Struct Multivector

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pub struct Multivector<const P: usize, const Q: usize, const R: usize> { /* private fields */ }

Expand description

A multivector in a Clifford algebra Cl(P,Q,R)

§Type Parameters

P: Number of basis vectors that square to +1
Q: Number of basis vectors that square to -1
R: Number of basis vectors that square to 0

Implementations§

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impl<const P: usize, const Q: usize, const R: usize> Multivector<P, Q, R>

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pub const DIM: usize

Total dimension of the algebra’s vector space

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pub const BASIS_COUNT: usize

Total number of basis blades (2^DIM)

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pub fn zero() -> Self

Create a new zero multivector

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pub fn scalar(value: f64) -> Self

Create a scalar multivector

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pub fn from_vector(vector: &Vector<P, Q, R>) -> Self

Create multivector from vector

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pub fn from_bivector(bivector: &Bivector<P, Q, R>) -> Self

Create multivector from bivector

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pub fn basis_vector(i: usize) -> Self

Create a basis vector e_i (i starts from 0)

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pub fn from_coefficients(coefficients: Vec<f64>) -> Self

Create a multivector from coefficients

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pub fn from_slice(coefficients: &[f64]) -> Self

Create a multivector from a slice (convenience for tests)

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pub fn get(&self, index: usize) -> f64

Get coefficient for a specific basis blade (by index)

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pub fn set(&mut self, index: usize, value: f64)

Set coefficient for a specific basis blade

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pub fn scalar_part(&self) -> f64

Get the scalar part (grade 0)

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pub fn set_scalar(&mut self, value: f64)

Set the scalar part

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pub fn vector_part(&self) -> Vector<P, Q, R>

Get vector part as a Vector type

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pub fn bivector_part(&self) -> Self

Get bivector part as a Multivector (grade 2 projection)

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pub fn bivector_type(&self) -> Bivector<P, Q, R>

Get bivector part as a Bivector type wrapper

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pub fn trivector_part(&self) -> f64

Get trivector part (scalar for 3D)

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pub fn set_vector_component(&mut self, index: usize, value: f64)

Set vector component

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pub fn set_bivector_component(&mut self, index: usize, value: f64)

Set bivector component

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pub fn vector_component(&self, index: usize) -> f64

Get vector component

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pub fn as_slice(&self) -> &[f64]

Get coefficients as slice for comparison

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pub fn to_vec(&self) -> Vec<f64>

Convert coefficients to owned vector

Useful for interoperability with other libraries and serialization.

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pub fn grade_project(&self, grade: usize) -> Self

Grade projection with alternative name for clarity

Alias for grade_projection that emphasizes the projection operation.

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pub fn wedge(&self, other: &Self) -> Self

Wedge product (alias for outer product)

The wedge product ∧ is the antisymmetric outer product, fundamental to exterior algebra and differential forms.

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pub fn dot(&self, other: &Self) -> Self

Dot product (alias for inner product)

The dot product · is the grade-lowering inner product, generalizing the vector dot product to all grades.

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pub fn add(&self, other: &Self) -> Self

Add method for convenience

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pub fn grade(&self) -> usize

Get the grade of a multivector (returns the highest non-zero grade)

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pub fn outer_product_with_vector(&self, other: &Vector<P, Q, R>) -> Self

Outer product with a vector (convenience method)

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pub fn geometric_product(&self, rhs: &Self) -> Self

Geometric product with another multivector

The geometric product is the fundamental operation in geometric algebra, combining both the inner and outer products.

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pub fn geometric_product_scalar(&self, rhs: &Self) -> Self

Scalar implementation of geometric product (fallback)

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pub fn inner_product(&self, rhs: &Self) -> Self

Inner product (grade-lowering, dot product for vectors)

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pub fn outer_product(&self, rhs: &Self) -> Self

Outer product (wedge product, grade-raising)

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pub fn scalar_product(&self, rhs: &Self) -> f64

Scalar product (grade-0 part of geometric product)

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pub fn reverse(&self) -> Self

Reverse operation (reverses order of basis vectors in each blade)

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pub fn grade_projection(&self, grade: usize) -> Self

Grade projection - extract components of a specific grade

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pub fn grade_magnitude(&self, grade: usize) -> f64

Compute the magnitude of a specific grade component.

This extracts the k-vector part and computes its magnitude. Useful for determining grade dominance in holographic representations.

§Arguments

grade - The grade to compute magnitude for (0=scalar, 1=vector, 2=bivector, etc.)

§Returns

The magnitude of the grade-k component, or 0.0 if no such component exists.

§Example

use amari_core::Multivector;
let mv = Multivector::<3,0,0>::basis_vector(0) + Multivector::<3,0,0>::scalar(2.0);
assert!((mv.grade_magnitude(0) - 2.0).abs() < 1e-10);  // scalar part
assert!((mv.grade_magnitude(1) - 1.0).abs() < 1e-10);  // vector part

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pub fn grade_spectrum(&self) -> Vec<f64>

Get all grade magnitudes as a vector.

Returns the magnitude of each grade component from 0 to DIM. This provides a “grade spectrum” view of the multivector.

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pub fn is_zero(&self) -> bool

Check if this is (approximately) zero

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pub fn norm_squared(&self) -> f64

Compute the norm squared (scalar product with reverse)

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pub fn magnitude(&self) -> f64

Compute the magnitude (length) of this multivector

The magnitude is defined as |a| = √(a·ã) where ã is the reverse of a. This provides the natural norm inherited from the underlying vector space.

§Mathematical Properties

Always non-negative: |a| ≥ 0
Zero iff a = 0: |a| = 0 ⟺ a = 0
Sub-multiplicative: |ab| ≤ |a||b|

§Examples

use amari_core::Multivector;
let v = Multivector::<3,0,0>::basis_vector(0);
assert_eq!(v.magnitude(), 1.0);

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pub fn norm(&self) -> f64

Compute the norm (magnitude) of this multivector

Note: This method is maintained for backward compatibility. New code should prefer magnitude() for clarity.

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pub fn abs(&self) -> f64

Absolute value (same as magnitude/norm for multivectors)

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pub fn approx_eq(&self, other: &Self, epsilon: f64) -> bool

Approximate equality comparison

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pub fn normalize(&self) -> Option<Self>

Normalize this multivector

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pub fn inverse(&self) -> Option<Self>

Compute multiplicative inverse if it exists

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pub fn exp(&self) -> Self

Exponential map for bivectors (creates rotors)

For a bivector B, exp(B) creates a rotor that performs rotation in the plane defined by B.

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pub fn left_contraction(&self, other: &Self) -> Self

Left contraction: a ⌟ b

Generalized inner product where the grade of the result is |grade(b) - grade(a)|. The left operand’s grade is reduced.

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pub fn right_contraction(&self, other: &Self) -> Self

Right contraction: a ⌞ b

Generalized inner product where the grade of the result is |grade(a) - grade(b)|. The right operand’s grade is reduced.

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pub fn hodge_dual(&self) -> Self

Hodge dual: ⋆a

Maps k-vectors to (n-k)-vectors in n-dimensional space. Essential for electromagnetic field theory and differential forms. In 3D: scalar -> pseudoscalar, vector -> bivector, bivector -> vector, pseudoscalar -> scalar