I have worked this out on paper and I'm hoping someone can point out my mistake.
A "look at" matrix is supposed to rotate you so your old $(x,y,z)$ coordinate axes become aligned with a new set of coordinate axes $(newx, newy, newz)$. Basically a change of basis.
Working in an LH coordinate system, using row major matrices, according to this document, this rotation matrix should be constructed as:
$$
\left[
\matrix
{
newx.x&newy.x&newz.x \\
newx.y&newy.y&newz.y \\
newx.z&newy.z&newz.z
}
\right]
$$
(Note I have left off the final translation row from the article – assuming we are working with 3×3 matrices here.)
Assume I pass in
- $eye=(0,0,0)$ (so there is no translation, only rotation of the coordinate axes),
- $at=(0,0,1)$ (so we are "looking down the +z axis"),
- $up=(\frac{-1}{2}, \frac{\sqrt{3}}{2}, 0)$ (so we basically are doing a 30 degree rotation about the z-axis).
Compute
- $newz=(0,0,1)$,
- $newx=( \frac{\sqrt{3}}{2}, \frac{1}{2}, 0 )$,
- $newy=(\frac{-1}{2}, \frac{\sqrt{3}}{2}, 0)$
So far we are on track: the $newz$ vector points in the same direction it did before, $newx$ is 30 degrees rotated about the z-axis, and $newy$ is 30 degrees rotated about the z-axis.
Form the rotation matrix:
$$
\left[
\matrix
{
\frac{\sqrt{3}}{2} & \frac{-1}{2} & 0 \\
\frac{1}{2} & \frac{\sqrt{3}}{2} & 0 \\
0 & 0 & 1
}
\right]
$$
(I tested this on a computer and this is exactly the matrix that the library will form, with these inputs),
Yet premultiply with a sample vector like $ \left[ \matrix{1&0&0} \right] $, and get $ \left[ \matrix{ \frac{\sqrt{3}}{2} & \frac{-1}{2}& 0 } \right] $ as the result, meaning $ \left[ \matrix{1&0&0} \right] $ lands rotated negative 30 degrees, not as expected.
If you transpose this rotation matrix though, you will get the expected answer of $ \left[ \matrix{ \frac{\sqrt{3}}{2} & \frac{1}{2}& 0 } \right] $.
Where have I gone wrong?
Best Answer
Ok, I get it. This is an unrotation matrix. The "look at" matrix unrotates things to align with the coordinate axes (ie its an inverse of the rotation matrix that would produce that rotation).