Developing an algorithm from differential calculs

Timber & Steel Pty Ltd is looking for a skilled mathematician to assist with some differential calculus for one of our current projects. It's been more than 15 years since any of us have done any university-level calculus, and we need a little assistance. We have two parts that we need assistance with: Part 1: For an unknown surface, S(x), which for every point x = (x,y), we define as S(x) = c(x) + u(x) r(x) c(x) and r(x) are vectors, both of which are known, and u(x) is an unknown scalar quantity — which I need to solve for. We specify c(x) = a + w(x) o, where both a and o are vectors and do not depend on x, and w(x) is a scalar that does vary with x. As additional inputs, we know the partial derivatives of c(x) and r(x) with respect to (x,y). There's also a number of different partial differentials that can be extracted for the intermediate values that are used to compute r(x), but these are likely not required, as we can simply assume that c(x) and r(x) are vector fields and I have values for them already. The normal to the surface is the cross product of the partial derivatives of S(x) with respect to x and y. Part 2: We now need to determine u(x), which is done with the cost function K(x) = Norm( n(x) x r(x) ) where we normalize the length of the cross product of the vectors between the normal and the directon vector r(x). Completing part 2 will provide the three terms for a quadratic equation of the form: Au(x)^2 + Bu(x) + C = 0 This is not a homework/academic project.