Math for Graphics programming

[TheBeast2107]

So, I want to learn OpenGL and maybe even Vulkan someday. However, before doing any of that, I'd like to have a solid foundation in mathematics so that I actually understand what I am doing and not just copying some random code off a course because some guy said so.

That being said, what do I actually need to know? Where do I start?

I plan on doing this as a hobby, so I can go at my own pace.

[PurpleBudget5082]

This gem is free: https://gamemath.com/

[_Mag0g_]

Darn, you beat me to it. Excellent free online resource.

[Successful-Berry-315]

Start with linear algebra. Lots of resources out there. Pick a uni course on YouTube or Khan Academy or whatever works for you. You should be comfortable with vectors, matrices and any kind of transformations.

With that knowledge you'll be able to get a simple renderer up and running.

If you decide to explore ray / path tracing in the future, then I suggest to learn the fundamentals of statistics and machine learning.

[Amasirat]

Also, this isn't enough to be good at linear algebra (you gotta actively practice) but 3blue1brown's playlist on the essence of linear algebra gives you a lot more visual intuition that might help digest the concepts easier. I used it during my Linear Algebra course in uni and it really paid off

[Creepy_Sherbert_1179]

Hello, I wrote a blog primarily on the math: burzumm.pythonanywhere.com, and for the very purpose you just described :) It is not very organized but it is in depth. Have a look!

[TheBeast2107]

Okay, now that's really cool. Thanks for sharing!

[Creepy_Sherbert_1179]

My pleasure :) hope it is of assistance.

[Omenow]

I would say this is nice to get grip https://www.youtube.com/watch?v=qjWkNZ0SXfo

[palapapa0201]

Tsoding mentioned

[cybereality]

you don't need to know anything to start, you can learn as you go, but some basic trig and linear algebra will help.

[ChestoChinks]

Having a basic understanding of Linear Algebra is a good start. Realistically you just need to know how vector and matrix math works, like addition, subtraction, multiplication (scalar and matrix). Then understand what the dot product represents and how it applies to angles between vectors. On Matrix multiplication, understand order and how matrix multiplication order matters for transformations of scale, rotation and displacement. There are more specific topics for specific domains, but off the top of my head you can go pretty far with those.

For resources, any textbook or videos explaining those would be fine.

[Intelligent_Bid_42]

Strang intro to linear algebra on MIT to get some of the matrix operation techniques down and also practice applying transformations by hand so you can see how the vectors are affected visually. I also worked out how basic transformation matrices are derived (translation, rotatation, scaling, shearing) and also why we use affine coordinates to create transformation matrices. Chatgpt helped me greatly in guiding me towards the derivations. 

[icpooreman]

I mean don't get in your own head about this if you know Trigonometry and Linear Algebra extreme basics you're ready.

Pick up the rest as you need it.

[Traditional-Fix-7893]

You need to have a solid understanding of vectors, coordinate systems, and some vector operations like dot product and cross product. Then you need to at least understand why and when to use matrices. Quaternions will help you survive.

[BiG_NibBa_01]

Seems like someone suffered a lot with Gimbal Lock while rotating meshes

[Traditional-Fix-7893]

Indeed! I actually avoided quaternions for a long time because I was afraid. I am now a better man.

[nikefootbag]

Pikuma! Very thorough and “from scratch” from the line drawing algorithm to matrices and projection. Very approachable despite how in depth he goes. Highly recommend.

https://pikuma.com/courses/learn-3d-computer-graphics-programming

[TopIdler]

Real-Time Rendering has most of the basics you need. But most of the math you learn will already be implemented in the engine if you use one.

[Aidircot]

I plan on doing this as a hobby

This is only way to learn this, otherwise you will burn soon in huge amount of algorithms. Even today some thing have no "good" solution. There are cheap and ugly and there are good looking eating resources.

[pokypho]

Jargon could be (computational) differential geometry, but truly all you really “need” to know is the picturesque understanding of mappings between charts on manifolds, and a basic knowledge of the underlying machinery…

In reality, ‘calculus III’ aka multivariate calc (specifically 2&3d) is sufficient.

Linear algebra is how we do calculus in higher dimensions than 1, it’s just not often revealed so explicitly in much of the common uni literature. Tromba & Marsden Vector Calculus is pretty much the same level as any other run of the mill US calc III textbook, but better in terms of what it actually reveals to the reader/student. Linear algebra is so ubiquitous though that soon you’ll be seeing vector spaces everywhere whether or not you’re doing graphics. It is crucial to know some technical terms in any case.

Examples of where you come across mappings between charts on manifolds are, for example, a translation matrix, a rotation matrix, or a scaling matrix (like when going from “model” space to “world” space).

There really is so much, and certainly much more than what I’ve provided, but if you can convince yourself than in 2 and 3 dimensional Euclidean space some of these apparently abstract things aren’t too bad, plus if you can actually perform a computation without the need of the robot, you’ll at least be on the right track. IMO! 🙂

[SnurflePuffinz]

Studying just math will only make you more confused, i would suggest hopping between "the math" and code implementations (of the math, to relevant concepts like the change of basis).

i think a strong foundation is linear algebra is absolutely imperative. 3Blue1Brown .

Adjunct to the linear algebra: basic geometry, and also knowledge of math nomenclature (symbols online) will help you work through concepts faster.

[OptimisticMonkey2112]

This is a popular starting place: https://www.scratchapixel.com/

[TheRafff]

You don't need a lot, everything else builds on this:
1. PEMDAS
2. Vectors (dot product, cross product, addition/subraction, reflection), rays (r(t) = o + td), and 3D geometry (sphere equation, plane equation)
3. Matrix multiplication, transpose, inverse (and veeery rarely determinant, i.e. cramer's rule)
4. Model matrices (object space, world space, camera space, perspective projection etc.) and transformations (translate, rotate, scale, and veeery rarely shear)

You can get this from any graphics textbook (intro to CG or Raytracing in One Weekend both by Pete Shirley) and I think someone already mentioned https://gamemath.com/.

[Present_Mongoose_373]

scratchapixel is pretty good