Thanks to you, I just spent half my morning trying to solve this without getting anywhere. I tried using vectors a, b and c that i project unto the xOy plane to get the sum of three parallelograms.
But then the problem is to parametrize these three vectors' positions. The first one can be define in spherical coords as (acos(phi)sin(psi), asin(phi)sin(psi), a*cos(psi)) but then I need to introduce a third parameter for vector b and write both b and c using this system...
I'll give it another try this afternoon using matrices.
EDIT: Alright, I guess you can write a, b and c as a rotational matrix using Euler angles ( http://en.wikipedia.org/wiki/Euler_angles ). You can probably also cancel one of the angles (let's say theta) by rotating your xOy plane around which doesn't change the problem.
You then just have to calculate S using phi and psi.
S = ||a^b||+||b^c||+||c^a||
and solve
dS/d(psi) = 0 and dS/d(phi) = 0
However, that's way too much trig for me for today so I'll leave it to someone more courageous than me.
But then the problem is to parametrize these three vectors' positions. The first one can be define in spherical coords as (acos(phi)sin(psi), asin(phi)sin(psi), a*cos(psi)) but then I need to introduce a third parameter for vector b and write both b and c using this system...
I'll give it another try this afternoon using matrices.
EDIT: Alright, I guess you can write a, b and c as a rotational matrix using Euler angles ( http://en.wikipedia.org/wiki/Euler_angles ). You can probably also cancel one of the angles (let's say theta) by rotating your xOy plane around which doesn't change the problem.
You then just have to calculate S using phi and psi.
S = ||a^b||+||b^c||+||c^a||
and solve
dS/d(psi) = 0 and dS/d(phi) = 0
However, that's way too much trig for me for today so I'll leave it to someone more courageous than me.