Derive cubic Bezier equation and properties

Cubic Bezier equation

A cubic Bezier curve has four control points P0, P1, P2, P3 (n = 3). The general Bezier curve is P(u) = Σ P_i · C(n,i) u^i (1−u)^(n−i), 0 ≤ u ≤ 1. For n = 3:

C(3,0) = 1, C(3,1) = 3, C(3,2) = 3, C(3,3) = 1

P(u) = (1−u)³ P0 + 3u(1−u)² P1 + 3u²(1−u) P2 + u³ P3

Blending functions: B0 = (1−u)³, B1 = 3u(1−u)², B2 = 3u²(1−u), B3 = u³ (they add to 1).

Expanding in powers of u:

P(u) = (−P0 + 3P1 − 3P2 + P3) u³ + (3P0 − 6P1 + 3P2) u² + (−3P0 + 3P1) u + P0

Matrix form:

                  | -1  3 -3  1 | | P0 |
P(u) = [u³ u² u 1]|  3 -6  3  0 | | P1 |
                  | -3  3  0  0 | | P2 |
                  |  1  0  0  0 | | P3 |

Check at u = 0: P = P0. At u = 1: P = P3.

Properties

1. Starts at P0 and ends at P3; other points are not on the curve.
2. Tangent at the ends: P'(0) = 3(P1 − P0), P'(1) = 3(P3 − P2).
3. Lies inside the convex hull of the control points.
4. Blending functions are positive and sum to 1.
5. Affine invariance.
6. Global control: moving a point changes the whole curve.
7. Symmetric: reversing the point order gives the same curve.
8. Variation diminishing.
9. Adjacent sections have C⁰ continuity if P3 of the first = P0 of the second, and C¹ if also the points around the join are collinear with equal spacing.