Comment by dgacmu

Comment by dgacmu 9 hours ago

You can do rotation with a 3x3 matrix.

The first lecture was using a 4x4 matrix because you can use it for a more general set of transformations, including affine transforms (think: translating an object by moving it in a particular direction).

Since you can combine a series of matrix multiplications by just pre-multiplying the matrix, this sets you up for doing a very efficient "move, scale, rotate" of an object using a single matrix multiplication of that pre-calculated 4x4 matrix.

If you just want to, e.g., scale and rotate the object, a 3x3 matrix suffices. Sounds like your first lecture jumped way too fast to the "here's the fully general version of this", which is much harder for building intuition for.

Sorry you had a bad intro to this stuff. It's actually kinda cool when explained well. I think they probably should have started by showing how you can use a matrix for scaling:

    [[2, 0, 0],
     [0, 1.5, 0],
     [0, 0, 1]]

for example, will grow an object by 2x in the x dimension, 1.5x in the y dimension, and keep it unchanged in the z dimension. (You'll note that it follows the pattern of the identity matrix). The derivation of the rotation matrix is probably best first derived for 2d; the wikipedia article has a decentish explanation:

https://en.wikipedia.org/wiki/Rotation_matrix

kevindamm 9 hours ago

The first time I learned it was from a book by LaMothe in the 90s and it starts with your demonstration of 3D matrix transforms, then goes "ha! gimbal lock" then shows 4D transforms and the extension to projection transforms, and from there you just have an abstraction of your world coordinate transform and your camera transform(s) and most everything else becomes vectors. I think it's probably the best way to teach it, with some 2D work leading into it as you suggest. It also sets up well for how most modern game dev platforms deal with coordinates.

Reply View 2 replies

jesuslop 6 hours ago

Think OpenGL used all those 2,3,4D critters at API level. It must be very hardware friendly to reduce your pipeline to matrix product. Also your scene graph (tree) is just this, you attach relative rotations and translations to graph nodes. You push your mesh (stream of triangles) at tree nodes, and composition of relative transforms up to the root is matrix product (or was the inverse?) that transform the meshes that go to the pipeline. For instance character skeletons are scene subgraphs, bones have translations, articulations have rotations. That's why it is so convenient to have rotations and translations in a common representation, and a linear one (4D matrix) is super. All this excluding materials, textures, and so on, I mean.

Reply View | 0 replies
z0r 7 hours ago

Tricks of the * Game Programming Gurus :)

Reply View | 0 replies

motorest 5 hours ago

> The first lecture was using a 4x4 matrix because you can use it for a more general set of transformations, including affine transforms (think: translating an object by moving it in a particular direction).

I think this is mixing up concepts that are orthogonal to linear spaces, linear transformations, and even specific operations such as rotations.

The way you mention "more general set of transformations" suggests you're actually referring to homogeneous coordinates, which is a trick that allows a subset of matrix-vector multiplication and vector addition in 3D spaces to be expressed as a single matrix-vector multiplication in 4D space.

This is fine and dandy if your goal is to take a headstart to 3D programming, where APIs are already designed around this. This is however a constrained level of abstraction above actual linear algebra, which may be and often is more confusing.

Reply View 0 replies

nyrikki 7 hours ago

> You can do rotation with a 3x3 matrix.

You can do a rotation or some rotations but SO(3) is not simply connected.

It mostly works for rigid bodies centered on the origin, but gimbal lock or Dirac's Plate Trick are good counter example lenses. Twirling a baton or a lasso will show that 720 degrees is the invariant rotation in SO(3)

The point at infinity with a 4x4 matrix is one solution, SU(3), quaternions, or recently geometric product are other options with benefits at the cost of complexity.

Reply View 2 replies

Ono-Sendai 7 hours ago

I think you are confused about what 'simply connected' means. A 3x3 matrix can represent any rotation. Also from a given rotation there is a path through the space of rotations to any other rotation. It's just that some paths can't be smoothly mapped to some other paths.

Reply View | 1 reply
- nyrikki 2 hours ago
  
  SO(3) contains all of the orthogonal 3x3 matrices of determinant 1.
  If you are dealing with rigid bodies rotated though the origin like with the product of linear translations you can avoid the problem. At least with an orthonormal basis R^3 with an orthogonal real valued 3x3 matrix real entries which, where the product of it with its transpose produces the identity matrix and with determinant 1
  But as soon as you are dealing with balls, where the magnitude can be from the origin to the radius, you run into the issue that the antipodes are actually the same point, consider the north and south poles being the same point, that is what I am saying when the topology is not simply connected.
  The rigid body rotation about the origin is just a special case.
  Twist a belt twice and tape one end to the table and you can untwist it with just horizontal translation, twist it once (360deg) and you cannot.
  
  Reply View | 0 replies