On a tolerance problem of parametric curves and surfaces (Q1195069)

A ground problem in Computer Aided Geometric Design is handling tolerances. Thus, a common instance of this problem is the following: given a parametric surface \(S\subset\mathbb{R}^ 3\) defined over a domain \(D\) and a tolerance value \(\varepsilon\), how to choose \(\delta\) in \(D\) such that a parameter change smaller than \(\delta\) will produce a variation smaller than \(\varepsilon\) in the surface? This kind of question is studied in the form of bounds for the modulus of continuity of a function. For a function \(f\) defined on [0,1] and a real value \(h\) its modulus of continuity is defined to be the number \[ \omega(f,h)=\sup\{| f(u)-f(v)|:0\leq u,\quad v\leq 1,\;| u- v|\leq h\} \] For parametric polynomial curves, i.e. curves whose coordinates are given by polynomial functions the author shows the following bounds. Given a polynomial by its Bézier representation \(p(u)=\sum^ n_{i=0}d_ ib_{n,i}(u)\) and a real number \(h\) we have that \[ h[1-(n-1)^ 2h]n\max| d_{i+1}-d_ i|\sigma_{n- 1}\leq\omega(p,h)\leq hn\max| d_{i+1}-d_ i| \] where \(\sigma_ n\sim 2^{1/2-n}\). For the case of parametric rational curves (this time the coordinates are given by rational functions) the author first gives a counterexample showing that a preceding upper bound given by \textit{A. P. Rockwood} [IEEE Comput. Graphics Appl. 1987, 7, No. 8, 15-26 (1987)] was erroneous and then provides a new upper bound having a rather complicated form but being computable. Finally, these results are extended to the case of parametric surfaces.