Coordinates of linked equidistant points on R3
This is my first post here. I hope someone can give me some light on this. I've been scratching my head with this for a few weeks.
I need to find the coordinates of 3 points:
These points are linked to 3 other points (there are fixed):
I mean, the point Px can rotate around point Cx by a fixed distance rx. So, the positions of P1..3 can be described as 3 spheres, and they can be defined with polar coordinates:
The distance from P1 to P2, P2 to P3 and P3 to P1 are the same ( ). From this I have 3 equations:
Since the three distances are the same, these 3 points define a equilateral triangle. It's geometric center must lie on the z axis so:
This is where i got today. I've been on this problem for almost a month, I started using cartesian coordinates but I need a lot more equations because I have 3 unknowns for each point. With polar coordinates I have only 2 unknowns for each point, but they're inside trigonometric functions.
I modeled the system with cartesian coordinates with 9 nonlinear equations with 9 unknowns and tried to solve it on Maple but it took 8 hours calculating and at the end said that the solutions may have been lost.
Things seems to be simpler with polar coordinates... but I got stucked here. The solution doesn't need to be analitycal, it could be a numeric one, but I looked for methods for solving non linear systems like this and couldn't find anything.
Does anyone have any idea?