archived in favor of three/four bundle "thor" to be published on a later date!

three

Roblox Lua library for 3D multivectors, the geometric product and associated operations

what/why

The space of 3D multivectors is a space of regular 3D vectors and an assortment of objects, all of which become transformations under the geometric product, a useful way to multiply vectors. Projections, rotations, reflections are defined with this product in a way that is practical, but also illuminating, suggestive, and privy to tinkering.

The language of geometric algebra is quickly taking the stage as a universal representation of transformations in space.

Please take some time to understand these captivating objects: https://www.youtube.com/watch?v=60z_hpEAtD8

usage

Place contents of three.lua into ReplicatedStorage
local three = require(game:GetService("ReplicatedStorage").three)
To specify a multivector directly: local m1 = three(a, x, y, z, xy, xz, yz, im)

• a describes the scalar component m1.scalar (returns a number)
• x, y, z describes the vector component m1.vector (returns a Vector3)
• xy, xz, yz describes the pseudovector component m1.pseudovector (returns a pure bivector Three object)
• im describes the pseudoscalar component m1.pseudoscalar (returns a pure pseudoscalar Three object)

If you have two multivectors m1, m2, you can...

• Add / subtract them
• Take their geometric product: m1 * m2 (m2 can also be a scalar, Vector3, or Vector2)
• Take their inner product: m1 .. m2 OR m1:inner(m2) (wrapper for 0.5 * (m1 * m2 + m2 * m1))
• Take their outer product: m1:outer(m2) (wrapper for 0.5 * (m1 * m2 - m2 * m1))
• Take a quotient (if m2 is invertible): m1 / m2 (wrapper for m1 * m2.inverse)
• Take the scalar product m1:scalarprod(m2), fat dot product m1:fatdot(m2), m1:left(m2) and m1:right(m2) contractions
• Take their exterior (wedge) product: m1 ^ m2 or m1:exterior(m2)
• Take a regressive product: m1:regressive(m2) / m1:meet(m2) / m1:join(m2)
• Take a generalized reflection: m1:reflect(m2)

If you have one multivector m1, you can...

• Take its reverse conjugate m1.reversion, which is the multivector such that, if m1 = v1 * v2 * ... * v_n for vectors v_i, then m1.reversion = v_n * v_(n-1) * ... * v_1
• Take its Clifford conjugate m1.bar, which is useful to provide a value analogous to magnitude (by m1 * m1.bar)
• Take its Hodge dual: m1.dual
• Take its "magnitude" m1.magnitude, which is vector magnitude if the multivector is a vector, but is defined by sqrt(sqrt(b.scalar ^ 2 + b.pseudoscalar ^ 2)), where b = m1 * m1.bar
• Take its "unit" multivector m1.unit, simply m1 / m1.magnitude
• Take its inverse m1.inverse, defined by m1 * m1.inverse = id = unit scalar = three(1)
  See an interesting paper on generalized multivector inverse: https://arxiv.org/abs/1712.05204v2 "INVERSE OF MULTIVECTOR: BEYOND P+Q=5 THRESHOLD" (A. ACUS AND A. DARGYS)

not yet implemented

• rotations, multivector SQRT
  Motivating paper on generalized multivector SQRT: https://arxiv.org/abs/2003.06873v1 "Square root of a multivector of Clifford algebras in 3D: A game with signs" (A. Acus, A. Dargys)

Warning: May Not Be Suitable for Production

• This code is superbly undertested: I have only verified it with bouts of trial-and-error. Use at your own risk.
• There is a concern with floating-point error here in that some operations that should yield a zero value in a component of a multivector, such as inverse, yields an epsilon value. This is resolvable by, for example, implementing a symbolic equation solver that would destroy cancelling terms before they are evaluated. (toeventuallydo)