I'm tryting to find a function: $z=f(x,y), x,y,z\in[0,1]$, based on the knowledge of what the the outermost corners are $[0,0], [1,0], [0,1], [1,1]$ and what the midpoint is $[0.5, 0.5]$. Furthermore, I know that $[0,0]$ is the lowest value and that $[1,1]$ is the largest value, and that increasing either $x$ or $y$ should increase $z$. The function should preferably be somewhat smooth.
Does anyone know a suitable function to use? Or of other approaches to approximate the function? It's okay if the function cant fit the values exactly.
I found the following function to behave in the way I wanted (when assuming that the corner [0,0] is 0 and [1,1] is 1): $$f(x,y)=((a_1x^{\gamma_1}+a_2y^{\gamma_2}+(xy)^{\gamma_3})/(a_1+a_2+1))^{\gamma_4}$$
The parameters can then be found by minimizing the cost function $$C(P)=(f(X,Y)-C)^\top(f(X,Y)-Z)$$ Where $P=[a_2,a_2,\gamma_1,\gamma_2,\gamma_3,\gamma_4]$, and $X$,$Y$,$Z$ are vectors of known x, y, and corresponding z values.