Second partial derivatives from first and mixed derivative of bicubic Bezier Surface Patch? Given the definition of the bicubic Bezier Surface Patch function:
$$f(u,v) = \begin{bmatrix}
u^3 & u^2 & u & 1
\end{bmatrix}
\begin{bmatrix}
-1 & 3 & -3 & 1 \\
3 & -6 & 3 & 0 \\
-3 & 3 & 0 & 0 \\
1 & 0 & 0 & 0 
\end{bmatrix}
\begin{bmatrix}
P_{00} & P_{01} & P_{02} & P_{03} \\
P_{10} & P_{11} & P_{12} & P_{13} \\
P_{20} & P_{21} & P_{22} & P_{23} \\
P_{30} & P_{31} & P_{32} & P_{33}
\end{bmatrix}
\begin{bmatrix}
-1 & 3 & -3 & 1 \\
3 & -6 & 3 & 0 \\
-3 & 3 & 0 & 0 \\
1 & 0 & 0 & 0 
\end{bmatrix}^T
\begin{bmatrix}
v^3 \\
v^2 \\
v \\
1
\end{bmatrix},$$
is it possible to compute the second partial derivatives $f_{uu}$ and  $f_{vv}$ knowing the values for $f$, $f_{u}$, $f_{v}$ and $f_{uv}$ for each of the 4 corners?
I have reasons to believe that it is indeed possible, but I do not know how.
Thank you!
 A: Yes. If you know the values of $f$, $f_u$, $f_v$ and $f_{uv}$, then you can write the equation of the patch in Hermite form, and you can then use this to obtain derivatives of any order.
Thinking of the problem another way ... you can used the 16 known values to compute the 16 control points $P_{ij}$ you used in your equation, and you can then use that equation to calculate anything you like. Computing the $P_{ij}$ from the known 16 quantities is easy; you just have to solve linear equations. In fact, the solution can be written explicitly: we have
$$
P_{00} = F_{00} \; ; \; P_{03} = F_{01} \; ; \; \text{etc.}
$$
$$
P_{01} = F_{00} + \tfrac13 F_{00}^{u}\; ; \;
P_{02} = F_{01} - \tfrac13 F_{01}^{u}\; ; \; \text{etc.}
$$
$$
P_{10} = F_{10} + \tfrac13 F_{10}^{v}\; ; \;
P_{20} = F_{10} - \tfrac13 F_{10}^{v}\; ; \; \text{etc.}
$$
$$
P_{11} = F_{00} + \tfrac13 F_{00}^{u} + \tfrac13 F_{00}^{v} - \tfrac19 F_{00}^{uv}\; ; \; \text{etc.}
$$
The notation here is the obvious one:
$$
F_{00}^{uv} = \frac{\partial^2 F}{\partial u \partial v}(u=0, v=0)
$$
and so on.
If possible, I suggest you rearrange your optimization problem so that the $P_{ij}$ are the independent variables. This will probably improve numerical stability, and will remove the need for the conversions outlined above.
