Bezier Surface:

• It is a surface that approximates or interpolates given input data.
• It is different from the previous surfaces in that it is a synthetic surface.
• It extends the Bezier curve to surfaces.
• It is a general surface that permits twists, and kinks.
• Bezier surface allows only global control of the surface

Properties of a Bezier Surface:

1) The boundary curves of a Bezier surface are Bezier curve.

2) Only the corner vertices are interpolated but the shape of the surface closely follows the control points p.

3) The vectors generate the tangent plane at with similar facts at the other corner points.

4) The Bezier patch lies in the convex hull of its control points

The patch (surface) is constructed from an $n \times m$ array of control points. $\left(P_{i j}, 0 \leq i \leq n, 0 \leq j \leq\right. m)$

The Bezier surface is parameterized by two variables, is given by the equation,

$\begin{aligned} P(u, v) &=\sum_{j=0}^{m} \sum_{i=0}^{n} P_{i j} B_{i n}(u) B_{j m}(0) \\ &=\sum_{i=0}^{n} P_{i 0} B_{i n}(u) \end{aligned}$

$\left.\text { (since } B_{o m}(0)=1 \text { and } B_{j m}(0)=0 \text { for } j=1,2, \ldots . m\right)$

Therefore, $\mathrm{P}(\mathrm{u}, 0)$ is a Bezier Curve.

Similarly, $\mathrm{P}(\mathrm{u}, 1), \mathrm{P}(1, \mathrm{v})$ and $\mathrm{P}(0, \mathrm{v})$ are Bezier Curve.

For Quadratic Bezier Surface,

$P(u, v)=\Sigma_{j=0}^{2} \Sigma_{i=0}^{2} P_{i j} B_{i 2}(u) B_{j 2}(v)$

$\quad \quad \quad =\sum_{j=0}^{2}\left[\sum_{i=0}^{2} P_{i j} B_{i 2}(u)\right] B_{j 2}(v)$

$\quad \quad \quad =\sum_{j=0}^{2}\left[P_{o j}(1-u)^{2}+2 P_{i j}(1-u)+P_{2 j}(u)^{2}\right] B_{j u}(v)$

$\quad \quad \quad =\sum_{j=0}^{2}\left[P_{o j}(1-2 u+u)^{2}+P_{i j}\left(2 u-2 u^{2}\right)+P_{2 j}\left(u^{2}\right)\right] B_{j u}(v)$

$\quad \quad \quad =\sum_{j=0}^{2}\left[\begin{array}{ccc}{1} & {u} & {u^{2}}\end{array}\right]\left[\begin{array}{ccc}{1} & {0} & {0} \\ {-2} & {2} & {0} \\ {1} & {-2} & {1}\end{array}\right]\left[\begin{array}{c}{P_{1 j}} \\ {P_{i j}} \\ {P_{2 j}}\end{array}\right] B_{j u}(v)$

$\quad \quad \quad =\left[\begin{array}{lll}{1} & {u} & {u^{2}}\end{array}\right]\left[\begin{array}{ccc}{1} & {0} & {0} \\ {-2} & {2} & {0} \\ {1} & {-2} & {1}\end{array}\right]\left[\begin{array}{ccc}{P_{00}} & {P_{01}} & {P_{02}} \\ {P_{10}} & {P_{11}} & {P_{12}} \\ {P_{20}} & {P_{21}} & {P_{22}}\end{array}\right]\left[\begin{array}{ccc}{1} & {-2} & {1} \\ {0} & {2} & {-2} \\ {0} & {0} & {1}\end{array}\right]\left[\begin{array}{c}{1} \\ {v} \\ {v^{2}}\end{array}\right]$

$P(u, v)=\left[\begin{array}{lll}{1} & {u} & {u^{2}}\end{array}\right] M\left[\begin{array}{ccc}{P_{00}} & {P_{01}} & {P_{02}} \\ {P_{10}} & {P_{11}} & {P_{12}} \\ {P_{20}} & {P_{21}} & {P_{22}}\end{array}\right] M^{T}\left[\begin{array}{l}{1} \\ {v} \\ {v^{2}}\end{array}\right]$

Where, $M=\left[\begin{array}{ccc}{1} & {0} & {0} \\ {-2} & {2} & {0} \\ {1} & {-2} & {1}\end{array}\right]$

For Cubic Bezier Surface,

$P(u, v)=\left[\begin{array}{llll}{1} & {u} & {u^{2}} & {u^{3}}\end{array}\right] M\left[\begin{array}{cccc}{P_{00}} & {P_{01}} & {P_{02}} & {P_{03}} \\ {P_{10}} & {P_{11}} & {P_{12}} & {P_{13}} \\ {P_{20}} & {P_{21}} & {P_{22}} & {P_{23}} \\ {P_{30}} & {P_{31}} & {P_{32}} & {P_{33}}\end{array}\right] M^{T}\left[\begin{array}{l}{1} \\ {v} \\ {v^{2}} \\ {v^{3}}\end{array}\right]$

Where, $\mathrm{M}=\left[\begin{array}{cccc}{1} & {0} & {0} & {0} \\ {-3} & {3} & {0} & {0} \\ {3} & {-6} & {3} & {0} \\ {-1} & {3} & {-3} & {1}\end{array}\right]$