Research On The Theory And Applications Of Exponential Polynomial Curves

In CAD/CAM modeling system, the Bézier method, B-spline method,NURBS method can describe and represent free curves and surfaces perfectly,however, NURBS method has some historical deficiency in the shape of the design,and in engineering quadratic curves and surfaces such as circles, ellipses andcatenaries can only be represented by approximate method. Considering the wideuse of C-Bézier and H-Bézier curves in CAD/CAM curve and surface modelingsystem, we do some research on curves and surfaces in the exponential space. Themain results are summarized as follows:1. In the algebraic and exponential space Γ_n=span{1,t,t²,…t^n-2,e^tw,e^-tw}（n≥2）, we firstly define two initial functions by introducing a non-zero complexparameter ω, and give the recurrence definition of degree-n Bézier-like basis byan integral approach, and then analyze the properties of Bézier-like basis.2. The Bézier-like curves in the above exponential space are defined, and theshape of curves can be adjusted by changing the value of parameterω. As anapplication, the geometric conditions for two Bézier-like curves to be ofG⁰-connection and G¹-connection are discussed. Using tensor product, we canconstruct biquadratic Bézier-like surface, and explore the G⁰-connection andG¹-connection conditions between two biquadratic Bézier-like surface.3. In the exponential space Γ_n, we construct E（Exponential）-Bézier basis byborrowing the idea from Bernstein basis, and analyze their properties, then definethe E-Bézier curves, which can serve to adjust the shape of curves in terms of thevalue of parameter ω, and analyze the basic properties.