3 ms·

Say you know the coordinates of all the corners of a cube in three dimensions. If you wanted to move that cube to the side and turn it, you could first add some

Say you know the coordinates of all the corners of a cube in three dimensions. If you wanted to move that cube to the side and turn it, you could first add some fixed number to all the x coordinates of every point, then do some trig to find the coordinates again after a rotation. Turns out you can produce the same output with fewer steps using matrix math. That is, you can produce a single matrix which is representative of the transform "move right m units and turn d degrees". And you can apply that transform matrix to any set of points you want. At least, this is what I've gleaned so far as someone that doesn't really understand. I welcome corrections.

ColinWright 14y ago
That's the best answer I've had so far, but it seems like it's just talking about matrix operations. There doesn't really seem to be any "proper" Linear Algebra in here. Nothing about bases, change of bases, null spaces, proofs that all bases have the same size, eigen-values and eigen-vectors, none of the juicy stuff. Just matrix multiplication. So I was wondering if the more advanced stuff is used, or if it really is just the dabbling in the shallows. Thanks.
- mjn 14y ago
  I wrote a short introduction to some of the projection math used in 3d graphics, which gets slightly into it, though not into greatly juicy stuff: http://www.kmjn.org/notes/3d_rendering_intro.html http://www.kmjn.org/notes/3d_rendering_intro.html Beyond just matrix arithmetic, a change of basis is usually used in viewing, to change an x-y-z "world" coordinate space into a u-v-w camera-relative coordinate space (discussed in the "View space" section). The homogeneous coordinates used to encode affine transforms into matrix multiplications also have a rigorous linear-algebra background, but admittedly they can be treated as just a strange hack (use a "dummy" 4th dimension for 3d coordinates) without really needing the theory.

ColinWright 14y ago

That's the best answer I've had so far, but it seems like it's just talking about matrix operations. There doesn't really seem to be any "proper" Linear Algebra in here. Nothing about bases, change of bases, null spaces, proofs that all bases have the same size, eigen-values and eigen-vectors, none of the juicy stuff. Just matrix multiplication. So I was wondering if the more advanced stuff is used, or if it really is just the dabbling in the shallows. Thanks.

mjn 14y ago
I wrote a short introduction to some of the projection math used in 3d graphics, which gets slightly into it, though not into greatly juicy stuff: http://www.kmjn.org/notes/3d_rendering_intro.html http://www.kmjn.org/notes/3d_rendering_intro.html Beyond just matrix arithmetic, a change of basis is usually used in viewing, to change an x-y-z "world" coordinate space into a u-v-w camera-relative coordinate space (discussed in the "View space" section). The homogeneous coordinates used to encode affine transforms into matrix multiplications also have a rigorous linear-algebra background, but admittedly they can be treated as just a strange hack (use a "dummy" 4th dimension for 3d coordinates) without really needing the theory.