This is an experimental Geometric Algebra library
hendecahedron/taraxacum {:git/url "https://github.com/hendecahedron/taraxacum.git" :sha "0b6fab772eb5577069c2b1eabfa9fa9e62f60066" in your deps.edn
Basis elements are of typeBasiswhich represents oriented volumes of space
(require '[hendecahedron.taraxacum.core :refer :all])
Taraxacum defaults to an unconventional 0-based indexing for bases, so null vectors start at p, but one can use the conventional 1-based indexing too.
Multivectors are linear combinations of bases and are the general objects used in GA expressions. Multivectors are represented with vectors of bases or vectors of coefficients and bases
The macro in-ga evaluates GA expressions in the context of a given GA, e.g.
(in-ga 4 0 1
(∧ [0.1 e0 0.2 e1 0.3 e2 -1.5 e3 1 e4]
[0.7 e0 0.4 e1 0.1 e2 9e-7 e3 1 e4]))
returns the intersection of two planes. (in-ga p q r body) evaluates the body in the context of a GA of the specified metric.
This multivector representation only works inside the macro in-ga, otherwise use (G basis scale) and make a vector of bases.
One can also create an algebra with ga e.g. (let [ga3d (ga 3 0 1)]) for 3d PGA, or (ga prefix p q r) where p q r is the metric signature: p number dimensions squaring to +1, q number dimensions squaring to -1 and r number dimensions squaring to 0.
This algebra is a map containing the basis elements and operations for working with multivectors
(:basis ga3d) is a map of basis elements
(let [{{:syms [e_ e0 e1 e2 e12 e01 e02]} :basis} (ga 3 0 1))
(:ops ga3d) is a map of operations for that GA
(let [{{:syms [* + - • ∨ ∧ ⁻ ∼ 𝑒]} :ops} ga3d]) the main GA operations:
* geometric product
• interior product
∧ exterior product
∼ dual
∨ regressive product
𝑒 exponential
- negate
+ bisection
⁻ invert
other keys: :help help, & :specials various distinguished objects of the algebra like I
(give more examples)
Experimental and definitely going to change, however I'm using the current version for my projects and it works fine.
This library is the result of my notes taken while learning GA, which I'm still learning.
I changed the name from abl-ajr to Taraxacum which is also a corruption of the original Arabic, and dandelion is a corruption of the original French.
My first implementation of this used multimethods, then I rewrote it using an experimental dispatch system and I'll probably rewrite it again.
This library uses the symbols • ∼ 𝑒 ⍣ ∧ ∨ so a GA-specific input source is needed for MacOS -- I used Ukelele to create one and mapped a key to switch sources (System Settings>Keyboard>Shortcuts>Input Sources). For Linux, there'll be a way to do it.
Speed - this implementation isn't slow but it's not optimised either
Symbolic implementation - the problem with floats is you get tiny elements as a result of near-cancellations which I currently filter out but a symbolic version would deal with this. I have tried using rationals instead but they're much slower.
Tests - I know I should add some tests. I'm using this for my other projects so it gets tested by being used but it ought really to have some tests nonetheless.
Geometric Algebra for Computer Science, Leo Dorst, Daniel Fontijne, Stephen Mann — main reference for this code, along with https://geometricalgebra.org, ganja.js and the Java reference implementation
A Guided Tour to the Plane-Based Geometric Algebra PGA, Leo Dorst
The Meet and Join, Constraints Between Them, and Their Transformation under Outermorphism, A supplementary discussion by Greg Grunberg of Sections 5.1—5.7 of the textbook Geometric Algebra for Computer Science, by Dorst, Fontijne, & Mann
Universal Geometric Algebra, David Hestenes
Geometric Algebra Primer, Jaap Suter
and the bivector discord
Thanks to Georgiana and Alex for listening to my ideas. Thanks to Dr Geoff James who many years ago gave an inspiring presentation on Clifford Algebra. And thanks to all in the bivector discord
Copyright © 2021 Matthew Chadwick
Distributed under the Eclipse Public License either version 1.0 or (at your option) any later version.