# Algebras¶

## Introduction¶

The Algebras module for SymPy provides support for basic algebraic operations on Quaternions.

## Quaternion Reference¶

This section lists the classes implemented by the Algebras module.

class sympy.algebras.Quaternion(
a=0,
b=0,
c=0,
d=0,
real_field=True,
norm=None,
)[source]

Provides basic quaternion operations. Quaternion objects can be instantiated as Quaternion(a, b, c, d) as in $$q = a + bi + cj + dk$$.

Parameters:

norm : None or number

Pre-defined quaternion norm. If a value is given, Quaternion.norm returns this pre-defined value instead of calculating the norm

Examples

>>> from sympy import Quaternion
>>> q = Quaternion(1, 2, 3, 4)
>>> q
1 + 2*i + 3*j + 4*k


Quaternions over complex fields can be defined as:

>>> from sympy import Quaternion
>>> from sympy import symbols, I
>>> x = symbols('x')
>>> q1 = Quaternion(x, x**3, x, x**2, real_field = False)
>>> q2 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
>>> q1
x + x**3*i + x*j + x**2*k
>>> q2
(3 + 4*I) + (2 + 5*I)*i + 0*j + (7 + 8*I)*k


Defining symbolic unit quaternions:

>>> from sympy import Quaternion
>>> from sympy.abc import w, x, y, z
>>> q = Quaternion(w, x, y, z, norm=1)
>>> q
w + x*i + y*j + z*k
>>> q.norm()
1


References

Parameters:

other : Quaternion

The quaternion to add to current (self) quaternion.

Returns:

Quaternion

The resultant quaternion after adding self to other

Examples

>>> from sympy import Quaternion
>>> from sympy import symbols
>>> q1 = Quaternion(1, 2, 3, 4)
>>> q2 = Quaternion(5, 6, 7, 8)
6 + 8*i + 10*j + 12*k
>>> q1 + 5
6 + 2*i + 3*j + 4*k
>>> x = symbols('x', real = True)
(x + 1) + 2*i + 3*j + 4*k


Quaternions over complex fields :

>>> from sympy import Quaternion
>>> from sympy import I
>>> q3 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
(5 + 7*I) + (2 + 5*I)*i + 0*j + (7 + 8*I)*k

angle()[source]

Returns the angle of the quaternion measured in the real-axis plane.

Explanation

Given a quaternion $$q = a + bi + cj + dk$$ where $$a$$, $$b$$, $$c$$ and $$d$$ are real numbers, returns the angle of the quaternion given by

$\theta := 2 \operatorname{atan_2}\left(\sqrt{b^2 + c^2 + d^2}, {a}\right)$

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(1, 4, 4, 4)
>>> q.angle()
2*atan(4*sqrt(3))

arc_coplanar(other)[source]

Returns True if the transformation arcs represented by the input quaternions happen in the same plane.

Parameters:

other : a Quaternion

Returns:

True : if the planes of the two quaternions are the same, apart from its orientation/sign.

False : if the planes of the two quaternions are not the same, apart from its orientation/sign.

None : if plane of either of the quaternion is unknown.

Explanation

Two quaternions are said to be coplanar (in this arc sense) when their axes are parallel. The plane of a quaternion is the one normal to its axis.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q1 = Quaternion(1, 4, 4, 4)
>>> q2 = Quaternion(3, 8, 8, 8)
>>> Quaternion.arc_coplanar(q1, q2)
True

>>> q1 = Quaternion(2, 8, 13, 12)
>>> Quaternion.arc_coplanar(q1, q2)
False

axis()[source]

Returns $$\mathbf{Ax}(q)$$, the axis of the quaternion $$q$$.

Explanation

Given a quaternion $$q = a + bi + cj + dk$$, returns $$\mathbf{Ax}(q)$$ i.e., the versor of the vector part of that quaternion equal to $$\mathbf{U}[\mathbf{V}(q)]$$. The axis is always an imaginary unit with square equal to $$-1 + 0i + 0j + 0k$$.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(1, 1, 1, 1)
>>> q.axis()
0 + sqrt(3)/3*i + sqrt(3)/3*j + sqrt(3)/3*k

exp()[source]

Returns the exponential of $$q$$, given by $$e^q$$.

