Comment by zelphirkalt | Hacker News Reader

Comment by zelphirkalt | Hacker News Reader

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When I was studying and made the mistake of choosing 3D computer graphics as a lecture, I remember some 4x4 matrix that was used for rotation, with all kinds of weird terms in it, derived only once, in a way I was not able to understand and that didn't relate to any visual idea or imagination, which makes it extra hard for me to understand it, because I rely a lot on visualization of everything. So basically, there was a "magical formula" to rotate things and I didn't memorize it. Exam came and demanded having memorized this shitty rotation matrix. Failed the exam, changed lectures. High quality lecturing.

Later in another lecture at another university, I had to rotate points around a center point again. This time found 3 3x3 matrices on wikipedia, one for each axis. Maybe making at least seemingly a little bit more sense, but I think I never got to the basis of that stuff. Never seen a good visual explanation of this stuff. I ended up implementing the 3 matrices multiplications and checked the 3D coordinates coming out of that in my head by visualizing and thinking hard about whether the coordinates could be correct.

I think visualization is the least of my problems. Most math teaching sucks though, and sometimes it is just the wrong format or not visualized at all, which makes it very hard to understand.

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

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.

Tricks of the * Game Programming Gurus :)

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.

> 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.

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.

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.

The belt trick is about smoothly mapping paths through SO(3) to each other, not about if any point in SO(3) is reachable from any other point. I think you are confusing the notion of 'simple-connected' with just 'connected'.

No, the belt trick is about the fundamental group of SO(n) where n>2 is Z_2and the fact that a 4pi rotation is homotopic to the identity.

It is often introduced as RP^3 being isomorphic to SO(3) and as RP^3 is not simply being connected thus SO(3) cannot be.

They don’t explain why until students have the ability to deal with 4D 3-spheres being projected on 4D hyperplanes.

You can build an intuition with a 2-sphere being projected through the origin onto a 2d plane. This will demonstrate how the antipodes are not uniquely identified.

If you go from SU(2), which is the double cover of SO(3) you can use basic algebra to reduce it to a^2+b^2+c^2+d^2=1 where a,b, and c are your free variables and figure it out.

It works well without that complexity at lower levels because people are actually rotating the clopen set, which is only allowed to be the entire body or the empty set.

The 4pi symmetry applies to the topological properties of neurons and the plate trick in the real world because you are translating a portion or a property of the body and not the entire body.

Same thing on how they don’t introduce Tori topology when using Euler angles.

Pick up basically any topology or Lie algebra book targeted at first year grad students and the topic will be covered.

Bredon’s Topology and Geometry is one I personally like.

I should add that “connectedness” is a local property that has to be everywhere.

The antipode pairs don’t have unique identities they are identified together, but IIRC they are never close enough to break that local condition of connectivity.

You could use the unit circle and a line at x=2, connecting points of that line through the origin and see how that identification of two points on the unit circle happens with that one line.

That may help but it wouldn’t explain the issues with so(3) that results in a 720 deg rotation being equivalent to a 0 deg rotation.

It would explain the point at infinity concepts as the poles are indescribable with a line from X=2 through the origin.

> 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.

In computer graphics, 4x4 matrices let you do a rotation and a translation together (among other things). There's the 3x3 rotation block you found later as well as a translation vector embedded in it. Multiplying a sequence of 4x4 matrices together accumulates the rotations and translations appropriately as if they were just a bunch of function applications. i.e. rotate(translate(point)) is just rotation_matrix * translation_matrix * point_vector if you construct your matrices properly. Multiplying a 4x4 matrix with another 4x4 matrix yields a 4x4 matrix result, which means that you can store an arbitrary chain of rotations and translations accumulated together into a single matrix...

Yeah you need to build up the understanding so that you can re-derive those matrices as needed (it's mostly just basic trigonometry). If you can't, that means a failure of your lecturer or a failure in your studying.

The mathematical term for the four by four matrices you were looking at is "quaternion" (I.e. you were looking at a set of four by four matrices isomorphic to the unit quaternions).

Why use quaternions at all, when three by three matrices can also represent rotations? Three by three matrices contain lots of redundant information beyond rotation, and multiplying quaternions requires fewer scalar additions and multiplications than multiplying three by three matrices. So it is cheaper to compose rotations. It also avoids singularities (gimbal lock).

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