3D Camera coordinates to world coordinates (change of basis?)

Question

Suppose I have the coordinates X, Y, Z and orientation Rx, Ry, Rz of an object with respect to a camera. In addition, I have the coordinates U, V, W and orientation Ru, Rv, Rw of this camera in the world.

How do I transform the position (location and rotation) of the object to its position in the world?

It sounds like a change of basis to me, but I haven't found a clear source yet.

Answer 1

In fact you have two basis : one relative to the camera, the other is absolute (the world). So you basically want to transform your relative data into absolute data.

Location

This is the easiest one. You have to translate the (X,Y,Z) position by the vector t(U,V,W). So all your positions in absolute are (Ax, Ay, Az) = (X,Y,Z)+t = (X+U,Y+V,Z+W).

Orientation

This is a bit more difficult. You have to find the rotation matrix that rotate your camera from (I assume) (0,0,1) to (Ru,Rv,Rw). You should look at Basic Rotation Matrices in order to decompose the 2 rotations that take (0,0,1) to (Ru,Rv,Rw) (one according to X axis, one according to Z axis for example). I advise you to draw the absolute basis and the vector (Ru,Rv,Rw) on a sheet of paper, it is the simplest way to get the right result.

So you have 2 basic rotations matrices r1 and r2. The resultant rotation matrix r = r1*r2 (or r2*r1, it doesn't matter). So the absolute orientation of your object is (ARx, ARy, ARz) = r*(Rx,Ry,Rz).

Hope this helps !