Generalization

Recursive definition

A recursive analogue for the Bézier ambit of amount n expresses it as a point-to-point beeline aggregate of a brace of agnate credibility in two Bézier curves of amount n − 1.

Let denote the Bézier ambit bent by the credibility P0, P1, ..., Pn. Then

to start, and

This recursion is elucidated in the animations below.

editExplicit definition

The blueprint can be bidding absolutely as follows:

where are the binomial coefficients.

For example, for n = 5:

editTerminology

Some analogue is associated with these parametric curves. We have

where the polynomials

are accepted as Bernstein base polynomials of amount n.

Note that t0 = 1 , (1 − t)0 = 1, and that the binomial coefficient, , aswell bidding as or is:

The credibility Pi are alleged ascendancy credibility for the Bézier curve. The polygon formed by abutting the Bézier credibility with lines, starting with P0 and finishing with Pn, is alleged the Bézier polygon (or ascendancy polygon). The arched bark of the Bézier polygon contains the Bézier curve.

editProperties

The ambit begins at P0 and ends at Pn; this is the alleged endpoint departure property.

The ambit is a beeline band if and alone if all the ascendancy credibility are collinear.

The alpha (end) of the ambit is departure to the aboriginal (last) area of the Bézier polygon.

A ambit can be breach at any point into two subcurves, or into arbitrarily abounding subcurves, anniversary of which is aswell a Bézier curve.

Some curves that assume simple, such as the circle, cannot be declared absolutely by a Bézier or piecewise Bézier curve; admitting a four-piece cubic Bézier ambit can almost a amphitheater (see Bézier spline), with a best adorable absurdity of beneath than one allotment in a thousand, if anniversary close ascendancy point (or offline point) is the ambit angular or angular from an alien ascendancy point on a assemblage circle. Added generally, an n-piece cubic Bézier ambit can almost a circle, if anniversary close ascendancy point is the ambit from an alien ascendancy point on a assemblage circle, area t is 360/n degrees, and n > 2.

The ambit at a anchored account from a accustomed Bézier curve, about alleged an account ambit (lying "parallel" to the aboriginal curve, like the account amid balustrade in a railroad track), cannot be absolutely formed by a Bézier ambit (except in some atomic cases). However, there are heuristic methods that usually accord an able approximation for applied purposes.citation needed

Every boxlike Bézier ambit is aswell a cubic Bézier curve, and added generally, every amount n Bézier ambit is aswell a amount m ambit for any m > n. In detail, a amount n ambit with ascendancy credibility P0, …, Pn is agnate (including the parametrization) to the amount n + 1 ambit with ascendancy credibility P'0, …, P'n + 1, area .