Using algebraic and geometric methods, functional relationships between a point on a conic segment and its corresponding parameter are derived when the conic mappings of the mappings represented by the expressions of rational conic segments are given. These formulae relate some triangular areas or some angles, determined by the selected point on the curve and the control points of the curve, as well as by the weights parametric angles of the selected point and two endpoints on the conic segment, as
This paper introduces the algebraic property of bivariate orthonormal Jacobi polynomials into geometric approximation. Based on the latest results on the transformation formulae between bivariate Bernstein polynomials and Jacobi polynomials, we naturally deduce a novel algorithm for multi-degree reduction of triangular Bézier surfaces. This algorithm possesses four characteristics: ability of error forecast, explicit expression, less time consumption, and best precision. That is, firstly, whether there exists a multi-degree reduced surface within a prescribed tolerance is judged beforehand; secondly, all the operations of multi-degree reduc- tion are just to multiply the column vector generated by sorting the series of the control points of the original surface in lexicographic order by a matrix; thirdly, this matrix can be computed at one time and stored in an array before processing de- gree reduction; fourthly, the multi-degree reduced surface achieves an optimal ap- proximation in the norm L2. Some numerical experiments are presented to validate the effectiveness of this algorithm, and to show that the algorithm is applicable to information processing of products in CAD system.
