How can I find Va given Vo and angle A on a 3D sphere?

[Ed]

Hi folks, I'm hoping for a little help with something which I'm sure should be pretty easy... but I don't have access to any kind of maths textbooks to look anything up, sadly, so I'm hoping you guys might help?

Let's consider a point P on the surface of a sphere, and define Vo as the unit vector from the origin of the sphere to this point. This vector also describes the normal to the plane which is tangential to the sphere at P.

Now let Va be a (unit) vector within this plane, i.e. Va is perpendicular to Vo, and the angle A describes the angle subtended by Va and the point within the plane which intersects the sphere's +Z axis.

Another way of looking at this would be: if Vz is the vector between P and the Z axis, within the plane normal to Vo, the angle A is between Va and Vz.

So now the question... if Vo and A are known... how do I get Va? (in terms of cartesian co-ords)

edit: made it a little clearer, I hope, and added a diagram drawn in MS Word which I hope should also help.

 

[kishtik]

It looked easy using an angle transformation... (You know the polar coors? Good.) Just assume that P is on the x-axis (or y if you want) and Va is now perpendicular to x right? Proceed after this and finally transfer (xva, 0, 0) to your P point. (Later, you can find the carthesian coordinates by tr.(so that wasnt `finally`) Sorry for not being clear about the transformations as i have little time.

 

[Ed]

You're quite right, by assuming it's on an axis it makes it possible to reduce the problem to a couple of 2D basic trig problems. Thanks for the reply.