How does glm::lookAt produce a View Matrix?

Question

Say we do this:

glm::mat4 View = glm::lookAt(glm::vec3(4,3,-3), glm::vec3(0,0,0),glm::vec3(0,1,0));

And after printing to the console with glm tostring (column major order):

View = -0.600000 -0.411597 0.685994 0.000000 0.000000 0.857493 0.514496 0.000000 -0.800000 0.308697 -0.514496 0.000000 -0.000000 -0.000000 -5.830953 1.000000

This matrix works as intended. I multiplied it by a perspective projection matrix and passed the resulting matrix in to my vertex shader to be multiplied by the vertices of a cube with side length = 1. So the full expression for each projected vertex is:

VERTEX VIEW PERSPECTIVE PROJECTION p.x -0.600000 -0.411597 0.685994 0.000000 0.629325 0.000000 0.000000 0.000000 p.y * 0.000000 0.857493 0.514496 0.000000 * 0.000000 0.839100 0.000000 0.000000 p.z -0.800000 0.308697 -0.514496 0.000000 0.000000 0.000000 -1.002002 -1.000000 1.0 -0.000000 -0.000000 -5.830953 1.000000 0.000000 0.000000 -0.200200 0.000000

Which produces this:

However, I was under the impression that to account for the translation of the camera, the camera's coordinates should be inverted and placed in the fourth column of the View Matrix. So I tried this:

 VERTEX VIEW PERSPECTIVE PROJECTION p.x -0.600000 -0.411597 0.685994 -4.000000 0.629325 0.000000 0.000000 0.000000 p.y * 0.000000 0.857493 0.514496 -3.000000 * 0.000000 0.839100 0.000000 0.000000 p.z -0.800000 0.308697 -0.514496 3.000000 0.000000 0.000000 -1.002002 -1.000000 1.0 -0.000000 -0.000000 -0.000000 1.000000 0.000000 0.000000 -0.200200 0.000000

Which produces this:

Why does the first View Matrix work but not the second one?

Ian · Accepted Answer · 2021-02-09 19:56:00Z

Alright so upon looking into it further, here's how glm::lookAt produces a View Matrix:

Make a row-major ordered 4x4 Translation Matrix by negating the camera position, c:

 1 0 0 0 0 1 0 0 0 0 1 0 -c.x -c.y -c.z 1

Multiply it by a column-major ordered 4x4 Rotation Matrix that you make using the formula here: https://www.khronos.org/registry/OpenGL-Refpages/gl2.1/xhtml/gluLookAt.xml

 TRANSLATION ROTATION 1 0 0 0 x.x y.x z.x 0 VIEW = 0 1 0 0 • x.y y.y z.y 0 0 0 1 0 x.z y.z z.z 0 -c.x -c.y -c.z 1 0.0 0.0 0.0 1

So what I think is going on is you're effectively inverting the camera-to-world matrix to get the world-to-camera matrix. The above equation produces a column-major ordered View Matrix.

And for people like me who like row-major ordered matrices (contiguous memory gang), you can do this with a row-major ordered rotation matrix to produce a row-major ordered View Matrix. Notice the multiplication order must be swapped:

 ROTATION TRANSLATION x.x x.y x.z 0 1 0 0 -c.x VIEW = y.x y.y y.z 0 • 0 1 0 -c.y z.x z.y z.z 0 0 0 1 -c.z 0.0 0.0 0.0 1 0 0 0 1

"So what I think is going on is you're effectively inverting the camera-to-world matrix to get the world-to-camera matrix." Bingo. — DMGregory
– DMGregory ♦, Commented Feb 9, 2021 at 18:43
On step 2. you should add that after the TRANSLATION * ROTATION multiplication the final matrix is transposed in order to get the world-to-camera result — PentaKon
– PentaKon, Commented Jan 10, 2022 at 11:06

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