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True, matrices are convenient for notation, and they tend to be confusing at first. A tutorial like this is better off without the detour through linear algebra.

But without them you lose the ability to smoosh multiple transforms together, making it harder to do anything hierarchical or animated. Personally, I'd place their utility higher than "just" notational convenience.

I feel like we haven't figured out how to teach matrices, that they're inherently easy after you understand them, but we don't know how to introduce or explain them easily. Do you introduce them later & do you have any good resources for them once you do broach the subject with your students?

I like your identity example. I think I really started to "get" matrices after realizing they are literally vectors put in a stack, and that the vectors represent the state of transforming from the identity to those vectors. With that in mind, a matrix can feel much easier than a rotation involving trig functions and hand-coded dot products. You can do all kinds of rotating and other transforming without any trig once you see matrices as transforms rather than an opaque and mysterious brick of numbers. But I admit it took me years to feel that way after my first encounters with matrices.

There's nothing magical about matrix multiplication. Smooshing multiple transforms together just comes down to taking those systems of equations and combining them. If I have one transform that scales space by 2, and another that scales translates space +5 (OK, yeah, this is an affine transform, not a linear one, but I wanted to focus on one coordinate for now), then I have:

    f(x) = 2*x

and

    g(x) = x+5

Using basic high school algebra we can find out that:

    f(g(x)) = 2*x+10

and

    g(f(x)) = 2*x+5

This applies all the way down. The magic isn't in numbers in a square shape, it's in basic algebra. Matrix multiplication comes down to composition of multiple systems of equations like this, and if you start from that, and nothing more than the distributive property of multiplication, you can easily derive "matrix multiplication".

Don't get me wrong. I use matrices in everything "production" for 3D graphics -- the compact notation is extremely convenient. I just wish we'd stop ascribing magical properties to matrices like "without them you lose the ability to smoosh multiple transforms together" because that's clearly false.

I would quickly start drifting if I attended a class on 3D graphics and the teacher started writing everything open without matrices like that. You must admit the usefulness of a short notation when communicating ideas.

While understanding and talking 3D graphics is certainly possible without matrix notation, I really see no reason to purposefully omit it in teaching or other communication. Teaching situation is also a good place to practice standard notation.

I appreciate the response, but I find it rather strange, if you are really teaching 3d graphics to students. I mean, you are technically right at a pedantic level, sure, yes, you can hand-code your matrix multiplies. I didn't mean to suggest that it was impossible, and I'd appreciate some benefit of the doubt before you claim I'm making false statements. I was trying to communicate that it's not practical to do so once you start using animation or hierarchies, and despite your objections and example, I still believe that's true.

You've given a 1-d example. If you tried to do what you're talking about with a character rig, it becomes unwieldy and unfeasible almost immediately. On top of that, the very second you try to do this in 3d with 3+ transforms, you will end up with a square of numbers, you will have derived the concept of a matrix just by trying to avoid them.

I've written lots of 3d transforms by hand using hard-coded dot products just like in your example, and I've written lots of code with matrices too. I've seen others do the same. What I haven't seen is hard-coded matrix multiplies to combine transforms. It is not practical to combine more than a single pair of transforms using manual dot products. In every practical way, you do lose the ability to smoosh transforms together if you don't use matrices. And I never said it was magical, it is simply one of the algebraic benefits of using matrices, and it is easy to understand, and easy to work out the mechanics of. That doesn't mean that it's equivalent in practice, and as it turns out, it is not equivalent in practice. There are significant practical benefits to using matrices once your needs grow past the level of complexity of a basic tutorial.

Re. how to teach matrices: First teach computer graphics, then teach linear algebra :)

I switched universities after the first year, and subsequently I took a lot of courses in the "wrong order". I did computer graphics first and then linear algebra. That's the wrong way around as defined in the curriculum, but it was clearly the right way for me. And for most people I believe too.

When the matrices came, they were clearly a practical vessel for the practical things we needed to to in computer graphics and were getting a little unwieldly and we had an inkling there was a pattern lying underneath for.

Sometimes it is both possible and better to teach the practical hands-on craft first, to create hunger for and an appreciation of the theory, and then teach the theory.

(And then go back and blow up the "craft" ideas and go further still.)

> I prefer to teach 3D graphics without matrices because it's really not anything more complicated than a compact notation.

From a programming perspective it's much less verbose. Also the 1:1 mapping of "spaces" (camera space, world space) makes it very clean to manage transformations between them.

> and isn't easily separable because it steals the "w=1" constant for depth remap and also adjusts the "w" afterwards

I'm sure you already know this but x,y,z,w == x/w,y/w,z/w. It's easy enough to move between a 3D rep and a 4D rep of the same coordinates.

Using 4D coordinates may be somewhat awkward as they're not necessary for representations of points in 3D space but it is a very elegant solution for lighting where it is necessary to represent infinity.

[Edit]

Sorry, the point I was making is that I was also bummed to not see matrices in the article. They are a little bit to wrap you head around but they've a very elegant solution to a lot of stuff in 3D.

Totally, I just think that matrices are a great abstraction around transformations. As long as certain constraints are followed you don't have to care what the previous operations are.

Yes, and you can learn them with your run-of-the-mill Metal tutorial that this tutorial was trying hard not to be.

Look up projective geometry -- homogeneous coordinates are points in 3-d projective space. The defining characteristic of projective points is you can multiply all their coordinates by the same (nonzero) factor without changing the underlying point. (1 0 2 1) equals (2 0 4 2) equals (0.001 0 0.002 0.001). A cool fact about projective space is that every two distinct lines intersect in a point, including parallel lines!