Returns:

Quaternion

The exponential of the quaternion.

Examples

>>> from sympy import Quaternion
>>> q = Quaternion(1, 2, 3, 4)
>>> q.exp()
E*cos(sqrt(29))
+ 2*sqrt(29)*E*sin(sqrt(29))/29*i
+ 3*sqrt(29)*E*sin(sqrt(29))/29*j
+ 4*sqrt(29)*E*sin(sqrt(29))/29*k

classmethod from_Matrix(elements)[source]

Returns quaternion from elements of a column vector. If vector_only is True, returns only imaginary part as a Matrix of length 3.

Parameters:

elements : Matrix, list or tuple of length 3 or 4. If length is 3,

assume real part is zero. Default value: False

Returns:

Quaternion

A quaternion created from the input elements.

Examples

>>> from sympy import Quaternion
>>> from sympy.abc import a, b, c, d
>>> q = Quaternion.from_Matrix([a, b, c, d])
>>> q
a + b*i + c*j + d*k

>>> q = Quaternion.from_Matrix([b, c, d])
>>> q
0 + b*i + c*j + d*k

classmethod from_axis_angle(vector, angle)[source]

Returns a rotation quaternion given the axis and the angle of rotation.

Parameters:

vector : tuple of three numbers

The vector representation of the given axis.

angle : number

The angle by which axis is rotated (in radians).

Returns:

Quaternion

The normalized rotation quaternion calculated from the given axis and the angle of rotation.

Examples

>>> from sympy import Quaternion
>>> from sympy import pi, sqrt
>>> q = Quaternion.from_axis_angle((sqrt(3)/3, sqrt(3)/3, sqrt(3)/3), 2*pi/3)
>>> q
1/2 + 1/2*i + 1/2*j + 1/2*k

classmethod from_euler(angles, seq)[source]

Returns quaternion equivalent to rotation represented by the Euler angles, in the sequence defined by seq.

Parameters:

angles : list, tuple or Matrix of 3 numbers

seq : string of length 3

Represents the sequence of rotations. For extrinsic rotations, seq must be all lowercase and its elements must be from the set {'x', 'y', 'z'} For intrinsic rotations, seq must be all uppercase and its elements must be from the set {'X', 'Y', 'Z'}

Returns:

Quaternion

The normalized rotation quaternion calculated from the Euler angles in the given sequence.

Examples

>>> from sympy import Quaternion
>>> from sympy import pi
>>> q = Quaternion.from_euler([pi/2, 0, 0], 'xyz')
>>> q
sqrt(2)/2 + sqrt(2)/2*i + 0*j + 0*k

>>> q = Quaternion.from_euler([0, pi/2, pi] , 'zyz')
>>> q
0 + (-sqrt(2)/2)*i + 0*j + sqrt(2)/2*k

>>> q = Quaternion.from_euler([0, pi/2, pi] , 'ZYZ')
>>> q
0 + sqrt(2)/2*i + 0*j + sqrt(2)/2*k

classmethod from_rotation_matrix(M)[source]

Returns the equivalent quaternion of a matrix. The quaternion will be normalized only if the matrix is special orthogonal (orthogonal and det(M) = 1).

Parameters:

M : Matrix

Input matrix to be converted to equivalent quaternion. M must be special orthogonal (orthogonal and det(M) = 1) for the quaternion to be normalized.

Returns:

Quaternion

The quaternion equivalent to given matrix.

Examples

>>> from sympy import Quaternion
>>> from sympy import Matrix, symbols, cos, sin, trigsimp
>>> x = symbols('x')
>>> M = Matrix([[cos(x), -sin(x), 0], [sin(x), cos(x), 0], [0, 0, 1]])
>>> q = trigsimp(Quaternion.from_rotation_matrix(M))
>>> q
sqrt(2)*sqrt(cos(x) + 1)/2 + 0*i + 0*j + sqrt(2 - 2*cos(x))*sign(sin(x))/2*k

index_vector()[source]

Returns the index vector of the quaternion.

