Calculating a LookAt matrix

Question

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?

Answer 1

That translation component helps you by creating an orthonormal basis with your "eye" at the origin and everything else expressed in terms of that origin (your "eye") and the three axes.

The concept isn't so much that the matrix is adjusting the camera position. Rather, it is trying to simplify the math: when you want to render a picture of everything that you can see from your "eye" position, it's easiest to pretend that your eye is the center of the universe.

So, the short answer is that this makes the math much easier.

Answering the question in the comment: the reason you don't just subtract the "eye" position from everything has to do with the order of the operations. Think of it this way: once you are in the new frame of reference (i.e., the head position represented by xaxis, yaxis and zaxis) you now want to express distances in terms of this new (rotated) frame of reference. That is why you use the dot product of the new axes with the eye position: that represents the same distance that things need to move but it uses the new coordinate system.

Answer 2

Dot product simply projects a point to an axis to get the x-, y-, or z-component of the eye. You are moving the camera backwards so looking at (0, 0, 0) from (10, 0, 0) and from (100000, 0, 0) would have different effect.