How to calculate center coordinates of two reverse arcs in 3D space

[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.

 

[blue_raver22]

Calculating the center coordinates of two reverse arcs in 3D space involves using mathematical equations and geometric principles. Here are the steps to follow:

1. Find the equations of the two arcs: The equations of the arcs can be found using the given points P1, P2, P3, and P4 and the radius of the arcs. This can be done using the general equation for a circle in 3D space: (x-x0)^2 + (y-y0)^2 + (z-z0)^2 = r^2, where (x0,y0,z0) is the center of the circle and r is the radius.

2. Set the equations of the two arcs equal to each other: Since the two arcs are tangent to each other at point M, the centers of the arcs C1 and C2 will have the same distance from point M. This can be represented by setting the equations of the arcs equal to each other and solving for x, y, and z.

3. Use the distance formula: The distance between point M and the center of arc C1 can be calculated using the distance formula: d = √[(x-x0)^2 + (y-y0)^2 + (z-z0)^2]. This distance should be equal to the distance between point M and the center of arc C2. Use this information to solve for x, y, and z.

4. Find the position of points Q1 and Q2: Once the center coordinates of C1 and C2 are found, the position of points Q1 and Q2 can be calculated by moving points P1 and P3 in the direction shown in the picture. This can be done by adding or subtracting a certain amount from the x, y, and z coordinates of points P1 and P3.

In summary, calculating the center coordinates of two reverse arcs in 3D space involves finding the equations of the arcs, setting them equal to each other, using the distance formula, and finally finding the position of points Q1 and Q2. It is a complex mathematical process, but with the given information and equations, it can be solved accurately.