Back
PDF Version (70 KB)

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

## Quadratic Bézier curves

 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