We would like to have a way to find specific $x$ & $y$ values on a surface described by two polynomials where the $z$-axis value intersects with a plane. In our case, the plane is a single $z$-axis value. The surface is fairly "smooth".
Before, this would be done with a lookup table but it's less precise.
As a programmer, this could be determined by using the $x$ & $y$ values to compute possible z values at across the range of the $x$ and $y$ axis. We could devise a way to determine which areas of the surface we don't need to check. for areas that are found to be of "interest", the algorithm could use more $x$ & $y$ values to get a more precise list of possible points we can use.
Even so, since we plan to put this in a battery powered embedded system, we wish to minimize the amount of floating point math.
Today, we can use MatLab's "solve" function but this "solve" function is more of a black box and we're not sure the best way to replicate the functionality.
we expect there must be a more elegant, cleaner algorithm to find these points but don't see a way...
I'm not a mathematician so I might not be using the right terms to describe what we wish to do.
Extra information from colleague:
In Matlab I have a set of two, non-linear equations with two unknowns. The function “solve” does the work in a nice way but is an expensive blackbox within the symbolic math toolbox.
The surface functions we work with may look similar to the following: f(x,y) = ax^3 + bx^2*y + cy^3 + dxy + e g(x,y) = fx^2 + gy^3 + hy^2*x + i
The task is to find pairs of x and y when f(x,y) is a known value.
Something with the approach of finding the “roots” of the equations did not work out the way I hoped. Here I assumed that the function value is known, i.e. f(x,y) = F and g(x,y) = G. Then, f(x,y) = 0 = ax^3 + bx^2*y + cy^3 + dxy + e – F g(x,y) = 0 = fx^2 + gy^3 + hy^2*x + I – G
It should not be that difficult with 2 equations and 2 unknowns but in a way, I cannot figure out how to solve this problem outside of Matlab. As we said, there is a simple numerical approach to solving the equations but I would like to investigate whether a more algebraic approach could help solving the problem more efficiently and elegant.