Utility Functions

Basic Functions

real()

real(object)

The same as object.real().

imag()

imag(object)

The same as object.imag().

trace()

trace(object)

The same as object.trace().

conj()

conj(object)

The same as object.conj().

norm()

norm(object)

The same as object.norm().

delta()

delta(object)

The same as object.delta().

inverse()

inverse(object[, module])

The same as object.inverse([module]).

unitary()

unitary(object)

The same as object.unitary().

sqrt()

sqrt(object)

The same as object.sqrt() (see [1] and [2] for details on roots extraction).

croot()

croot(object)

The same as object.croot() (see [1] and [2] for details on roots extraction).

Additional Functions

QuaternionToRotation()

QuaternionToRotation(object)

The same as object.QuaternioinToRotation().

RotationToQuaternion()

RotationToQuaternion(angle, vector)

Converts a pair angle-vector into a quaternion.

StringToQuaternion()

StringToQuaternion(string)

Converts a string into a quaternion.

MatrixToEuler()

MatrixToEuler(matrix)

Converts a 3X3 (numpy) matrix into a vector having Euler angles as components.

EulerToQuaternion()

EulerToQuaternion(list)

Converts a vector whose components are Euler angles into a quaternion.

identity()

identity()

Returns 1 as a quaternion.

>>> qmath.identity()
(1.0)

zero()

zero(object)

Returns 0 as a quaternion.

>>> qmath.zero()
(0.0)

dot()

dot(object_1, object_2)

Returns the dot product of two quaternions.

>>> a = qmath.quaternion('1+2i-2k')
>>> b = qmath.quaternion('3-2i+8j')
>>> qmath.dot(a,b)
-1.0

CrossRatio()

CrossRatio(a, b, c, d)

Returns the cross ratio of four quaternions.

If a and d or b and c are equal, returns the string 'Infinity'.

The arguments of CrossRatio can be passed as a tuple.

>>> a = qmath.quaternion([1,0,1,0])
>>> b = qmath.quaternion([0,1,0,1])
>>> c = qmath.quaternion([-1,0,-1,0])
>>> d = qmath.quaternion([0,-1,0,-1])
>>> qmath.CrossRatio(a,b,c,d)
(2.0)
>>> tpl = a,b,c,d
>>> qmath.CrossRatio(tpl)
(2.0)
>>> qmath.CrossRatio(a,b,b,d)
'Infinity'
>>> qmath.CrossRatio(a,a,a,d)
(1.0)
>>> qmath.CrossRatio(a,b,a,b)
(0.0)

Moebius()

Moebius(z, a, b, c, d)

Returns the Moebius transformation with parameters a,b,c and d.

If c * z + d == 0 returns the string 'Infinity'.

The arguments of Moebius can be passed as a tuple.

>>> a = qmath.quaternion([1,1,1,0])
>>> b = qmath.quaternion([-2,1,0,1])
>>> c = qmath.quaternion([1,0,0,0])
>>> d = qmath.quaternion([0,-1,-3,-4])
>>> z = qmath.quaternion([1,1,3,4])
>>> qmath.Moebius(z,a,b,c,d)
(-5.0+7.0i+7.0k)
>>> d = - z
>>> z = qmath.Moebius(z,a,b,c,d)
>>> z
'Infinity'
>>> qmath.Moebius(z,a,b,c,d)
(1.0+1.0i+1.0j)

References

[1]	(1, 2) Abrate, Quadratic Formulas for Generalized Quaternions, The Journal of Algebra and its Applications, Vol. 8, Issue 3 (2009), pp. 289-306.

[2]	(1, 2) Abrate The Roots of a Generalized Quaternion, Congressus Numeratium, Vol. 201 (2010), pp. 179-186, Kluwer Academic Publishers.