Two Kinds Of Geometric Approximation And Two Kinds Of Algebraic Representation Of Curves And Surfaces

Approximation and representation of curves and surfaces are the two fundamental theoretical questions in computer aided geometric design(CAGD).Therein approximation of degree reduction and derivatives for curves and surfaves,rational representation of conics and boundary representation of ball surface have become one of the hotspots of investigation, since they directly relate to the function,quality,precision,and efficiency of geometric design systems.However,up to now there is still no theoretical breakthrough in these areas.Facing these challenges,we take applied mathematics as a tool and rely on modern industry as the background to do an in-depth study.Then we overcome these key technological problems and provide a series of convenient and efficient geometric algorithms.These abundant and innovative results are represented as follows:1.In respect of degree-reduced approximation of surfaces:we discover a sharp tool, triangular Jacobi basis,to uniformly achieve explicit,optimal,and constrained multi-degree reduction of triangular surfaces,and then apply it into algorithm design.The algebraic property of triangular orthonormal Jacobi polynomials is introduced into geometric approximation.And based on the transformation formulae between bivariate Bernstein basis and bivariate Jacobi basis,a simple and intuitive algorithm is deduced for multi-degree reduction of triangular Bézier surfaces with or without corner constraints. The algorithm has following four advantages which known methods for multi-degree reduction of surfaces cannot own at the same time.That is,ability of error forecast, explicit expression,less time consumption,and best precision.It means,firstly,we can judge beforehand that whether there exists a multi-degree reduced surface within a prescribed tolerance in order to avoid useless operations;Secondly,all the operations of multi-degree reduction are just to multiply the column vector generated by arranging the series of control points of the surface in iexicographic order by a simple matrix;Thirdly, this matrix can be computed at one time and then stored in database before processing degree reduction to be summon for using at any time;Fourthly,the multi-degree reduced surface achieves an optimal approximation in the norm L₂.Specially,for the case of corner constraints,this algorithm can keep the boundary curves of degree-reduced surface high order continuity at corners.Also applying Foley-Opitz averaging scheme,the piecewise degree-reduced patches possess global C¹ continuity.Combined with surface subdivision technique,it will suit better for modeling demands in CAGD.2.In respect of degree-reduced approximation of curves:we invent an algebraic method combining generalized inverse matrix with partitioned matrix and an optimized method combining orthonormal basis operations with constrained quadratic programming.Then we achieve constrained multi-degree reduction of parametric curves or disk curves with high precision and high efficiency.As for the Said-Bézier generalized Ball(SBGB) curve, we deduce a matrix for its degree elevation,and present some formulae for its derived vectors at two endpoints and the corresponding matrix representations according to the piecewise expressions of SBGB basis.Based on these matrices,we apply the principles of both generalized inverse matrix and partitioned matrix to deduce an explicit algorithm for multi-degree reduction of SBGB curves with endpoints constraints of arbitrary order continuity.As for the disk Bézier curve,applying the orthogonality of Jacobi polynomials, we find out an optimal multi-degree reduced polynomial approximation of the center curve of the original disk Bézier curve in the norm L₂,and regard it as the center curve of the degree-reduced disk Bézier curve;Next,applying the conversion formulae between Bernstein basis and Legendre basis as well as the orthogonality of Legendre basis,we transform the task,optimally multi-degree reduced approximating the error radius curve of the disk Bézier curve,into solving a constraned quadratic programming(QP) problem. Both above-mentioned techniques have entirely three fine characters,easy processing, high precision and high speed.3.In respect of derivative approximation of triangular surfaces:we find two acuminous implements,degree elevation and difference operators,to successfully deduce the results for derivative approximation of triangular parameter surfaces.Applying a series of identical formulae and abbreviated notations also combined with refined inequality skills, we derive sharp bound estimates on first and second partial derivatives of rational triangular Bézier surfaces.We also prove that the new bounds are superior to the known ones in terms of precision and validity.The results have practical value for promoting and reinforcing functions in geometric design systems.4.In respect of rational representations of conics:we present a new idea to study geometric characters of rational quartic Bézier conics according to two kinds of algebraic class conditions,i.e.,degree-reducible and improperly parameterized.We first regard rational quartic Bézier conics as two special kinds,i.e.,degree-reducible and improperly parameterized;and then on the basis of strictly analyzing the algebraic quantity of linear convex combinations and geometric quantity of triangle areas,we derive the necessary and sufficient conditions for rational quartic Bézier representation of conics.They can be divided into two parts:Bézier control points and weights.By using these conditions,two algorithms are provided to construct and design rational conics.One is to judge whether a rational quartic Bézier curve is a conic section,and which type it is.Another is to present positions of the control points and values of the weights of a given conic section in form of a rational quartic Bézier curve.These results not only enrich the theory of CAGD,but also enlarge applied range of geometric modeling and geometric design systems.Based on these results,we apply the transformation relations between lower degree Bernstein basis and Said-Ball basis or DP-NTP basis to deduce the necessary and sufficient conditions for rational lower degree Said-Ball or DP-NTP representations of conics respectively.Also the corresponding algorithms for curve modeling are given.5.In respect of boundary representation of ball surfaces:a new analytical method is created which is combining the envelop theory in differential geometry and the orthogonality of Legendre algebraic expression.Recurring to the envelope theory of the family of bivariate parametric surfaces,spherical coordinates and Cramer's rule,we give an explicit representation of the exact boundary of a ball Bézier surface.Then according to the orthogonality of Legendre polynomials,the exact boundary of the ball Bézier surface is optimal squarely approximated in polynomial form.Furthermore,by the conversion formulae between Legendre basis and Bernstein basis,the approximate boundary of the ball Bézier surface is expressed in Bernstein form.It is more suited for use in shape design systems.