I asked this question earlier, but I got no answer, maybe due to my bad English and bad explanation. Now, trying to ask in a different way, hope you can help me:
In a surface, the form of $z(x,y=\text{constant)}$ is known, say (For Example: $z=ax+b$, where $a$ and $b$ depend on the constant $y$).
Also, the form of $z(x=\text{constant},y)$ is known, say (For Example: $z=cy^2+d$, where $c$ and $d$ depend on the constant $x$).
How to find the from of $z(x,y)$ where both $x$ and $y$ are variables?
See the illustration:
For a fixed $y$, the relation between $x$ and $z$ is $z=ax+b$ (Green straight line).
And for a fixed $x$, the relation between $y$ and $z$ is $z=cy^2+d$ (Red parabola).
Is the form $z(x,y)=\sqrt{c_1x+c_2y^2+c_3xy+c_4}$? [This is not my actual problem].
My actual forms of $z(x,y=\text{constant})$ and $z(x=\text{constant},y)$ are much more complicated than a straight line and a parabola, but to make my problem easier to understand, I chose this example.
To summarize my problem; Knowing the form of $z(x,y=\text{constant})$ and the form of $z(x=\text{constant},y)$, how to find the form of $z(x,y)$ where both $x$ and $y$ are variables?
Any general/procedural way for any two known forms?
Your help would be appreciated. THANKS!
The simplest form could be $$z=c y^2+d \qquad \text{with} \qquad c=c_0+c_1x \quad \text{and}\quad d=d_0+d_1x$$ which would make $$z=(c_1 y^2+d_1)x + (c_0 y^2+d_0)$$ $$z=(c_0+c_1x) y^2+(d_1x+d_0)$$