face to control the approximation of the mesh by changing the value of the parameters without changing the control points.In order to discuss the connection problem of the new surface,the traditional Bezier triangle expression of the new surface is analyzed,and the G1 smooth connection conditions of the new surface are given,which have clear geometric significance.The geometric iteration of the new surface is discussed,when the parameterization is uniform,the geometric iteration algorithm is convergent,and the convergence speed is accelerated with the increase of parameter value.