Solid angles and position vectors

[Mattew]

I've already posted this question in the math section, but since I got no reply I'll try it here (sorry for the cross-posting).
I'm using solid angles to define directions of objects moving from the centre of the sphere towards all points in the space around, which means I divides the (4pi) solid space around the centre in K-> infinity directions, each one defined by a solid angle w(i). It is a procedure commonly used in 2D, where each object departing from the centre of a disk chooses its direction in [0, 2pi]...the only difference appears to be the magnitude of the entire space, which is 4pi (solid angle of the sphere) in this case.
Now my problem is: If I have two position vectors defininig two of the objects movements in directions w1 and w2, how do I find the angle between them whithout introducing further coordinates (polar or x,y,z axis?). Is there a possibility to find the relative direction (each one defined by a soli angle) of the two vectors based on the only w parameter?
In 2D, calling teta1 and teta2 (in [0,2pi]) the directions of the two objects, i would graphically represent them on a x-y cartesian system and find the angle between them as teta2 - teta1, so the sum vector of the two would be sqrt[(v1cos(teta2-teta1))^2 + v2^2 sin(teta2 - teta1)^2 ], but I can't figure out how it works I am my 3D framework.
Can anybody give me some hints?
Thanks

 

[Doc Al]

How are you specifying the direction of your position vectors without a coordinate system?

 

[Mattew]

Well, that's mainly what I'm concerned about: if if you think about it geometrically, the spherical space around p (the origin of our coordinate system) is divided into cones with vertex in P, with the solid angle of the cone determining a direction, then passing to the limit (-> infinity) you should have enough infinitesimal solid angles to cover all the space around p, which means all the directions. Now we have objects departing from p and choosing a direction omega(i) according to a uniform distribution in [0,4pi].
That's only a theoric point of view but it should work, then the problem is how to express distances between direction vectors with origin in P and direction omega(i)...which probably leads us back to express the solid angles with the two polar coordinates...or not?

 

[Doc Al]

Now we have objects departing from p and choosing a direction omega(i) according to a uniform distribution in [0,4pi].

A solid angle is not a direction. 4pi is just the total solid angle. I don't see how you can specify the position vector on the unit sphere with a single parameter--you need 2 dimensions. Why are you trying to avoid polar coordinates?

 

[Mattew]

To explain you you why I'll start from beginning: I have a 2D framework representing a Boolean Model distributed class of sensors and a a moving target, in which sensors choose their directions following a uniform distribution in [0,2pi] and so the target does...this means that we can compute relative velocity of a sensor (moving towards the angle teta1) and the target (moving towards teta2) according to the angle teta2-teta1. Stochastic geometry is applied to state the hit time between the two, which leads to quite complicated computations. I'm now transporting this model in 3D, and the simplest way to do it (considering stochastic and infinite directions) was to consider, as I told, the sphere instead of the disk and the solid angle instead of the planar one.
Actually, ad you said solid angles are not directions, but if you think of dividing the spherical space around the origin p in many identic cones with vertex in p, then the space defined by each cone and the sphere is a solid angle omega, a fraction of the total 4pi of the sphere. Intuitively, as the cone gets smaller (or if you prefer considering the axis of the cones), that angle omega could be seen as a direction followed by a sensor moving from p, more or less as teta was the direction in the planar case. That's the idea, which has worked as far as I didn't have to consider relative positions of two objects (the first part of my work was just stochastic applied to the class of sensor for determining area coverage). Going back to polar coordinates would add a lot of stuff in computing relative velocity of the two objects and applying to it stochastic theorems that I'm already using in the planar case...