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In computer aided geometric design (CAGD) and computer graphics, it is a general manipulation to approximate a surface by triangulation mesh. Thus a key problem is to estimate the error of the approximation. So far, many papers have given various estimate bounds of the distance between a parametric patch of a C2 surface and an arbitrary triangle whose vertices are on the patch, but these estimates are all imperfect, some of them have large error, some of them have complicated representation formulae. By using a succinct new method, a sharp upper estimate of the maximum distance between a patch and a triangle is obtained and a strict proof is given. This is very valuable for CAGD.

In computer aided geometric design, the degree reduction of the parameter curve is a key technique in data exchange and data compression. The various existing methods of degree reduction cannot decide whether the degree reduction curve satisfying the given tolerance exists beforehand, cannot give approximation of the best multi-degree reduction, or cannot provide explicit expression and error formula of the degree reduction curve. In this paper, we propose an entirely new method, which can hurdle the above flaws completely. For a given Bexier curve of degree n, we can easily decide whether a Bezier reduction curve of degree m exists, which has equal derivatives with the given curve up to (r−1)-th and (s−1)-th orders (r, s≤m