I want to solve a system of equations which will result in expressions for xs, ys, zs for a coordinate (xs, ys, zs) which corresponds to the point at which a vector, originating from the point (xp, yp, zp), intersects a paraboloid of equation zs = [ (xs^2 +ys^2) / 4f ] - f .
assumptions = {rps > 0, -Pi/2 < theta < Pi/2, rps \[Element] Real,
theta \[Element] Real, xp \[Element] Real, yp \[Element] Real ,
zp \[Element] Real, phi \[Element] Real, f > 0, f \[Element] real}
Solve[{rps^2 == (xs - xp)^2 + (ys - yp)^2 + (zs -
zp)^2,
rps Sin[theta] Cos[phi] + xp == Sqrt[4 f zs + 4 f^2 - ys^2],
rps Sin[theta] Sin[phi] + yp == Sqrt[4 f zs + 4 f^2 - xs^2],
rps Cos[theta] + zp == (xs^2 + ys^2 )/(4 f) - f}, {xs, ys, zs},
Assumptions -> assumptions ]
rps is the length of the vector (from the point (xp,yp,zp) to the intersection point on the paraboloid); for the last three equations i have used the spherical coordinate definitions of x, y ,z in terms of theta and phi (since my answer must be in terms of these variables) and equated them to the rearranged form of the expression of the paraboloid.
BUT when i run this code it will either run for hours (and i left it over night!) or simply crashes my computer.
What am i doing wrong?
I'm sure that this reasonable straight forward geometric problem must have a solution.