Returns:

Quaternion: representing index vector of the provided quaternion.

Explanation

The index vector is given by $$\mathbf{T}(q)$$, the norm (or magnitude) of the quaternion $$q$$, multiplied by $$\mathbf{Ax}(q)$$, the axis of $$q$$.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(2, 4, 2, 4)
>>> q.index_vector()
0 + 4*sqrt(10)/3*i + 2*sqrt(10)/3*j + 4*sqrt(10)/3*k


integrate(*args)[source]

Computes integration of quaternion.

Returns:

Quaternion

Integration of the quaternion(self) with the given variable.

Examples

Indefinite Integral of quaternion :

>>> from sympy import Quaternion
>>> from sympy.abc import x
>>> q = Quaternion(1, 2, 3, 4)
>>> q.integrate(x)
x + 2*x*i + 3*x*j + 4*x*k


Definite integral of quaternion :

>>> from sympy import Quaternion
>>> from sympy.abc import x
>>> q = Quaternion(1, 2, 3, 4)
>>> q.integrate((x, 1, 5))
4 + 8*i + 12*j + 16*k

inverse()[source]

Returns the inverse of the quaternion.

is_pure()[source]

Returns true if the quaternion is pure, false if the quaternion is not pure or returns none if it is unknown.

Explanation

A pure quaternion (also a vector quaternion) is a quaternion with scalar part equal to 0.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(0, 8, 13, 12)
>>> q.is_pure()
True

is_zero_quaternion()[source]

Returns true if the quaternion is a zero quaternion or false if it is not a zero quaternion and None if the value is unknown.

Explanation

A zero quaternion is a quaternion with both scalar part and vector part equal to 0.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(1, 0, 0, 0)
>>> q.is_zero_quaternion()
False

>>> q = Quaternion(0, 0, 0, 0)
>>> q.is_zero_quaternion()
True

log()[source]

Returns the logarithm of the quaternion, given by $$\log q$$.

Examples

>>> from sympy import Quaternion
>>> q = Quaternion(1, 2, 3, 4)
>>> q.log()
log(sqrt(30))
+ 2*sqrt(29)*acos(sqrt(30)/30)/29*i
+ 3*sqrt(29)*acos(sqrt(30)/30)/29*j
+ 4*sqrt(29)*acos(sqrt(30)/30)/29*k

mensor()[source]

Returns the natural logarithm of the norm(magnitude) of the quaternion.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(2, 4, 2, 4)
>>> q.mensor()
log(2*sqrt(10))
>>> q.norm()
2*sqrt(10)

mul(other)[source]

Multiplies quaternions.

Parameters:

other : Quaternion or symbol

The quaternion to multiply to current (self) quaternion.

Returns:

Quaternion

The resultant quaternion after multiplying self with other

Examples

>>> from sympy import Quaternion
>>> from sympy import symbols
>>> q1 = Quaternion(1, 2, 3, 4)
>>> q2 = Quaternion(5, 6, 7, 8)
>>> q1.mul(q2)
(-60) + 12*i + 30*j + 24*k
>>> q1.mul(2)
2 + 4*i + 6*j + 8*k
>>> x = symbols('x', real = True)
>>> q1.mul(x)
x + 2*x*i + 3*x*j + 4*x*k


Quaternions over complex fields :

>>> from sympy import Quaternion
>>> from sympy import I
>>> q3 = Quaternion(3 + 4*I, 2 + 5*I, 0, 7 + 8*I, real_field = False)
>>> q3.mul(2 + 3*I)
(2 + 3*I)*(3 + 4*I) + (2 + 3*I)*(2 + 5*I)*i + 0*j + (2 + 3*I)*(7 + 8*I)*k

norm()[source]

Returns the norm of the quaternion.

normalize()[source]

Returns the normalized form of the quaternion.

orthogonal(other)[source]

Returns the orthogonality of two quaternions.

Parameters:

other : a Quaternion

Returns:

True : if the two pure quaternions seen as 3D vectors are orthogonal.

False : if the two pure quaternions seen as 3D vectors are not orthogonal.

