Siggraph2019 Geometric Algebra

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.

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]

> 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

Maxwell's theory is hard to understand because it's based on an almost-but-not-quite-appropriate algebra. It's like stringly-typed programming. It works , but it's messy and hard to understand. Maxwell's set of coupled differential vector electromagnetic equations simplify to a hilariously short single equation in Geometric Algebra. A random Google turned up this paper comparing classic EM and the GA formulation. It's not even the simplest possible representation, because that uses natural units and a 4D GA to basically condense the entire set of EM theory into about 4 characters worth of equation that is fully relativistic for free: [link] It's almost a joke. To me it's reminiscent of looking at beginner programmers. You see them do crazy things like calculate a date "next month" by taking apart the pretty-printed date string, parsing to find the month, realising that sometimes the day part is one digit and sometimes it's two, then having to worry about m-d-y or d-m-y formats, building little tables of "days per month" and leap years... and so on. They can write pages and pages of error free code and it's still Wrong because the correct way is to just call "thedatevar.AddMonth(1);" and be done with it. PS: 3D game engine vector algebra libraries have all of this in common with the physics maths. Things like the cross product being bizarre, having to pick a basis, not being able to interpolate rotations, gymbal lock, rounding error, having different maths for 2D and 3D, a bun