Study On Weighted Lupa（？） Q-Bézier Curve And Surface

Curve and surface modeling is the core content of Computer Aided Geometer De-sign. Classical B′ezier method lays the foundation of parametric curve and surface, whichis the main representation of curves and surfaces. Totally positivity of basis functionoccupies an important position, it is highly related with variation diminishing propertyand shape preserving property. This paper constructs generalized rational B′ezier curvesand surfaces based on Lupa?s -analogue of Bernstein operator and totally positivity basis.The constructed curve and surface is called weighted Lupa?s -B′ezier curve and surface.This curve and surface can represent conic sections and quadric surfaces, accurately. Inthis paper, the main contributions include:Firstly, this paper demonstrates that Lupa?s -analogue of Bernstein functions are thenormalized totally positivity(NTP) basis of a rational space which is spanned by a set ofLupa?s -analogue of Bernstein functions with common denominator. Moreover, accordingto properties and conclusions of totally positive basis, we validate variation diminishingproperty, convexity and monotonicity of Lupa?s -analogue of Bernstein operator. Lupa?s-analogue of Bernstein basis function can be generated to weighted Lupa?s -analogueof Bernstein function by adding positive weights. Using the above mentioned method,we prove that weighted Lupa?s -analogue of Bernstein function is the normalized totallypositivity basis of a rational space. Meanwhile, this paper also validates the correctnessof the conclusion by means of the definitions of the rational Extented Chebyshev spaceand its normalized totally positivity(NTP) basis.Secondly, this paper uses weighted Lupa?s -analogue of Bernstein basis function toconstruct the corresponding weighted Lupa?s -B′ezier curve, and this curve can representconic sections, accurately. Weighted Lupa?s -B′ezier curve can be reduced to classicalB′ezier curve and Lupa?s -B′ezier curve, and it has geometric properties, such as a?neinvariance, convex hull property, reducibility and so on. Using the perspective projectiontransformation, the paper gives the homogeneous coordinate expression, the correspond-ing degree elevation and de Casteljau algorithm of this curve. Weighted Lupa?s -B′eziercurve contains two kinds of shape parameter, weights and -integer. Weights have theimportant geometric meanings of cross-ratio. Numerical examples show that weightedLupa?s -B′ezier curve have better shape-preserving property than classical rational B′eziercurve and rational Phillips -B′ezier curve. In order to design complex curves, this paperalso gives the smooth blending conditions of the two weighted Lupa?s -B′ezier curves fromparametric continuity and geometric continuity of curves, respectively.Finally, the paper constructs tensor product weighted Lupa?s -B′ezier surface overthe rectangular domain, and this surface can represent quadric surfaces, accurately. Thissurface has good geometric properties, such as a?ne invariance, convex hull propertyand so on. The degree elevation and de Casteljau algorithm are established based on thetechnique of homogeneous coordinate. Weights have the important geometric meanings ofcross-ratio. Numerical examples illustrate the e?ect of the two kinds of shape parameters,weights and -integer. Further more, this paper gives the condition of 1continuitybetween adjacent weighted Lupa?s -B′ezier surfaces.