# Bézier surface

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Bézier surfaces are a species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling. As with Bézier curves, a Bézier surface is defined by a set of control points. Similar to interpolation in many respects, a key difference is that the surface does not, in general, pass through the central control points; rather, it is "stretched" toward them as though each were an attractive force. They are visually intuitive and, for many applications, mathematically convenient.

## History

Bézier surfaces were first described in 1962 by the French engineer Pierre Bézier who used them to design automobile bodies. Bézier surfaces can be of any degree, but bicubic Bézier surfaces generally provide enough degrees of freedom for most applications.

## Equation

A given Bézier surface of degree (nm) is defined by a set of (n + 1)(m + 1) control points ki,j where i = 0, ..., n and j = 0, ..., m. It maps the unit square into a smooth-continuous surface embedded within the space containing the ki,j s – for example, if the ki,j s are all points in a four-dimensional space, then the surface will be within a four-dimensional space.

A two-dimensional Bézier surface can be defined as a parametric surface where the position of a point p as a function of the parametric coordinates u, v is given by:

$\mathbf {p} (u,v)=\sum _{i=0}^{n}\sum _{j=0}^{m}B_{i}^{n}(u)\,B_{j}^{m}(v)\,\mathbf {k} _{i,j}$ evaluated over the unit square, where

$B_{i}^{n}(u)={n \choose i}u^{i}(1-u)^{n-i}$ is a basis Bernstein polynomial, and

${n \choose i}={\frac {n!}{i!(n-i)!}}$ is a binomial coefficient.

Some properties of Bézier surfaces:

• A Bézier surface will transform in the same way as its control points under all linear transformations and translations.
• All u = constant and v = constant lines in the (u, v) space, and – in particular – all four edges of the deformed (u, v) unit square are Bézier curves.
• A Bézier surface will lie completely within the convex hull of its control points, and therefore also completely within the bounding box of its control points in any given Cartesian coordinate system.
• The points in the patch corresponding to the corners of the deformed unit square coincide with four of the control points.
• However, a Bézier surface does not generally pass through its other control points.

Generally, the most common use of Bézier surfaces is as nets of bicubic patches (where m = n = 3). The geometry of a single bicubic patch is thus completely defined by a set of 16 control points. These are typically linked up to form a B-spline surface in a similar way as Bézier curves are linked up to form a B-spline curve.

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