E1: Conformal Geometric Algebra (CGA)

Module: sci_form::experimental::cgaFeature flag: experimental-cga

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Overview

Replaces the internal geometry engine (separate quaternions + vectors) with Conformal Geometric Algebra $G(4,1)$, unifying rotations and translations into a single multiplicative operator called a Motor ($MX\tilde{M}$). This eliminates gimbal lock and enables massively parallelizable geometric operations for conformer generation and MOF assembly.

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Theory

G(4,1) Multivector

The conformal model embeds 3D Euclidean space into a 5D space with signature $(4,1)$:

$$e_1^2=e_2^2=e_3^2=e_{+}^2=1,e_{-}^2=-1$$

A general multivector has $2^5=32$ components spanning all grade combinations. The geometric product encodes both the inner and outer products:

$$ab=a\cdot b+a\wedge b$$

Motors

A Motor is the composition of a rotor (rotation) and translator:

• Rotor: $R=\cos (\theta /2)-\sin (\theta /2)\hat{L}$
• Translator: $T=1-\frac{1}{2}t\cdot e_{\mathrm{\infty }}$
• Motor: $M=TR$

Objects transform via the sandwich product $X^{\prime }=MX\tilde{M}$.

Point Embedding

3D coordinates lift to conformal points:

$$P=p+\frac{1}{2}|p{|}^2{e}_{\mathrm{\infty }}+e_o$$

Coordinates are recovered from the transformed multivector by extracting the Euclidean components and normalizing.

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API

use sci_form::experimental::cga::*;

// Multivector operations
let a = Multivector::basis(1);           // e₁
let b = Multivector::basis(2);           // e₂
let ab = a.geometric_product(&b);        // e₁e₂
let rev = ab.reverse();                  // ẽ₁₂ = e₂e₁

// Motor from axis-angle
let motor = Motor::from_axis_angle([0.0, 0.0, 1.0], std::f64::consts::FRAC_PI_2);

// Transform a point
let point = cga_point([1.0, 0.0, 0.0]);
let rotated = motor.apply(&point);
let coords = extract_point(&rotated);   // ≈ [0, 1, 0]

// Dihedral rotation
let dihedral_motor = Motor::dihedral([0.0, 0.0, 0.0], [1.0, 0.0, 0.0], 1.047);

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Applications

• Conformer generation: Dihedral torsion scans using CGA motors instead of rotation matrices
• MOF assembly: Unit cell lattice vectors as translation motors, SBU placement via motor composition
• Supercell construction: Motor exponentiation $M_{super}=M_{cell}^{n_a\cdot n_b\cdot n_c}$

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Tests

cargo test --features experimental-cga --test regression -- test_cga

Covers: multivector algebra, motor construction, point embedding round-trips, dihedral rotations, and materials assembly comparison.