I have an equation of body surface is polar coordinates defined as Fourier series: $$ r=r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] $$ Also I have a line equation in Cartesian coordinates: $y=kx+b$
I need to find their point of intersection. How can I do this? I guess, I should use a transformation matrix. But I don't understand, how it should be. I think, this matrix will help me with transformation a Fourier series equation to Cartesian coordinates, or vice versa line equation to polar coordinates and later solve the system of equations.
There is no hope for an analytic solution. I would transform the line equation into polar coordinates: $$ r = \frac{b}{\sin\phi-k\cos\phi} $$ and then use a numerical method to find an intersection point. This amounts to finding a root of $$ \left(r_{0} + \sum [a_{i}\cos(i\phi) + b_{i}\sin(i\phi)] \right)({\sin\phi-k\cos\phi}) - b = 0 $$ The expression on the left is a trigonometric sum, so it's not hard to find its derivative and use the Newton–Raphson method.