People must get taught math terribly if they think "I don't need to worry about piles of abstract math to understand a rotation, all I have to do is think about what happens to the XYZ axes under the matrix rotation". That is what you should learn in the math class!

Anyone who has taken linear algebra should know that (1) a rotation is a linear operation, (2) the result of a linear operation is calculated with matrix multiplication, (3) the result of a matrix multiplication is determined by what it does to the standard basis vectors, the results of which form the columns of the matrix.

This guy makes it sound like he had to come up with these concepts from scratch, and it's some sort of pure visual genius rather than math. But... it's just math.

▲ omnicognate an hour ago | parent | next [-]

A lot of people who find themselves having to deal with matrices when programming have never taken that class or learned those things (or did so such a long time ago that they've completely forgotten). I assume this is aimed at such people, and he's just reassuring them that he's not going to talk about the abstract aspects of linear algebra, which certainly exist.

I'd take issue with his "most programmers are visual thinkers", though. Maybe most graphics programmers are, but I doubt it's an overwhelming majority even there.

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foofoo12 an hour ago | parent [-]

> most programmers are visual thinkers

I remember reading that there's a link between aphantasia (inability to visualize) and being on the spectrum.

Being an armchair psychologist expert with decades of experience, I can say with absolute certainty that a lot of programmers are NOT visual thinkers.

	▲	voidUpdate an hour ago \| parent [-]
		Do you have anything I can read about that? I'm definitely on the spectrum and have whatever the opposite of aphantasia is, I can see things very clearly in my head

▲ zelphirkalt an hour ago | parent | prev | next [-]

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.

▲ dgacmu an hour ago | parent | next [-]

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 an hour ago \| parent [-]
		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.

▲ aqme28 15 minutes ago | parent | prev | next [-]

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.

▲ scarmig an hour ago | parent | prev [-]

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).

▲ vasco 2 hours ago | parent | prev [-]

Math is just a standardized way to communicate those concepts though, it's a model of the world like any other. I get what you mean, but these intuitive or visualising approaches help many people with different thinking processes.

Just imagine that everyone has equal math ability, except the model of math and representations of mathematical concepts and notation is more made for a certain type of brains than others. These kind of explanations allow bringing those people in as well.