Important point about the De casteljeau algorithm
1)
Bezier Curve: P (u) =
................ (1)
Here B_{n,i} (u) = ^{n}c_{i} u^{i} (1 - u) ^{n-i} .............. (2)
C (n, i) = nC_{i} = n!/i!(n - i)! 0 ≤ u ≤ 1
2) Cubic Bezier curve has n = 3 as:
------------------(3)
= p_{0} B_{3, 0} (u) + p_{1} B_{3, 1} (u) + p_{2} B_{3, 2} (u) + p_{3} B_{3, 3} (u)
Now, let's find B_{3}, 0 (u), B_{3}, _{1} (u), B_{3, 2 }(u) , B_{3, 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! . u^{2}. (1 - u) = 3u^{2}(1 - u)
2!(3 -2)!
B3,3(u) = 3C3 u3 (1 - u)3 - 3
= 3! . u^{3}. (1 - u)
3!(3 -3)!
By using (a), (b) , (c) and (d) in (5) we find here:
P (u) = p_{0} (1 - u)^{3} + 3p_{1}u (1 - u)^{2 }3p_{2}u^{2} (1 - u) + p_{3} u^{3}