numeric-prelude-0.2: An experimental alternative hierarchy of numeric type classes

Portabilityportable (?)
Stabilityprovisional
Maintainernumericprelude@henning-thielemann.de

Number.Quaternion

Contents

Description

Quaternions

Synopsis

Cartesian form

data T a Source

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

Instances

C T 
C a b => C a (T b)

The '(*>)' method can't replace scale because it requires the Algebra.Module constraint

C a b => C a (T b) 
(C a, Sqr a b) => C a (T b) 
Sqr a b => Sqr a (T b) 
Eq a => Eq (T a) 
Read a => Read (T a) 
Show a => Show (T a) 
C a => C (T a) 
C a => C (T a) 
C a => C (T a) 
C a => C (T a) 

fromReal :: C a => a -> T aSource

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

Construct a quaternion from real and imaginary part.

Conversions

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

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

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

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

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

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

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.

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

Revert toComplexMatrix.

Operations

scalarProduct :: C a => (a, a, a) -> (a, a, a) -> aSource

crossProduct :: C a => (a, a, a) -> (a, a, a) -> (a, a, a)Source

conjugate :: C a => T a -> T aSource

The conjugate of a quaternion.

scale :: C a => a -> T a -> T aSource

Scale a quaternion by a real number.

norm :: C a => T a -> aSource

normSqr :: C a => T a -> aSource

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

normalize :: C a => T a -> T aSource

scale a quaternion into a unit quaternion

similarity :: C a => T a -> T a -> T aSource

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.

slerpSource

Arguments

:: 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.