Equation

A accustomed Bézier apparent of adjustment (n, m) is authentic by a set of (n + 1)(m + 1) ascendancy credibility ki,j. It maps the assemblage aboveboard into a smooth-continuous apparent anchored aural a amplitude of the aforementioned ambit as { ki,j }. For example, if k are all credibility in a four-dimensional space, again the apparent will be aural a four-dimensional space.

A two-dimensional Bézier apparent can be authentic as a parametric apparent area the position of a point p as a action of the parametric coordinates u, v is accustomed by:

evaluated over the assemblage square, where

is a Bernstein polynomial, and

is the binomial coefficient.

Some backdrop of Bézier surfaces:

A Bézier apparent will transform in the aforementioned way as its ascendancy credibility beneath all beeline transformations and translations.

All u = connected and v = connected ambit in the (u, v) space, and, in particular, all four edges of the askew (u, v) assemblage aboveboard are Bézier curves.

A Bézier apparent will lie absolutely aural the arched bark of its ascendancy points, and accordingly aswell absolutely aural the bonds box of its ascendancy credibility in any accustomed Cartesian alike system.

The credibility in the application agnate to the corners of the askew assemblage aboveboard accompany with four of the ascendancy points.

However, a Bézier apparent does not about canyon through its added ascendancy points.

Generally, the a lot of accepted use of Bézier surfaces is as nets of bicubic patches (where m = n = 3). The geometry of a individual bicubic application is appropriately absolutely authentic by a set of 16 ascendancy points. These are about affiliated up to anatomy a B-spline apparent in a agnate way as Bézier curves are affiliated up to anatomy a B-spline curve.

Simpler Bézier surfaces are formed from biquadratic patches (m = n = 2), or Bézier triangles.