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Blade

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Blade is a geometric algebra library for Mojo 🔥.

It generates efficient geometric algebras from arbitrary signatures, enabling expressive, type-safe representations of geometry in any dimension.

What is Geometric Algebra?

Geometric algebra (GA) is a mathematical framework that unifies vectors, complex numbers, quaternions, rotations, reflections, and many other geometric concepts into a single algebraic system.

The fundamental type in GA is the multivector, which can represent:

  • Scalars
  • Vectors
  • Lines and planes
  • Rotations and reflections
  • Higher-dimensional geometric objects

Blade generates the geometric product from a given signature, allowing the same API to work across Euclidean, projective, conformal, spacetime, and custom algebras.

Signatures

A geometric algebra is defined by its signature:

ga(p, n, z)

where:

  • p = number of basis vectors that square to +1
  • n = number of basis vectors that square to -1
  • z = number of basis vectors that square to 0

For example, blade contains Complex, which is an alias of the signature ga(0, 1, 0):

from blade import Complex

var z = 1 + Complex.i
print(z * z)

# prints: 0 + 2e1

Here, e1 is the algebra's only basis vector, acting as the imaginary unit i.

Multivectors

Every multivector is parameterized by both:

  • its algebra signature
  • a compile-time basis mask

The basis mask allows Blade to eliminate unnecessary storage and arithmetic, producing smaller and more efficient operations.

comptime sig = Signature(2, 0, 1)
comptime Vector = Multivector[sig, sig.vector_mask()]

var v = Vector(1.0, 2.0, 1.0)

print(v * v)

You can also import ga, which makes multivector construction implicit, allowing things like:

comptime GA = ga(3, 0, 0)

var v = 2*GA.e1 + 5*GA.e2 + 3*GA.e3
var b = 2*GA.e12 + 2*GA.e13 + 2*GA.e23

Contributing

Contributions are welcome!

If you find a bug, have an idea, or want to discuss geometric algebra or Blade, feel free to open an issue or email:

helehex@gmail.com