Bivector
8 HN recommenders · recommended in 11 months, 2020–2022
1:37:22 · 87K views · Nov 6, 2019
A SIGGRAPH 2019 course on geometric algebra. Charles Gunn gives an intro, then Steven De Keninck gives a programmer-focused course with live examples in ganja.js.
HN commenters repeatedly point people to it as the most accessible, developer-friendly introduction to geometric algebra and its use in graphics and physics.
17 comments from Hacker News
Ok look just watch this mind blowing video: this video After that, you can’t argue that geometric algebra is not THE right way to do geometry. Projective geometry (in any dimension) is a subset of geometric algebra. Please watch it.
> Quite bluntly, this all sounds like an attempt to reinvent euclidean geometry following a convoluted way. I mean, what does all this buy you that applying a subset of affine transformations (scaling, translation, rotation) to an orthogonal coordinate system doesn't give you already? It is attempting to reinvent Euclidean geometry, yes. I don't think it's convoluted though. To give a prime example, take the article we're commenting on: interpolating rotations. Or more generally: interpolating transformations. Just doing this with rotations without suffering gimbal locks already brings you to quaternions. Are quaternions 'convoluted'? The fact that all objects are native to the algebra means they're composable. Take for example this slide of the formula of a 4D torus in coordinates, and in 4D PGA: [LINK] The talk in general has a bunch of example applications: this video Questions such as "the intersection of this line and this plane", "the line through two points" "the circle where these two spheres intersect", "the point at the intersection of three planes", "the projection of this line on this plane" and such are trivial, native (the resulting object is part of the algebra) and exception-free in geometric algebra. E.g. two planes always intersect, it just happens that the intersection is a line at infinity if they're parallel. The exact same code used to translate and rotate a point around the origin can be used to translate and rotate a line, or a plane around the origin. Al
Well I've been on a real Geometric Algebra (aka Clifford Algebra) kick lately, and ran across ganja.js [1]. It's a single no deps file that is...impressive. 120k uncompressed, and with it you can construct any degree algebra (including the more esoteric hyperbolic/parabolic ones), render to canvas, svg or webgl(!). It also includes a clever little DSL parser and interpreter (it overloads the scientific notation to name basis vectors!) that lets you construct more complex things from simple things using various kinds of products. The author, Steven De Keninck, is quite impressive as well, having got his start in the demoscene some time ago. He has a good video from SIGGRAPH 2019 [2] that explains why this algebra is better than [matrices, tensors, vectors, complex numbers]. Of particular interest (to me anyway) is the 2D projective geometry. I don't want to oversell it, but ganja is fucking amazing and there is a great deal I want to do with it. For one, I'd like to recapitulate my physics degree with it. [1] [LINK] [2] this video
I think that simple should be understood here as easy to use . As a crude analogy, the mechanical principles underlying a car with an automatic shift aren't simple, but the abstraction it provides makes driving much easier to learn. A more mathematically grounded example could be the complex plane compared to the real line: when encountered for the first time, it is not simple in any way, but it offers a clean solution to many (originally real valued) problems (perhaps by the fact that it is algebraically closed?). NB: the author of this library also made a very interesting presentation of geometric algebra (using this library as vehicle) for SIGGRAPH 2019: this video It has been posted a few times on HN before.
Here is a video this video and the associated HN thread [LINK] that provide some more background on GA and its applications. I think you're generally right that GA is a rebranding of concepts we already knew about, however representing them in a uniform manner seems like a useful thing to do. The at t=4083 in the video they show some interesting examples of projections, rotations, and other geometric operations that can be done quite simply this video