Tangent Distance

[Dr.Peterson]

Your P is my rho from here:

The elbow is part of a torus, which is most easily described in term of cylindrical coordinates; I'll define the distance from the origin in the xy-plane as [MATH]\rho = \sqrt{x^2 + y^2}[/MATH]. Then the equation of the torus is [MATH](\rho - R)^2 + z^2 = r^2[/MATH].

And by writing the equations as you have, you simply reveal that I somehow missed the easy way to solve the same equations! Specifically, your first equation is my second equation solved for rho. (P).