| Safe Haskell | None |
|---|---|
| Language | Haskell98 |
Math.Algebras.Quaternions
Description
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
- data HBasis
- type Quaternion k = Vect k HBasis
- i :: Num k => Quaternion k
- k :: Num k => Quaternion k
- j :: Num k => Quaternion k
- class Algebra k a => HasConjugation k a where
- scalarPart :: Num k => Quaternion k -> k
- vectorPart :: (Eq k, Num k) => Quaternion k -> Quaternion k
- (<.>) :: (Num k, Eq k) => Vect k HBasis -> Quaternion k -> k
- (^-) :: (Num a, Fractional a1, Eq a) => a1 -> a -> a1
- refl :: (HasConjugation k a, Show a, Ord a, Num k, Eq k) => Vect k a -> Vect k a -> Vect k a
- asMatrix :: (Num t, Eq t) => (Vect t HBasis -> Quaternion t) -> [Vect t HBasis] -> [[t]]
- reprSO3' :: Fractional a => a -> a -> a
- reprSO3 :: (Eq k, Fractional k) => Quaternion k -> [[k]]
- reprSO4' :: Fractional a => (a, a) -> a -> a
- reprSO4 :: (Eq k, Fractional k) => (Quaternion k, Quaternion k) -> [[k]]
- reprSO4d :: (Fractional k, Eq k) => Vect k (DSum HBasis HBasis) -> [[k]]
- one' :: Num k => Vect k (Dual HBasis)
- k' :: Num k => Vect k (Dual HBasis)
- j' :: Num k => Vect k (Dual HBasis)
- i' :: Num k => Vect k (Dual HBasis)
Documentation
type Quaternion k = Vect k HBasis Source
i :: Num k => Quaternion k Source
The quaternions have {1,i,j,k} as basis, where i^2 = j^2 = k^2 = ijk = -1.
k :: Num k => Quaternion k Source
The quaternions have {1,i,j,k} as basis, where i^2 = j^2 = k^2 = ijk = -1.
j :: Num k => Quaternion k Source
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 Source
Methods
conj :: Vect k a -> Vect k a Source
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 Source
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
| (Eq k, Num k) => HasConjugation k HBasis | |
| (Eq k, Num k) => HasConjugation k OBasis |
scalarPart :: Num k => Quaternion k -> k Source
The scalar part of the quaternion w+xi+yj+zk is w. Also called the real part.
vectorPart :: (Eq k, Num k) => Quaternion k -> Quaternion k Source
The vector part of the quaternion w+xi+yj+zk is xi+yj+zk. Also called the pure part.
(^-) :: (Num a, Fractional a1, Eq a) => a1 -> a -> a1 Source
reprSO3' :: Fractional a => a -> a -> a Source
reprSO3 :: (Eq k, Fractional k) => Quaternion k -> [[k]] Source
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 a => (a, a) -> a -> a Source
reprSO4 :: (Eq k, Fractional k) => (Quaternion k, Quaternion k) -> [[k]] Source
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.