Examination of cases

A Bézier ambit is authentic by a set of ascendancy credibility P0 through Pn, area n is alleged its adjustment (n = 1 for linear, 2 for quadratic, etc.). The aboriginal and endure ascendancy credibility are consistently the end credibility of the curve; however, the average ascendancy credibility (if any) about do not lie on the curve.

editLinear Bézier curves

Given credibility P0 and P1, a beeline Bézier ambit is artlessly a beeline band amid those two points. The ambit is accustomed by

and is agnate to beeline interpolation.

editQuadratic Bézier curves

A boxlike Bézier ambit is the aisle traced by the action B(t), accustomed credibility P0, P1, and P2,

,

which can be interpreted as the beeline interpolant of agnate credibility on the beeline Bézier curves from P0 to P1 and from P1 to P2 respectively. More absolutely it can be accounting as:

It departs from P0 in the administration of P1, again aeroembolism to access at P2 in the administration from P1. In added words, the tangents in P0 and P2 both canyon through P1. This is anon apparent from the acquired of the Bézier curve:

A boxlike Bézier ambit is aswell a emblematic segment. As a ambit is a cone-shaped section, some sources accredit to boxlike Béziers as "conic arcs".2

editCubic Bézier curves

Four credibility P0, P1, P2 and P3 in the even or in higher-dimensional amplitude ascertain a cubic Bézier curve. The ambit starts at P0 traveling against P1 and arrives at P3 advancing from the administration of P2. Usually, it will not canyon through P1 or P2; these credibility are alone there to accommodate directional information. The ambit amid P0 and P1 determines "how long" the ambit moves into administration P2 afore axis appear P3.

Writing BPi,Pj,Pk(t) for the boxlike Bézier ambit authentic by credibility Pi, Pj, and Pk, the cubic Bézier ambit can be authentic as a beeline aggregate of two boxlike Bézier curves:

The absolute anatomy of the ambit is:

For some choices of P1 and P2 the ambit may bisect itself, or accommodate a cusp.