None : if the two pure quaternions seen as 3D vectors are orthogonal is unknown.

Explanation

Two pure quaternions are called orthogonal when their product is anti-commutative.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(0, 4, 4, 4)
>>> q1 = Quaternion(0, 8, 8, 8)
>>> q.orthogonal(q1)
False

>>> q1 = Quaternion(0, 2, 2, 0)
>>> q = Quaternion(0, 2, -2, 0)
>>> q.orthogonal(q1)
True

parallel(other)[source]

Returns True if the two pure quaternions seen as 3D vectors are parallel.

Parameters:

other : a Quaternion

Returns:

True : if the two pure quaternions seen as 3D vectors are parallel.

False : if the two pure quaternions seen as 3D vectors are not parallel.

None : if the two pure quaternions seen as 3D vectors are parallel is unknown.

Explanation

Two pure quaternions are called parallel when their vector product is commutative which implies that the quaternions seen as 3D vectors have same direction.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(0, 4, 4, 4)
>>> q1 = Quaternion(0, 8, 8, 8)
>>> q.parallel(q1)
True

>>> q1 = Quaternion(0, 8, 13, 12)
>>> q.parallel(q1)
False

pow(p)[source]

Finds the pth power of the quaternion.

Parameters:

p : int

Power to be applied on quaternion.

Returns:

Quaternion

Returns the p-th power of the current quaternion. Returns the inverse if p = -1.

Examples

>>> from sympy import Quaternion
>>> q = Quaternion(1, 2, 3, 4)
>>> q.pow(4)
668 + (-224)*i + (-336)*j + (-448)*k

pow_cos_sin(p)[source]

Computes the pth power in the cos-sin form.

Parameters:

p : int

Power to be applied on quaternion.

Returns:

Quaternion

The p-th power in the cos-sin form.

Examples

>>> from sympy import Quaternion
>>> q = Quaternion(1, 2, 3, 4)
>>> q.pow_cos_sin(4)
900*cos(4*acos(sqrt(30)/30))
+ 1800*sqrt(29)*sin(4*acos(sqrt(30)/30))/29*i
+ 2700*sqrt(29)*sin(4*acos(sqrt(30)/30))/29*j
+ 3600*sqrt(29)*sin(4*acos(sqrt(30)/30))/29*k

property product_matrix_left

Returns 4 x 4 Matrix equivalent to a Hamilton product from the left. This can be useful when treating quaternion elements as column vectors. Given a quaternion $$q = a + bi + cj + dk$$ where a, b, c and d are real numbers, the product matrix from the left is:

$\begin{split}M = \begin{bmatrix} a &-b &-c &-d \\ b & a &-d & c \\ c & d & a &-b \\ d &-c & b & a \end{bmatrix}\end{split}$

Examples

>>> from sympy import Quaternion
>>> from sympy.abc import a, b, c, d
>>> q1 = Quaternion(1, 0, 0, 1)
>>> q2 = Quaternion(a, b, c, d)
>>> q1.product_matrix_left
Matrix([
[1, 0,  0, -1],
[0, 1, -1,  0],
[0, 1,  1,  0],
[1, 0,  0,  1]])

>>> q1.product_matrix_left * q2.to_Matrix()
Matrix([
[a - d],
[b - c],
[b + c],
[a + d]])


This is equivalent to:

>>> (q1 * q2).to_Matrix()
Matrix([
[a - d],
[b - c],
[b + c],
[a + d]])

property product_matrix_right

Returns 4 x 4 Matrix equivalent to a Hamilton product from the right. This can be useful when treating quaternion elements as column vectors. Given a quaternion $$q = a + bi + cj + dk$$ where a, b, c and d are real numbers, the product matrix from the left is:

$\begin{split}M = \begin{bmatrix} a &-b &-c &-d \\ b & a & d &-c \\ c &-d & a & b \\ d & c &-b & a \end{bmatrix}\end{split}$

Examples

>>> from sympy import Quaternion
>>> from sympy.abc import a, b, c, d
>>> q1 = Quaternion(a, b, c, d)
>>> q2 = Quaternion(1, 0, 0, 1)
>>> q2.product_matrix_right
Matrix([
[1, 0, 0, -1],
[0, 1, 1, 0],
[0, -1, 1, 0],
[1, 0, 0, 1]])


Note the switched arguments: the matrix represents the quaternion on the right, but is still considered as a matrix multiplication from the left.

