I'm in the midst of writing a 3d engine and I've come across the LookAt algorithm described in the DirectX documentation:
zaxis = normal(At - Eye)
xaxis = normal(cross(Up, zaxis))
yaxis = cross(zaxis, xaxis)
xaxis.x yaxis.x zaxis.x 0
xaxis.y yaxis.y zaxis.y 0
xaxis.z yaxis.z zaxis.z 0
-dot(xaxis, eye) -dot(yaxis, eye) -dot(zaxis, eye) l
Now I get how it works on the rotation side, but what I don't quite get is why it puts the translation component of the matrix to be those dot products. Examining it a bit it seems that it's adjusting the camera position by a small amount based on a projection of the new basis vectors onto the position of the eye/camera.
The question is why does it need to do this? What does it accomplish?