基于升阶矩阵的有理曲面之间L_2距离计算

The L_2 Distances for Rational Surfaces Based on Matrix Representation of Degree Elevation

计算曲线曲面之间的距离是几何设计与几何逼近的一个重要课题,如估计有理曲线曲面的降阶逼近和多项式逼近的误差时,需要一种简洁有效的方法来计算原曲线曲面和逼近曲线曲面间的距离.首先给出了基于升阶矩阵的两张有理B啨zier曲面的L2距离表示,然后利用这个L2距离表示和最小二乘法,对有理B啨zier曲面多项式逼近的误差作了明确而统一的度量.最后,基于Bernstein基与B样条基的相互转换,把有理B啨zier曲线曲面的L2距离表示简洁地推广到有理B样条曲线曲面.所得到的几个计算曲线曲面之间的L2距离的公式均可通过矩阵运算表示,十分利于程序的实现,有应用价值.最后还给了几个实例.

Computing the distance between curves（surfaces） is an important subject in the computer aided geometric design and the geometric approximation.For example,when estimating the errors for the approximation of rational curves（surfaces） by degree reduction or polynomial curves（surfaces）,the distance between the original curves（surfaces） and the approximating curves（surfaces） need to be calculated by an efficient way.In order to give a uniform measure for the distance between curves（surfaces）,the L2 distance between rational curves（surfaces） based on matrix representation of degree elevation is detailedly studied.Firstly,the L2 distance for two rational Bézier surfaces which is based on the degree elevation is presented.Then,using the L2 distance and the least-squares method,a clear and uniform measure for errors in the polynomial approximation of rational Bézier surfaces is derived.What is more,based on the conversion between Bézier bases and B-spline bases,the L2 distance for rational Bézier surfaces is generalized to that for rational B-spline surfaces.All the formulas for the L2 distance between curves and surfaces in this paper are presented through matrix operations,which is convenient for computer programs,so they are applicable and useful in practice.Finally,several examples are presented.