Surface algorithms using bounds on derivatives

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306 D. Filip et al. / <strong>Surface</strong> <strong>algorithms</strong> The derivative of a Brzier curve is a Brzier curve of lower degree and thus n-1 c'(t)= Y'~ n(bi+a-bi)B n-l, i=0 n-2 c"(t) = ~, n(n- 1)(6/+ 2 -- 2bi+a + bi)BF -1, i=0 sup Ic'(t)l~
neously by <strong>using</strong> the represented. The x(u, D. Filip et aL/ <strong>Surface</strong> <strong>algorithms</strong> 307 power basis form, by which all these other forms can also be exactly v) component of a degree n by m polynomial takes the form ), x(u, v)= ai,jv j u, ai, j~R. i=0 \j=O ] We can also write the surface in another way where x(u, o)= E c,(v)u, i=0 n i (4.1) rn ci(v)= E ai,j v'i. (4.2) y=O Using this form, it is easy to obtain the <strong>bounds</strong> needed. We first calculate the numbers (Ai, k } <strong>using</strong> the curve case, where Ai,k=su p dkc~(V)dv ~ , O
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Page 15 and 16: 5. Geometric vs. parametric distanc
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