The L\-2 Distances for Rational Surfaces Based on Matrix Representation of Degree Elevation

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 L\-2 distance between rational curves (surfaces) based on matrix representation of degree elevation is detailedly studied. Firstly, the L\-2 distance for two rational Bézier surfaces which is based on the degree elevation is presented. Then, using the L\-2 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 L\-2 distance for rational Bézier surfaces is generalized to that for rational B-spline surfaces. All the formulas for the L\-2 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.