Math.Algebras.Quaternions

A module defining the algebra of quaternions over an arbitrary field.

The quaternions are the algebra defined by the basis {1,i,j,k}, where i^2 = j^2 = k^2 = ijk = -1

Documentation

data HBasis

Constructors

One

I

J

K

Instances

Eq HBasis

Ord HBasis

Show HBasis

Num k => Algebra k HBasis

Num k => HasConjugation k HBasis

Num k => Coalgebra k (Dual HBasis)

type Quaternion k = Vect k HBasis

i, k, j :: Num k => Quaternion k

The quaternions have {1,i,j,k} as basis, where i^2 = j^2 = k^2 = ijk = -1.

class Algebra k a => HasConjugation k a where

Methods

conj :: Vect k a -> Vect k a

A conjugation operation is required to satisfy the following laws:

• conj (x+y) = conj x + conj y
• conj (x*y) = conj y * conj x (note the order-reversal)
• conj (conj x) = x
• conj x = x if and only if x in k

sqnorm :: Vect k a -> k

The squared norm is defined as sqnorm x = x * conj x. It satisfies:

• sqnorm (x*y) = sqnorm x * sqnorm y
• sqnorm (unit k) = k^2, for k a scalar

Instances

Num k => HasConjugation k HBasis

Num k => HasConjugation k OBasis

scalarPart :: Num k => Quaternion k -> k

The scalar part of the quaternion w+xi+yj+zk is w. Also called the real part.

vectorPart :: Num k => Quaternion k -> Quaternion k

The vector part of the quaternion w+xi+yj+zk is xi+yj+zk. Also called the pure part.

reprSO3 :: Fractional k => Quaternion k -> [[k]]

Given a non-zero quaternion q in H, the map x -> q^-1 * x * q defines an action on the 3-dimensional vector space of pure quaternions X (ie linear combinations of i,j,k). It turns out that this action is a rotation of X, and this is a surjective group homomorphism from H* onto SO3. If we restrict q to the group of unit quaternions (those of norm 1), then this homomorphism is 2-to-1 (since q and -q give the same rotation). This shows that the multiplicative group of unit quaternions is isomorphic to Spin3, the double cover of SO3.

reprSO3 q returns the 3*3 matrix representing this map.

reprSO4 :: Fractional k => (Quaternion k, Quaternion k) -> [[k]]

Given a pair of unit quaternions (l,r), the map x -> l^-1 * x * r defines an action on the 4-dimensional space of quaternions. It turns out that this action is a rotation, and this is a surjective group homomorphism onto SO4. The homomorphism is 2-to-1 (since (l,r) and (-l,-r) give the same map). This shows that the multiplicative group of pairs of unit quaternions (with pointwise multiplication) is isomorphic to Spin4, the double cover of SO4.

reprSO4 (l,r) returns the 4*4 matrix representing this map.

one', k', j', i' :: Num k => Vect k (Dual HBasis)