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)