Vertices of bezier curve find out 3 points on bezier curve, Computer Graphics

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Specified p0 (1, 1): p1 (2, 3); p2 (4, 3); p3 (3, 1) as vertices of Bezier curve find out 3 points on Bezier curve?

Solution: We consider Cubic Bezier curve as:

P (u) = p0 (1 - u)3 + 3p1 u (1 - u)2 + 3p2 u2 (1 - u) + p3u3

P (u) = (1, 1) (1 - u)3   + 3 (2, 3)u (1 - u)2 + 3 (4, 3) u2 (1 - u) + (3, 1)u3. we select various values of u from 0 to 1.

u = 0:   P (0) = (1, 1) (1 - 0)3 + 0 + 0 + 0 = (1, 1)

u = 0.5:  P (0.5) = (1, 1)(1 - 0.5)3+3(2, 3)(0.5) (1 - 0.5)2 + 3 (4, 3)(0.5)2(1 - 0.5)+(3,1) (0.5)3

= (1, 1) (0.5)3 + (2, 3) (0.375) + (0.375) (4, 3) + (3, 1) (0.125)

= (0.125, 0.125) +  (0.75, 1.125) + (1.5, 1.125) + (1.125, 0.125) P (0.5) = (3.5, 2.5)

u = 1:  P (1) = 0 + 0 + 0 + (3, 1). 13

= (3, 1)


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