Download An Introduction to NURBS: With Historical Perspective (The by David F. Rogers PDF

By David F. Rogers

For one week after receiving this booklet I agreed with an past very severe evaluation. I replaced my brain. the topic isn't really effortless yet written by way of somebody who understands his company. Having received used to his notation i locate this booklet progressively more priceless and refer again to it each time an issue arises and customarily find the solution or a few pointer to the reply.

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0)i(1 O) n-i i! (n - i)! = 0 - and Jn, i(0) - Thus P(0)- i # 0 BOJn, o(0) - Bo and the first point on the B6zier curve a n d on its control polygon are coincident, as previously claimed. (1)n(o) n-n n! (1) = 1 i - n n! t~(1 - 1) n - / - - 0 Jn, i(1) -- i! ( n - i)! Thus i ~ n P(1) -- B n J n , n(1) -- B n and the last point on the B6zier curve a n d the last point on its control polygon are coincident. T h e blending functions shown in Fig. 4 illustrate these results. 5) i=0 An example illustrates the technique for d e t e r m i n i n g a Bdzier curve.

For e x a m p l e , at t = 1/2 P ' ( 1 / 2 ) -- - 3 ( 1 - 1/2)2Bo -[- 3(1 - 3/2)(1 - 1/2)B1 q- (3/2)(2 - 3/2)B2 -[- 3/4B3 -- -3/4 B0 - 3/4 81 + 3/4 B2 + 3/4 B3 = - 3 / 4 (B0 + B1 - B2 - B3) C o m p l e t e results for B0, B1, B2, B3 given in Ex. 1 are s h o w n in Fig. 6. Similarly, t h e second derivatives are J~:0(t) = { ( - 3 t ) ~ - 3t~} (1 - t)~ = 6(1 - t) t 2 ( 1 - t) 2 J~',l(t) = {(1 - 3t) 2 - 3t 2 - (1 - 2t)} (3t)(1 - t) 2 = - 6 ( 2 - 3t) t2(1 - t)2 J~',2(t) = {(2 j, {(a a,a(t) = - - 3t) 2 3t 2 2(1 t2(1 - t ) 2 - - at) ~ 3t ~ a(1 t2(1 _ t ) 2 - - - - 2t)} (3t2)( 1 2 t ) } t ~ = 6t _ t) = 6(1 - 3t) 31 Using Eq.

1, t h e basic a l g o r i t h m for c a l c u l a t i n g p o i n t s on a Bdzier c u r v e is f o r each value of t h e p a r a m e t e r t P(t) - 0 f o r each value of i d e t e r m i n e t h e B e r n s t e i n basis f u n c t i o n J~,i(t) P(t) - P(t) + BiJn, i(t) loop loop N o t i c e t h a t t h e s u m m a t i o n sign in Eq. 1) c o r r e s p o n d s t o t h e i n n e r loop over i, 0 _< i _< n. T h e o u t e r loop c a l c u l a t e s i n d i v i d u a l p o i n t s on t h e c u r v e b a s e d on t h e p a r a m e t e r v a l u e t.