nblade 0.1.8

N-dimensional Blade - High-performance Geometric Algebra library powered by Rust

nblade - N-dimensional Blade

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Overview

nblade (N-dimensional Blade) is a high-performance geometric algebra library powered by Rust, with Python bindings. It supports arbitrary dimensions and arbitrary metric signatures G(p, q, r).

Features

• Arbitrary Dimensions: Up to 64-dimensional vector spaces (practical limit ~12D for dense operations)
• Arbitrary Signatures: Support for G(p, q, r) metric signatures (Euclidean, spacetime, conformal, etc.)
• High Performance: Rust backend with parallel computing and SIMD optimization
• Dual Representation: Automatic selection of dense or sparse representation
• Complete Operations: All standard geometric algebra operations
• NumPy Integration: Create vectors directly from NumPy arrays

Installation

Python

pip install nblade

From Source

# Install maturin
pip install maturin

# Build and install
maturin develop --release

Quick Start

import nblade

# Create 3D Euclidean geometric algebra G(3,0,0)
algebra = nblade.Algebra.euclidean(3)

# Create basis vectors
e1 = algebra.basis_vector(0)
e2 = algebra.basis_vector(1)
e3 = algebra.basis_vector(2)

# Create a vector from a list
v = algebra.vector([1, 2, 3])

# Geometric product
product = e1 * e2  # Results in bivector e12

# Outer product (wedge product)
wedge = e1 ^ e2  # Results in bivector e12

# Inner product
inner = e1 | e2  # Results in 0 (orthogonal vectors)

# Dual
I = algebra.config.volume_element()  # Pseudoscalar
v_dual = v.dual()

# Rotation using rotor
import math
plane = e1 ^ e2
rotor = algebra.rotor(plane, math.pi / 2)  # 90-degree rotation
rotated = e1.rotate_by(rotor)

API Reference

Algebra Class

algebra = nblade.Algebra(dimension, p=0, q=0, r=0)

# Factory methods
algebra = nblade.Algebra.euclidean(dimension)  # G(n, 0, 0)
algebra = nblade.Algebra.spacetime(dimension)  # G(1, n-1, 0)
algebra = nblade.Algebra.cga()                 # G(4, 1, 0)

# Properties
algebra.dimension   # Vector space dimension
algebra.signature   # (p, q, r) tuple

# Methods
algebra.basis_vector(i)      # Create e_i
algebra.vector([x, y, z])    # Create vector from list
algebra.scalar(value)        # Create scalar multivector
algebra.zeros()              # Create zero multivector
algebra.one()                # Create unit scalar
algebra.rotor(plane, angle)  # Create rotor

MultiVector Class

# Construction
mv = nblade.MultiVector.basis_vector(config, i)
mv = nblade.MultiVector.from_scalar(config, value)
mv = nblade.MultiVector.from_coefficients(config, coeffs)
mv = nblade.MultiVector.zeros(config)
mv = nblade.MultiVector.one(config)

# Products
mv.geometric_product(other)  # Geometric product
mv.outer_product(other)      # Outer/wedge product
mv.left_inner(other)         # Left contraction
mv.right_inner(other)        # Right contraction

# Involutions
mv.grade_involution()        # Grade involution (A*)
mv.reversion()               # Reversion (A†)
mv.clifford_conjugate()      # Clifford conjugate (A‡)

# Other operations
mv.dual()                    # Dual (A⊥)
mv.inverse_dual()            # Inverse dual (A⁻⊥)
mv.inverse()                 # Multiplicative inverse
mv.norm()                    # Norm |A|
mv.norm_squared()            # |A|²

# Grade operations
mv.grade(r)                  # r-grade part
mv.even_part()               # Even grades
mv.odd_part()                # Odd grades

# Geometric operations
mv.project_to(blade)         # Project onto blade
mv.reject_from(blade)        # Reject from blade
mv.reflect_in(blade)         # Reflect in blade
mv.rotate_by(rotor)          # Rotate by rotor

# Operator overloads
mv1 + mv2   # Addition
mv1 - mv2   # Subtraction
mv1 * mv2   # Geometric product
mv1 ^ mv2   # Outer product
mv1 | mv2   # Left inner product
~mv        # Grade involution
-mv        # Negation

Module Functions

# Create rotor for rotation in a plane
rotor = nblade.create_rotor(plane, angle)

# Reciprocal frame computation
reciprocal = nblade.reciprocal_frame(vectors)

# Basis expansion
coeffs = nblade.basis_expansion(multivector)

Supported Operations

Operation	Symbol	Formula
Geometric Product	AB	AB = A·B + A∧B
Outer Product	A∧B	Antisymmetric part
Left Inner	A⌋B	Left contraction
Right Inner	A⌊B	Right contraction
Grade Involution	A* (Python: ~A)	(-1)^r A_r
Reversion	A† (Python: A.reversion())	(-1)^(r(r-1)/2) A_r
Clifford Conjugate	A‡ (Python: A.clifford_conjugate())	(A*)†
Dual	A⊥	A·I or AI
Inverse Dual	A⁻⊥	A⌋I

Performance

nblade is optimized for performance:

• SIMD acceleration for 2D-4D operations (with simd feature)
• Adaptive parallelism for high-dimensional operations (≥6D)
• Memory pooling for reduced allocation overhead (with pool feature)
• Dense/Sparse auto-selection based on coefficient density

Benchmarks

Operation	Dimension	Time
Geometric Product	3D	~5ms
Geometric Product	5D	~15ms
Geometric Product (SIMD)	3D	~3ms

Examples

2D Rotation

import nblade
import math

# Create 2D Euclidean algebra
algebra = nblade.Algebra.euclidean(2)

# Create basis vectors
e1, e2 = algebra.basis_vectors()

# Create rotor for 45-degree rotation
plane = e1 ^ e2
rotor = algebra.rotor(plane, math.pi / 4)

# Rotate e1
rotated = e1.rotate_by(rotor)
print(rotated)  # ~ 0.707*e1 + 0.707*e2

3D Cross Product via Dual

import nblade

algebra = nblade.Algebra.euclidean(3)
e1, e2, e3 = algebra.basis_vectors()

# Cross product: a × b = (a ∧ b)*
# where * is the dual in 3D

a = e1 + 2*e2
b = 2*e1 + e3

# Compute cross product
a_cross_b = (a ^ b).dual()
print(a_cross_b)

Contributing

We welcome contributions! See CONTRIBUTING.md for guidelines.

Development Setup

# Clone the repository
git clone https://github.com/UynajGI/nblade.git
cd nblade

# Install Rust (if not already)
curl --proto '=https' --tlsv1.2 -sSf https://sh.rustup.rs | sh

# Build
cargo build --release

# Run tests
cargo test --all-features

# Build Python bindings
pip install maturin
maturin develop --release

# Run Python tests
pytest tests/test_python_api.py

Code Style

• Rust: Follow cargo fmt and cargo clippy
• Python: Follow PEP 8, use ruff format

License

MIT License