Bezier curve through given points

Two properties

1. The curve passes through the first and last control points (P(0) = P0, P(1) = Pn) and is tangent to the first and last edges of the control polygon there.
2. The curve lies inside the convex hull of its control points (the blending functions are non-negative and sum to 1).

Curve

The curve passes through the end points, and the other two points are control points:

P0 = (0,0,0), P1 = (7,5,2), P2 = (2,0,1), P3 = (−2,1,1)

Cubic Bezier: P(u) = (1−u)³P0 + 3u(1−u)²P1 + 3u²(1−u)P2 + u³P3

In power form: P(u) = a u³ + b u² + c u + P0, with

• a = −P0 + 3P1 − 3P2 + P3 = (21 − 6 − 2, 15 − 0 + 1, 6 − 3 + 1) = (13, 16, 4)
• b = 3P0 − 6P1 + 3P2 = (−42 + 6, −30 + 0, −12 + 3) = (−36, −30, −9)
• c = −3P0 + 3P1 = (21, 15, 6)

x(u) = 13u³ − 36u² + 21u y(u) = 16u³ − 30u² + 15u z(u) = 4u³ − 9u² + 6u

Check at u = 1: x = 13 − 36 + 21 = −2, y = 16 − 30 + 15 = 1, z = 4 − 9 + 6 = 1, which is P3. Sample point at u = 0.5: (3.125, 2, 1.25).

Answer: the Bezier curve above, with start (0,0,0) and end (−2,1,1).