Specified p_{0} (1, 1): p_{1} (2, 3); p_{2} (4, 3); p_{3} (3, 1) as vertices of Bezier curve find out 3 points on Bezier curve?
Solution: We consider Cubic Bezier curve as:
P (u) = p_{0} (1 - u)^{3} + 3p_{1} u (1 - u)^{2} + 3p_{2} u^{2} (1 - u) + p_{3}u^{3}
P (u) = (1, 1) (1 - u)^{3} + 3 (2, 3)u (1 - u)^{2} + 3 (4, 3) u^{2} (1 - u) + (3, 1)u^{3}. 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). 1^{3}
= (3, 1)