Surface algorithms using bounds on derivatives

296 D. Filip et aL / <strong>Surface</strong> <strong>algorithms</strong>
and a surface normal ratio factor to try to determine surface flatness. The theory in this paper
describes how to determine these factors very precisely.
In Section 2 we develop the general theory needed, in Section 3, we give applications of the
general theory, and in Section 4 we show how to compute the upper <strong>bounds</strong> needed for the
<strong>algorithms</strong>. Section 5 derives another theorem to improve the <strong>algorithms</strong> in certain cases.
2. General theory
In this section we present some new and old approximation theoretic results for curves and
surfaces.
2.1. Approximating curves
Wang used the following result for linearly approximating curves, which is proved in [de
Boor '78, p. 39]:
Theorem 1. Let f:[a, b]~ R be any C 2 function and let l(x)
l(a) =f(a) and l(b)=f(b). Then
sup If(x)-l(x)l