Bezier surface equation and properties

A Bezier surface is a parametric surface patch defined by a rectangular net (mesh) of control points. It is the tensor product of two Bezier curves, one in the $u$ direction and one in the $w$ direction.

Equation

For an $(m+1)\times(n+1)$ net of control points $\mathbf B_{ij}$:

$$\mathbf P(u,w)=\sum_{i=0}^{m}\sum_{j=0}^{n}\mathbf B_{ij}\,J_{m,i}(u)\,J_{n,j}(w),\qquad 0\le u,w\le1$$

where $J_{m,i}(u)=\binom mi u^i(1-u)^{m-i}$ is the Bernstein basis. The bicubic patch has $m=n=3$, so 16 control points:

$$\mathbf P(u,w)=\sum_{i=0}^{3}\sum_{j=0}^{3}\mathbf B_{ij}J_{3,i}(u)J_{3,j}(w)$$

Properties

1. The degree in $u$ is $m$ and in $w$ is $n$ (one less than the number of points in that direction).
2. The four corner points of the net lie on the surface: $\mathbf P(0,0)=\mathbf B_{00}$, $\mathbf P(1,0)=\mathbf B_{m0}$, etc.
3. The four boundary curves are Bezier curves defined by the boundary rows and columns of the net.
4. Tangent planes at the corners are spanned by the adjacent net edges.
5. Convex hull: the surface lies in the convex hull of the control net.
6. Global control: moving any control point changes the whole patch.
7. Invariant under affine transformation (transform the net only).
8. Two patches join with $C^0$ continuity if the edge rows coincide, and with $C^1$ if the adjacent rows are collinear with equal ratio.

Difference from curve

 Curve:   1 parameter u,   row of points B_i
 Surface: 2 parameters u,w, net of points B_ij

   B03---B13---B23---B33
    |     |     |     |      u -> (to the right)
   B02---B12---B22---B32
    |     |     |     |      w ^  (upwards)
   B01---B11---B21---B31
    |     |     |     |
   B00---B10---B20---B30

Only the four corner points lie on the surface; the other points only pull it towards them.