Back

Bézier Curves
Urs Oswald    osurs@bluewin.ch    http://www.ursoswald.ch
September 11, 2002

 Quadratic Bézier curves Cubic Bézier curves Bézier curves of arbitrary order

 Let be distinct points. If points divide the line segments by an equal ratio, then moves on a quadratic Bézier curve if moves about . Thus for some real number , where . Fig. 1: Quadratic Bézier curve

From the above equations, it follows that , therefore (1)

We equally find and . Substitution yields , and after expansion, (2)

## Cubic Bézier curves

 Let be distinct points, and let the points divide their respective line segments by an equal ratio: where . (In fig. 2 as well as in fig. 1, .) Fig. 2: Cubic Bézier curve

From (2), we get From (1), which, after expansion, yields (3)

In fig. 2, if moves about the segment , then moves on the quadratic Bézier curve determined by points , while moves on the quadratic Bézier curve determined by points .

## Bézier curves of arbitrary order

For distinct points , the Bézier curve of order ( ) can be recursively defined by (4)

where .

Theorem 1 (Bézier curves of order 1)   For points , the Bézier curve of order 1 is given by the equation PROOF: By the above definition, Theorem 2   For any non-negative integer , the Bézier curve of order is given by the equation PROOF: By induction on . For , the theorem claims which is correct by the first part of the definition.
For , we have by definition. By induction hypothesis,  We get   Substituting the last two results, we get  ,        as by the fundamental law of the binomial coefficients. Quadratic Bézier curves Cubic Bézier curves Bézier curves of arbitrary order