>>> q2.product_matrix_right * q1.to_Matrix()
Matrix([
[ a - d],
[ b + c],
[-b + c],
[ a + d]])


This is equivalent to:

>>> (q1 * q2).to_Matrix()
Matrix([
[ a - d],
[ b + c],
[-b + c],
[ a + d]])

static rotate_point(pin, r)[source]

Returns the coordinates of the point pin (a 3 tuple) after rotation.

Parameters:

pin : tuple

A 3-element tuple of coordinates of a point which needs to be rotated.

r : Quaternion or tuple

Axis and angle of rotation.

It’s important to note that when r is a tuple, it must be of the form (axis, angle)

Returns:

tuple

The coordinates of the point after rotation.

Examples

>>> from sympy import Quaternion
>>> from sympy import symbols, trigsimp, cos, sin
>>> x = symbols('x')
>>> q = Quaternion(cos(x/2), 0, 0, sin(x/2))
>>> trigsimp(Quaternion.rotate_point((1, 1, 1), q))
(sqrt(2)*cos(x + pi/4), sqrt(2)*sin(x + pi/4), 1)
>>> (axis, angle) = q.to_axis_angle()
>>> trigsimp(Quaternion.rotate_point((1, 1, 1), (axis, angle)))
(sqrt(2)*cos(x + pi/4), sqrt(2)*sin(x + pi/4), 1)

scalar_part()[source]

Returns scalar part($$\mathbf{S}(q)$$) of the quaternion q.

Explanation

Given a quaternion $$q = a + bi + cj + dk$$, returns $$\mathbf{S}(q) = a$$.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(4, 8, 13, 12)
>>> q.scalar_part()
4

set_norm(norm)[source]

Sets norm of an already instantiated quaternion.

Parameters:

norm : None or number

Pre-defined quaternion norm. If a value is given, Quaternion.norm returns this pre-defined value instead of calculating the norm

Examples

>>> from sympy import Quaternion
>>> from sympy.abc import a, b, c, d
>>> q = Quaternion(a, b, c, d)
>>> q.norm()
sqrt(a**2 + b**2 + c**2 + d**2)


Setting the norm:

>>> q.set_norm(1)
>>> q.norm()
1


Removing set norm:

>>> q.set_norm(None)
>>> q.norm()
sqrt(a**2 + b**2 + c**2 + d**2)

to_Matrix(vector_only=False)[source]

Returns elements of quaternion as a column vector. By default, a Matrix of length 4 is returned, with the real part as the first element. If vector_only is True, returns only imaginary part as a Matrix of length 3.

Parameters:

vector_only : bool

If True, only imaginary part is returned. Default value: False

Returns:

Matrix

A column vector constructed by the elements of the quaternion.

Examples

>>> from sympy import Quaternion
>>> from sympy.abc import a, b, c, d
>>> q = Quaternion(a, b, c, d)
>>> q
a + b*i + c*j + d*k

>>> q.to_Matrix()
Matrix([
[a],
[b],
[c],
[d]])

>>> q.to_Matrix(vector_only=True)
Matrix([
[b],
[c],
[d]])

to_axis_angle()[source]

Returns the axis and angle of rotation of a quaternion.

Returns:

tuple

Tuple of (axis, angle)

Examples

>>> from sympy import Quaternion
>>> q = Quaternion(1, 1, 1, 1)
>>> (axis, angle) = q.to_axis_angle()
>>> axis
(sqrt(3)/3, sqrt(3)/3, sqrt(3)/3)
>>> angle
2*pi/3

to_euler(
seq,
avoid_square_root=False,
)[source]

Returns Euler angles representing same rotation as the quaternion, in the sequence given by seq. This implements the method described in [R3].

For degenerate cases (gymbal lock cases), the third angle is set to zero.

