good day. Actually I'm stuck with this problem
I want to get the 2 points (vertex coordinates) in a 3d circle circle intersection
actually I know a lot of data,
Circle 1 center c1c = (-0.23103,-0.12451,1.78538)
Circle 2 center c2c = (0.56227,1.38846, 2.82537)
Circle 1 radius* c1r = 2
Circle 2 radius* c2r = 2
Circle 1 point c1p** = (-1.40115, -0.58086, 3.34184)
Circle 2 point c2p** = (1.87197, 2.8998, 2.82537)
Circles plane normal*** Cn = (-0.7073, 0.6130, -0.3520)
*in this case both circles have the same radius, but this can change in other problems.
** Both additional points, each by his circle, are given randomly.
**** I calc the Circles plane normal Cn using the fourth points that I have (c1c, c2c, c1p,c2p).
Actually I'm trying to apply the math from
http://paulbourke.net/geometry/circlesphere/ "Intersection of two circles" but that is only for 2D and I need for 3D; and for more that tried to calculate the Z axis not achieved.
two years a go I ask some similar question that I solve using some advices and this triangle idea: Z coordinates of 3rd point (vertex) of a right triangle given all data in 3D , but today I don't have any data of new point, BUT I have the normal.
I get two posibble solutions:
the first one may be is all that I need, BUT mathematics is beyond me pitifully.
The second one is conected with a software called Geometric Tools Engine and I can't get the math or the logic behind that solution.
can you help me with a clear and specific solution?, understanding that I am not a mathematician
thank you.
Best Answer
Define two unit vectors $v$ and $w$ as follows: $$ v={C_{2C}-C_{1C}\over|C_{2C}-C_{1C}|}, \quad w=C_n\times v. $$ You already know how to compute $a$ and $h$, so intersection points are given by: $$ C_{1C}+av\pm hw. $$