## Coordinates of linked equidistant points on R3

Hi everyone...

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:
$\displaystyle {P1 = (x1, y1, z1)}$
$\displaystyle {P2 = (x2, y2, z2)}$
$\displaystyle {P3 = (x3, y3, z3)}$

These points are linked to 3 other points (there are fixed):

$\displaystyle {C1 = (c1x, c1y, c1z)}$
$\displaystyle {C2 = (c2x, c2y, c2z)}$
$\displaystyle {C3 = (c3x, c3y, c3z)}$

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:

$\displaystyle {x1 = c1x+r1*sin(\theta 1)*cos(\phi 1)}$
$\displaystyle {y1 = c1y+r1*sin(\theta 1)*sin(\phi 1)}$
$\displaystyle {z1 = c1z+r1*cos(\theta 1)}$

$\displaystyle {x2 = c2x+r2*sin(\theta 2)*cos(\phi 2)}$
$\displaystyle {y2 = c2y+r2*sin(\theta 2)*sin(\phi 2)}$
$\displaystyle {z2 = c2z+r2*cos(\theta 2)}$

$\displaystyle {x3 = c3x+r3*sin(\theta 3)*cos(\phi 3)}$
$\displaystyle {y3 = c3y+r3*sin(\theta 3)*sin(\phi 3)}$
$\displaystyle {z3 = c3z+r3*cos(\theta 3)}$

The distance from P1 to P2, P2 to P3 and P3 to P1 are the same ( $\displaystyle {\sqrt{3}*Rsw}$). From this I have 3 equations:

$\displaystyle {(x1-x2)^2+(y1-y2)^2+(z1-z2)^2=3*Rsw^2}$
$\displaystyle {(x3-x2)^2+(y3-y2)^2+(z3-z2)^2=3*Rsw^2}$
$\displaystyle {(x1-x3)^2+(y1-y3)^2+(z1-z3)^2=3*Rsw^2}$

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:

$\displaystyle {(x1+x2+x3)/3=0}$
$\displaystyle {(y1+y2+y3)/3=0}$

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?

Best regards,
Rodrigo Basniak