It's Friday and I've gotten hyper-fixeated by #geometricalgebra
Specifically, I'm trying to figure out what directions an ant on the surface of a unit sphere would have to travel to intercept another one moving with a constant know speed.
There is a really nice relationship between tangent vectors and rotors, but solving for the time to intercept involves, among other things, taking the log of a bivector, which I didn't even think was possible until today
@gnomekat
There must be something in the air - I was thinking recently what use an infinite dimensional Clifford algebra might be put to. Not quite sure yet. Still mulling that one. LOL
#maths #GeometricAlgebra
If functions can be represented as vectors and you can do the dot product between them...
What happens if you try to use the wedge product between two functions?
This question just popped into my brain
#geometricalgebra #math
Wow, a riveting tale of sunset geometry 🌅📐—because who doesn't want to spend their evening calculating the Earth's radius with a single sunset photo? 🤓🔍 Clearly, classical #trigonometry just wasn't making math nerdy enough, so let's sprinkle in some Geometric Algebra for extra pizzazz. 🎉
https://www.shapeoperator.com/2016/12/12/sunset-geometry/ #sunsetgeometry #mathnerd #GeometricAlgebra #STEMeducation #HackerNews #ngated
Trying to explain to people why #geometricalgebra is so much better is really frustrating
I watched this excellent Swift Introduction to Geometric Algebra, https://youtu.be/60z_hpEAtD8?si=Gv_rHCafbNXj3WXr . I enjoyed it! Geometric algebra allows you to multiply and divide scalars, vectors, and other geometric objects. It's fascinating and helps make sense of concepts like imaginary numbers and quaternions.
#geometricAlgebra
#quaternions
#vectors
I also want to explore more #GeometricAlgebra applied to #physics and #ComputationalPhysics (not necessarily for #FluidDynamics and #CFD, but that would be preferable as it's obviously our primary topic of relevance). I can't seem to find anything that combines #SPH and GA, so it might even lead to some new interesting venues to explore.
3/n
On a more practical note, I think #GeometricAlgebra may be useful for the redesign of the #ConstructiveSoldiGeometry subsystem that we used in #GPUSPH to allow programmatic definition of the test case geometry (where the fluid is, the shape of the boundaries, how they move, etc). We do need to redesign it, and from what I've learned about GA so far, it would really help us “do it right”, in a relatively simple way.
2/n
I'm (slowly) learning #GeometricAlgebra. There's several reasons for that. (The slowly there's only one: time. I mean several reason why I want to learn GA.)
The first is, shall we say, philosophical: I find it almost insulting that, despite by otherwise pretty solid background in mathematics, I only ever heard for the first time about the existence of GA a couple of years ago. That such a powerful mathematical framework is still largely unknown is … inappropriate.
1/n
@internic
'Outer product' and 'tensor product' are soooo last millennium - all the cool kids are doing geometric algebra these days.
#maths #algebra #mathematics #GeometricAlgebra
After my initial frenzy of reading and watching content (and taking copious notes) about #GeometricAlgebra when I first heard about it, I've settled on a list of heavier resources to consume (mostly books) and haven't made much progress, as usual with books these days 😅 (1/3)
ok I think I got vanilla #geometricalgebra properly learned... now time to see about the projective and conformal models .. more fun to be had I think
ᕙ༼◕ ᴥ ◕༽ᕗ
Welp. I guess it's time for codegen.
Today I crashed clang trying to evaluate my generic functions.
Last couple days I've been working on doing rigidbody dynamics in plane-based #GeometricAlgebra which has been really cool.
I've learned about a bunch of new stuff, like lie algebra interpolators and integrators.
Turns out that PGA makes interpolating between orientations basically trivial, and it integrates rotation and translation together so I don't have to do any extra work to support torques. The only thing from angular dynamics I have to do is use an inertia tensor.
Solving my compile time issues wasn't so hard. Just changed the implementation file for my projective #GeometricAlgebra library to not use generic dispatch. Using a ton of generic dispatch in user code will of course slow things back down, but I'm less worried about that because it can be alleviated as needed.
If I were using a codegen approach this wouldn't have been a concern because I could just monomorphize all the functions and only use _Generic for function dispatch in two layers. Oh well
Well, that's done now. :)
Finished my _Generic support for my 3d plane-based #GeometricAlgebra library in C.
Now I have another issue. Gotta figure out how I can reduce build times. All these inline functions are making me feel like I'm compiling Rust or something, lol.
Today I got started doing the 3d plane-based #GeometricAlgebra library and getting it finished up with _Generic support. This mostly wraps up the math library for my game engine, at which point I can move on from this.
If I end up feeling like I want to optimize things I might eventually follow along with Look, Ma, No Matrices! and work to reduce the number of multiplies required, but honestly I think that's mostly useful for shaders. DCE and inlining work well enough for CPU side.
see a Michael Penn video about why the cross product doesn't exist in 4d and all the comments are about #geometricalgebra
yess yess it's spreading.. good..
As of today I think my 2d projective #GeometricAlgebra implementation in C is done!
I'll be moving on to my 3d implementation to flesh it out and add type generic functionality to it, but for now just 2d PGA is enough.
Now on to the next steps in my engine, working on the systems part of my species-based #ECS. That will include component queries, system initialization and deinitialization, and more.
In spite of me railing on _Generic in C these last few days, it's been a quite useful tool.
Today I'm getting very close to a complete implementation of a 2d type-generic plane-based #GeometricAlgebra library for the C #programming language, primarily useful for my game engine which has been the focus of my #GameDev work for the last while.
Next steps for me after this will be doing the same for my partially-complete implementation of a 3d version.
