Maintainer	numericprelude@henning-thielemann.de
Stability	provisional
Portability	portable (?)
Safe Haskell	None
Language	Haskell98

Number.Quaternion

Contents

Cartesian form
Conversions
Operations

Description

Quaternions

Synopsis

data T a
fromReal :: C a => a -> T a
(+::) :: a -> (a, a, a) -> T a
toRotationMatrix :: C a => T a -> Array (Int, Int) a
fromRotationMatrix :: C a => Array (Int, Int) a -> T a
fromRotationMatrixDenorm :: C a => Array (Int, Int) a -> T a
toComplexMatrix :: C a => T a -> Array (Int, Int) (T a)
fromComplexMatrix :: C a => Array (Int, Int) (T a) -> T a
scalarProduct :: C a => (a, a, a) -> (a, a, a) -> a
crossProduct :: C a => (a, a, a) -> (a, a, a) -> (a, a, a)
conjugate :: C a => T a -> T a
scale :: C a => a -> T a -> T a
norm :: C a => T a -> a
normSqr :: C a => T a -> a
normalize :: C a => T a -> T a
similarity :: C a => T a -> T a -> T a
slerp :: C a => a -> (a, a, a) -> (a, a, a) -> (a, a, a)

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 Source #
Instance details Defined in Number.Quaternion Methods zero :: C a => T a Source # (<+>) :: C a => T a -> T a -> T a Source # (*>) :: C a => a -> T a -> T a Source #
C a b => C a (T b) Source #	The '(*>)' method can't replace `scale` because it requires the Algebra.Module constraint
Instance details Defined in Number.Quaternion Methods (*>) :: a -> T b -> T b Source #
C a b => C a (T b) Source #
Instance details Defined in Number.Quaternion
(C a, Sqr a b) => C a (T b) Source #
Instance details Defined in Number.Quaternion Methods norm :: T b -> a Source #
Sqr a b => Sqr a (T b) Source #
Instance details Defined in Number.Quaternion Methods normSqr :: T b -> a Source #
Eq a => Eq (T a) Source #
Instance details Defined in Number.Quaternion Methods (==) :: T a -> T a -> Bool # (/=) :: T a -> T a -> Bool #
Read a => Read (T a) Source #
Instance details Defined in Number.Quaternion Methods readsPrec :: Int -> ReadS (T a) # readList :: ReadS [T a] # readPrec :: ReadPrec (T a) # readListPrec :: ReadPrec [T a] #
Show a => Show (T a) Source #
Instance details Defined in Number.Quaternion Methods showsPrec :: Int -> T a -> ShowS # show :: T a -> String # showList :: [T a] -> ShowS #
C a => C (T a) Source #
Instance details Defined in Number.Quaternion Methods zero :: T a Source # (+) :: T a -> T a -> T a Source # (-) :: T a -> T a -> T a Source # negate :: T a -> T a Source #
C a => C (T a) Source #
Instance details Defined in Number.Quaternion Methods isZero :: T a -> Bool Source #
C a => C (T a) Source #
Instance details Defined in Number.Quaternion Methods (*) :: T a -> T a -> T a Source # one :: T a Source # fromInteger :: Integer -> T a Source # (^) :: T a -> Integer -> T a Source #
C a => C (T a) Source #
Instance details Defined in Number.Quaternion Methods (/) :: T a -> T a -> T a Source # recip :: T a -> T a Source # fromRational' :: Rational -> T a Source # (^-) :: T a -> Integer -> T a Source #

fromReal :: C a => a -> T a Source #

(+::) :: a -> (a, a, a) -> T a infix 6 Source #

Construct a quaternion from real and imaginary part.

Conversions

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

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

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

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

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 a Source #

Revert toComplexMatrix.

Operations

scalarProduct :: C a => (a, a, a) -> (a, a, a) -> a Source #

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

conjugate :: C a => T a -> T a Source #

The conjugate of a quaternion.

scale :: C a => a -> T a -> T a Source #

Scale a quaternion by a real number.

norm :: C a => T a -> a Source #

normSqr :: C a => T a -> a Source #

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

normalize :: C a => T a -> T a Source #

scale a quaternion into a unit quaternion

similarity :: C a => T a -> T a -> T a Source #

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.

slerp Source #

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.