Number.Quaternion · numeric-prelude-0.4.4

Cartesian form

5 declarations

datadata T a

Quaternions could be defined based on Complex numbers. However quaternions are often considered as real part and three imaginary parts.

Instances12Sqr, Eq, Read, Show, C, …

C TDefined in numeric-prelude-0.4.4 · Number.Quaternion
C a b => C a (T b)Defined in numeric-prelude-0.4.4 · Number.Quaternion
C a b => C a (T b)Defined in numeric-prelude-0.4.4 · Number.Quaternion
The (*>) method can't replace scale because it requires the Algebra.Module constraint
(C a, Sqr a b) => C a (T b)Defined in numeric-prelude-0.4.4 · Number.Quaternion
Sqr a b => Sqr a (T b)Defined in numeric-prelude-0.4.4 · Number.Quaternion
Eq a => Eq (T a)Defined in numeric-prelude-0.4.4 · Number.Quaternion
Read a => Read (T a)Defined in numeric-prelude-0.4.4 · Number.Quaternion
Show a => Show (T a)Defined in numeric-prelude-0.4.4 · Number.Quaternion
C a => C (T a)Defined in numeric-prelude-0.4.4 · Number.Quaternion
C a => C (T a)Defined in numeric-prelude-0.4.4 · Number.Quaternion
C a => C (T a)Defined in numeric-prelude-0.4.4 · Number.Quaternion
C a => C (T a)Defined in numeric-prelude-0.4.4 · Number.Quaternion

valuereal :: T a -> a

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real part

valueimag :: T a -> (a, a, a)

#

imaginary parts

valuefromReal :: C a => a -> T a

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value(+::) :: a -> (a, a, a) -> T a

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Construct a quaternion from real and imaginary part.

Conversions

5 declarations

valuetoRotationMatrix :: C a => T a -> Array (Int, Int) a

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Let c be a unit quaternion, then it holds similarity c (0+::x) == toRotationMatrix c * x

valuefromRotationMatrix :: C a => Array (Int, Int) a -> T a

#

valuefromRotationMatrixDenorm :: C a => Array (Int, Int) a -> T a

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The rotation matrix must be normalized. (I.e. no rotation with scaling) The computed quaternion is not normalized.

valuetoComplexMatrix :: C a => T a -> Array (Int, Int) (T a)

#

Map a quaternion to complex valued 2x2 matrix, such that quaternion addition and multiplication is mapped to matrix addition and multiplication. The determinant of the matrix equals the squared quaternion norm (normSqr). Since complex numbers can be turned into real (orthogonal) matrices, a quaternion could also be converted into a real matrix.

valuefromComplexMatrix :: C a => Array (Int, Int) (T a) -> T a

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Revert toComplexMatrix.

Operations

9 declarations

valuescalarProduct :: C a => (a, a, a) -> (a, a, a) -> a

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valuecrossProduct :: C a => (a, a, a) -> (a, a, a) -> (a, a, a)

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valueconjugate :: C a => T a -> T a

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The conjugate of a quaternion.

valuescale :: C a => a -> T a -> T a

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Scale a quaternion by a real number.

valuenorm :: C a => T a -> a

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valuenormSqr :: C a => T a -> a

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the same as NormedEuc.normSqr but with a simpler type class constraint

valuenormalize :: C a => T a -> T a

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scale a quaternion into a unit quaternion

valuesimilarity :: C a => T a -> T a -> T a

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similarity mapping as needed for rotating 3D vectors

It holds similarity (cos(a/2) +:: scaleImag (sin(a/2)) v) (0 +:: x) == (0 +:: y) where y results from rotating x around the axis v by the angle a.

valueslerp

:: C a
=> a
For 0 return vector v, for 1 return vector w
-> (a, a, a)
vector v, must be normalized
-> (a, a, a)
vector w, must be normalized
-> (a, a, a)

#

Spherical Linear Interpolation

Can be generalized to any transcendent Hilbert space. In fact, we should also include the real part in the interpolation.