Bézier curve

Given a set of n+1 control points P_i the Bézier curve is written in terms of Bernstein polynomials:

The curve has the following properties:

• there is no line that has more intersections with a Bézier curve than with the curve composed by the line segments through the points.
• the curve can be translated and rotated by performing these operations on the control points.
• a numerical instability for large numbers of control points.
• moving a single control point changes the global shape of the curve.
  This can be avoided by smoothly patching together low-order Bézier curves.

The rational Bézier curve is a generalization of the curve:

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Another generalization of the curve is the B-spline.