Bezier curve and its properties

A Bezier curve is a parametric curve defined by a set of control points P0 … Pn. It starts at P0, ends at Pn, and is pulled towards the other points, which it generally does not pass through.

P(u) = Σ (i = 0 to n) P_i · B_{i,n}(u), 0 ≤ u ≤ 1

where B_{i,n}(u) = C(n,i) · u^i · (1 − u)^(n−i) are the Bernstein blending functions.

Properties

1. The curve passes through the first and last control points: P(0) = P0, P(1) = Pn.
2. The degree is one less than the number of control points (4 points give a cubic).
3. Tangent at the start is along P0P1 and at the end along Pn−1Pn: P'(0) = n(P1 − P0), P'(1) = n(Pn − Pn−1).
4. The curve lies inside the convex hull of its control points.
5. Blending functions are non-negative and sum to 1 (partition of unity).
6. Affine invariance: transforming the control points transforms the curve in the same way.
7. Global control: moving one control point changes the whole curve (no local control).
8. Symmetry: reversing the control point order gives the same curve.
9. Variation diminishing: the curve does not cross a line more often than the control polygon does.

Example (quadratic): control points P0(0,0), P1(2,4), P2(4,0). At u = 0.5: P = 0.25(0,0) + 0.5(2,4) + 0.25(4,0) = (2, 2). The curve starts at (0,0), ends at (4,0) and does not reach P1(2,4); it stays inside the triangle.

        P1
       /  \
      / .-.\
     / /   \\
   P0        P2