Geometric Theory, Algorithm And Application Of A Class Of Αβ-bézier Curves

Spline curves with shape parameters have stronger expressive power than traditional spline curves,and constructing curve theory with good properties is of great significance in the field of CAD/CAM.A class of cubic with two shape parameters based on endpoint constraints αβ-Bézier curves have the same excellent properties as traditional cubic Bézier curves.This type of curve has independent shape parameters.Without changing the control vertices of the curve,changing the shape parameter values can achieve the adjustment of the curve shape.However,there are still some shortcomings in the theory and application of this type of curve,such as the degree of curve is limited to cubic curves,and there is no relevant algorithm.Based on this situation,this paper investigates a class of cubic αβ-Bézier curve with two shape parameters has been studied,including the following research contents and achievements:(1)Extended cubic αβ-Bézier curve to degree-n αβ-Bézier curve,discussing the influence of shape parameters on the curve.By giving definition of recursive relationships of Bézier basis functions,and then define degree-n αβ-Bézier curve,defined to the degree-n αβ-Bézier curves have excellent properties like traditional degree-n Bézier curves.Finally,using the relationship between degree-n αβ-Bézier curves and traditional degree-n Bézier curves,three situations of the influence of shape parameters on the shape of degree-n αβ-Bézier curves are given,the examples show that the curve can effectively modify the shape of the curve.(2)Given a geometric drawing algorithm for degree-n αβ-Bézier curves.According to recursive relation of degree-n αβ-Bézier basis function,the geometric drawing algorithm of degree-n αβ-Bézier curve is obtained accordingly.The algorithm is implemented through a series of weighting operations on control vertices,which is simpler and more efficient than the method based on directly calculating a point on a curve through a curve expression.A series of calculation examples also demonstrate the feasibility and effectiveness of the algorithm.(3)For degree-n αβ-Bézier curves presents a segmentation algorithm.Firstly,a condition for the segmentation algorithm based on geometric drawing algorithm to hold is given.Validated the degree-n αβ-Bézier curve does not satisfy the segmentation algorithm based on geometric drawing algorithms.For this situation,combine the relationship between degree-n αβ-Bézier curves and traditional degree-n Bézier curves A segmentation algorithm based on the recombination of control vertices is proposed,and the expression of the sub curve is given after the segmentation of the degree-n αβ-Bézier curves.Finally,an example is given to demonstrate the feasibility of the algorithm.(4)Studied an algorithm for upgrading the degree of degree-n αβ-Bézier curves.By giving the relationship between αβ-Bézier higher degree basis functions and lower degree basis functions gives the algorithm for upgrading lower degree αβ-Bézier curves to higher degree curves is presented.Finally,a concrete example of curve upgrading is given,which shows that the algorithm can achieve the degree-n degree elevation of αβ-Bézier curves.