Bezier properties and order of continuity

Properties of the Bezier curve

1. Passes through the first and last control points: P(0) = P0 and P(1) = Pn.
2. Degree = number of control points − 1.
3. Tangent at the ends is along the first and last control polygon edges: P'(0) = n(P1 − P0), P'(1) = n(Pn − Pn−1).
4. The curve lies inside the convex hull of the control polygon.
5. Blending (Bernstein) functions are non-negative and sum to 1.
6. Affine invariance: the curve can be transformed by transforming the control points.
7. Global control: moving one control point changes the whole curve.
8. Symmetry: the same curve is obtained when the point order is reversed.
9. Variation diminishing: the curve oscillates no more than the control polygon.

Order of continuity

When two curve sections are joined, the order of continuity tells how smoothly they meet. There are two kinds.

Parametric continuity

• C⁰: the sections meet: the end point of the first equals the start point of the second.
• C¹: the first derivatives (tangent vectors) are equal at the join.
• C²: the first and second derivatives (rate of change of tangent) are equal at the join.

Geometric continuity

• G⁰: same as C⁰.
• G¹: tangent directions are equal, but magnitudes may differ (proportional).
• G²: curvature is equal.

 C0: joined      C1: joined, same slope     C2: also same curvature
  ___/\___          ___/---___

Example: for two Bezier sections, C⁰ needs P0' = Pn; C¹ needs P1' = Pn + (Pn − Pn−1).