Bézier surface

Bézier surfaces are a type 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 (n, m) 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 points – for example, if the ki,j points 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:

p

(

u

,

v

)

=

∑

i

=

0

n

∑

j

=

0

m

B

i

n

(

u

)

B

j

m

(

v

)

k

i

,

j

{\displaystyle \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

i

)

u

i

(

1

−

u

)

n

−

i

{\displaystyle B_{i}^{n}(u)={n \choose i}u^{i}(1-u)^{n-i}}

is a basis Bernstein polynomial, and

(

n

i

)

=

n

!

i

!