Parameters:

seq : string of length 3

Represents the sequence of rotations. For extrinsic rotations, seq must be all lowercase and its elements must be from the set {'x', 'y', 'z'} For intrinsic rotations, seq must be all uppercase and its elements must be from the set {'X', 'Y', 'Z'}

When True, first and third angles are given as an addition and subtraction of two simpler atan2 expressions. When False, the first and third angles are each given by a single more complicated atan2 expression. This equivalent expression is given by:

$\operatorname{atan_2} (b,a) \pm \operatorname{atan_2} (d,c) = \operatorname{atan_2} (bc\pm ad, ac\mp bd)$

Default value: True

avoid_square_root : bool

When True, the second angle is calculated with an expression based on acos, which is slightly more complicated but avoids a square root. When False, second angle is calculated with atan2, which is simpler and can be better for numerical reasons (some numerical implementations of acos have problems near zero). Default value: False

Returns:

Tuple

The Euler angles calculated from the quaternion

Examples

>>> from sympy import Quaternion
>>> from sympy.abc import a, b, c, d
>>> euler = Quaternion(a, b, c, d).to_euler('zyz')
>>> euler
(-atan2(-b, c) + atan2(d, a),
2*atan2(sqrt(b**2 + c**2), sqrt(a**2 + d**2)),
atan2(-b, c) + atan2(d, a))


References

to_rotation_matrix(
v=None,
homogeneous=True,
)[source]

Returns the equivalent rotation transformation matrix of the quaternion which represents rotation about the origin if v is not passed.

Parameters:

v : tuple or None

Default value: None

homogeneous : bool

When True, gives an expression that may be more efficient for symbolic calculations but less so for direct evaluation. Both formulas are mathematically equivalent. Default value: True

Returns:

tuple

Returns the equivalent rotation transformation matrix of the quaternion which represents rotation about the origin if v is not passed.

Examples

>>> from sympy import Quaternion
>>> from sympy import symbols, trigsimp, cos, sin
>>> x = symbols('x')
>>> q = Quaternion(cos(x/2), 0, 0, sin(x/2))
>>> trigsimp(q.to_rotation_matrix())
Matrix([
[cos(x), -sin(x), 0],
[sin(x),  cos(x), 0],
[     0,       0, 1]])


Generates a 4x4 transformation matrix (used for rotation about a point other than the origin) if the point(v) is passed as an argument.

classmethod vector_coplanar(q1, q2, q3)[source]

Returns True if the axis of the pure quaternions seen as 3D vectors q1, q2, and q3 are coplanar.

Parameters:

q1

A pure Quaternion.

q2

A pure Quaternion.

q3

A pure Quaternion.

Returns:

True : if the axis of the pure quaternions seen as 3D vectors

q1, q2, and q3 are coplanar.

False : if the axis of the pure quaternions seen as 3D vectors

q1, q2, and q3 are not coplanar.

None : if the axis of the pure quaternions seen as 3D vectors

q1, q2, and q3 are coplanar is unknown.

Explanation

Three pure quaternions are vector coplanar if the quaternions seen as 3D vectors are coplanar.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q1 = Quaternion(0, 4, 4, 4)
>>> q2 = Quaternion(0, 8, 8, 8)
>>> q3 = Quaternion(0, 24, 24, 24)
>>> Quaternion.vector_coplanar(q1, q2, q3)
True

>>> q1 = Quaternion(0, 8, 16, 8)
>>> q2 = Quaternion(0, 8, 3, 12)
>>> Quaternion.vector_coplanar(q1, q2, q3)
False


vector_part()[source]

Returns $$\mathbf{V}(q)$$, the vector part of the quaternion $$q$$.

Explanation

Given a quaternion $$q = a + bi + cj + dk$$, returns $$\mathbf{V}(q) = bi + cj + dk$$.

Examples

>>> from sympy.algebras.quaternion import Quaternion
>>> q = Quaternion(1, 1, 1, 1)
>>> q.vector_part()
0 + 1*i + 1*j + 1*k

>>> q = Quaternion(4, 8, 13, 12)
>>> q.vector_part()
0 + 8*i + 13*j + 12*k
`