Finding centers of two equal-radius tangent arcs in 3D

[Brad_]

Hi,

Given 3D points P1(200,60,140), P2(300,120,110), P3(3,0,-1), P4(-100,0,-1) and the radius of
arc C1MP3 is equal to radius of arc C2MP1. How do I calculate coordinates x, y, z of
points C1 and C2? See this image.

Points C1 and C2 are centers of two reverse arcs which are tangent to each other at point M which lies on ray Q1Q2.
Arc C1MP3 is tangent to ray P3P4 and arc C2MP1 is tangent to ray P1P2.



Points Q1 and Q2 emerge as a result of moving points P1 and P3 in the direction obvious from picture.
It is easy to calculate centers of arcs with different radius. But how to calculate centers of arcs
with equal radius. How to find the position of points Q1 and Q2?

 

[Simon Bridge]

Welcome to PF;
We do not know the curves are in the same plane?

 

[SteamKing]

And it doesn't appear from the diagram that the arcs are necessarily circular, either. That is, it doesn't appear that the entire curve lies within a single plane.

 

[Stephen Tashi]

Perhaps you can phrase the problem as asking for the centers of two spheres of equal radius whose surfaces are tangent at some point and such that line P1P2 is tangent to the surface of one sphere and line P3P4 is tangent to the surface of the other.