Important point about the de casteljeau algorithm, Computer Graphics

Important point about the De casteljeau algorithm

1)     

1690_Important point about the De casteljeau algorithm.png

Bezier Curve: P (u) =

   ................     (1)

Here Bn,i (u) = nci ui (1 - u) n-i       ..............        (2)

C (n, i) = nCi = n!/i!(n - i)!      0 ≤ u ≤ 1

2)      Cubic Bezier curve has n = 3 as:

1691_Important point about the De casteljeau algorithm 1.png

------------------(3)

= p0 B3, 0 (u) + p1 B3, 1 (u) + p2 B3, 2 (u) + p3 B3, 3 (u)

Now, let's find B3, 0 (u), B3, 1 (u), B3, 2 (u) , B3, 3 (u) by using equation as in above:

Bn, i (u) = nci u (1 - u)

a)  B3, 0(u) = 3C0   u0   (1 - u)3 - 0

=          3!       . 1. (1 - u)3 = (1 - u)3

       0!(3 - 0)!

 

b)  B3, 1(u) = 3C1   u1   (1 - u)3 - 1

=          3!       . u. (1 - u)2 = 3u (1 - u)2

1!(3 - 1)!

c) B3,2(u) = 3C2   u2   (1 - u)3 - 2

=          3!       . u2. (1 - u) = 3u2(1 - u)

       2!(3 -2)!

B3,3(u) = 3C3   u3   (1 - u)3 - 3

=          3!       . u3. (1 - u) 

         3!(3 -3)!

By using (a), (b) , (c) and (d) in (5) we find here:

P (u) = p0 (1 - u)3 + 3p1u (1 - u)2 3p2u2 (1 - u) + p3 u3

Posted Date: 4/4/2013 5:13:58 AM | Location : United States







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