I have been working on a research idea for sometime where I need to optimize a given situation. I have reduced it down to finding the closest point on a curve from a given point(P). The curve is made from the intersection of a given sphere(S) and a given cone(C). P lies on the surface of the S.
I have been working on this problem statement and I simplified this problem into finding the minimum distance from a point(centre of sphere S) to the surface of cone along a plane. However, I am unable to find a generalized method of solving this problem. Will be great if someone could help :D
In this I know the plane equation,cone equation,sphere equation and the point P. It will be great even if you can only provide a symbolic algorithm.
Given
$$ p = (x,y,z)\\ p_0 = (x_0,y_0,z_0)\\ C\to x^2+y^2 = z^2\\ S\to (x - 3)^2 + y^2 + (z - 1)^2 = 10 $$
Using the Lagrange multipliers, this problem can be stated as
$$ L(x,y,\lambda,\mu) = ||p-p_0||^2+\lambda(x^2+y^2 - z^2)+\mu( (x - 3)^2 + y^2 + (z - 1)^2 - 10) $$
with stationary points given by the solutions of
$$ \nabla L = \left\{ \begin{array}{rcl} 2 \mu (x-3)+2 (x-3)+2 \lambda x=0 \\ 2 (y-5)+2 \lambda y+2 \mu y=0 \\ 2 (z-5)+2 \mu (z-1)-2 \lambda z=0 \\ x^2+y^2-z^2=0 \\ (x-3)^2+y^2+(z-1)^2-10=0 \\ \end{array} \right. $$
Attached a plot showing the solutions.
$$ \left( \begin{array}{cccc} x & y & z & ||p-p_0|| \\ 0.206925 & 0.380531 & -0.433153 & 7.65898 \\ 0.475273 & -1.73045 & 1.79453 & 7.87072 \\ 1.55558 & -0.727696 & -1.71737 & 8.94516 \\ 2.63556 & 2.07762 & 3.35599 & 3.37282 \\ \end{array} \right) $$