ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

class Quaternion(*args, **kwargs)¶

Bases: pybind11_object

Overloaded function.

__init__(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, first_component: ostk.core.type.Real, second_component: ostk.core.type.Real, third_component: ostk.core.type.Real, fourth_component: ostk.core.type.Real, format: ostk::mathematics::geometry::d3::transformation::rotation::Quaternion::Format) -> None
Create a quaternion from four components and format.
Args:
first_component (float): First component (x or w depending on format). second_component (float): Second component (y or x depending on format). third_component (float): Third component (z or y depending on format). fourth_component (float): Fourth component (w or z depending on format). format (Quaternion.Format): The quaternion format (XYZS or SXYZ).

Example:
>>> q = Quaternion(0.0, 0.0, 0.0, 1.0, Quaternion.Format.XYZS) >>> q = Quaternion(1.0, 0.0, 0.0, 0.0, Quaternion.Format.SXYZ)
__init__(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, vector: numpy.ndarray[numpy.float64[4, 1]], format: ostk::mathematics::geometry::d3::transformation::rotation::Quaternion::Format) -> None
Create a quaternion from a 4D vector and format.
Args:
vector (numpy.array): The 4D vector containing quaternion components. format (Quaternion.Format): The quaternion format.

Example:
>>> vector = numpy.array([0.0, 0.0, 0.0, 1.0]) >>> q = Quaternion(vector, Quaternion.Format.XYZS)
__init__(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, vector_part: numpy.ndarray[numpy.float64[3, 1]], scalar_part: ostk.core.type.Real) -> None
Create a quaternion from vector and scalar parts.
Args:
vector_part (numpy.array): The vector part (x, y, z components). scalar_part (float): The scalar part (s component).

Example:
>>> vector_part = numpy.array([0.0, 0.0, 0.0]) >>> scalar_part = 1.0 >>> q = Quaternion(vector_part, scalar_part)
__init__(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion) -> None
Create a quaternion by copying another quaternion.
Args:
quaternion (Quaternion): The quaternion to copy.

Example:
>>> original = Quaternion(0.0, 0.0, 0.0, 1.0, Quaternion.Format.XYZS) >>> copy = Quaternion(original)

Methods

`angular_difference_with`	Compute the angular difference with another quaternion.
`conjugate`	Conjugate the quaternion in-place.
`cross_multiply`	Perform cross multiplication with another quaternion.
`dot_multiply`	Perform dot multiplication with another quaternion.
`dot_product`	Compute the dot product with another quaternion.
`euler_angle`	Create a quaternion from Euler angles.
`exp`	Compute the exponential of the quaternion.
`get_scalar_part`	Get the scalar part of the quaternion.
`get_vector_part`	Get the vector part of the quaternion.
`inverse`	Invert the quaternion in-place.
`is_defined`	Check if the quaternion is defined.
`is_near`	Check if the quaternion is near another quaternion.
`is_unitary`	Check if the quaternion is unitary.
`lerp`	Linear interpolation between two quaternions.
`log`	Compute the natural logarithm of the quaternion.
`nlerp`	Normalized linear interpolation between two quaternions.
`norm`	Compute the norm (magnitude) of the quaternion.
`normalize`	Normalize the quaternion in-place.
`parse`	Parse a quaternion from a string.
`pow`	Raise the quaternion to a power.
`rectify`	Rectify the quaternion in-place (ensure positive scalar part).
`rotate_vector`	Rotate a vector using this quaternion.
`rotation_matrix`	Create a quaternion from a rotation matrix.
`rotation_vector`	Create a quaternion from a rotation vector.
`s`	Get the s-coordinate of the quaternion.
`shortest_rotation`	Create a quaternion representing the shortest rotation between two vectors.
`slerp`	Spherical linear interpolation between two quaternions.
`to_conjugate`	Get the conjugate of the quaternion.
`to_inverse`	Get the inverse of the quaternion.
`to_normalized`	Get a normalized copy of the quaternion.
`to_string`	Overloaded function.
`to_vector`	Convert the quaternion to a 4D vector.
`undefined`	Create an undefined quaternion.
`unit`	Create a unit quaternion (identity rotation).
`x`	Get the x-coordinate of the quaternion.
`xyzs`	Create a quaternion in XYZS format.
`y`	Get the y-coordinate of the quaternion.
`z`	Get the z-coordinate of the quaternion.

class Format( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion.Format, value: int, )¶

Bases: pybind11_object

Members:

XYZS

SXYZ

property name¶

__add__( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, arg0: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

__mul__(*args, **kwargs)¶

Overloaded function.

__mul__(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, arg0: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion) -> ostk.mathematics.geometry.d3.transformation.rotation.Quaternion
__mul__(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, arg0: numpy.ndarray[numpy.float64[3, 1]]) -> numpy.ndarray[numpy.float64[3, 1]]
__mul__(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, arg0: float) -> ostk.mathematics.geometry.d3.transformation.rotation.Quaternion

angular_difference_with( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk::mathematics::geometry::Angle¶

Compute the angular difference with another quaternion.

Parameters:: quaternion (Quaternion) -- The quaternion to compare with.
Returns:: The angular difference.
Return type:: Angle

Example

>>> q1 = Quaternion.unit()
>>> q2 = Quaternion(0.1, 0.2, 0.3, 0.9, Quaternion.Format.XYZS)
>>> angle = q1.angular_difference_with(q2)

conjugate( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → None¶

Conjugate the quaternion in-place.

Example

>>> q = Quaternion(1.0, 2.0, 3.0, 4.0, Quaternion.Format.XYZS)
>>> q.conjugate()

cross_multiply( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Perform cross multiplication with another quaternion.

Parameters:: quaternion (Quaternion) -- The quaternion to multiply with.
Returns:: The result of cross multiplication.
Return type:: Quaternion

Example

>>> q1 = Quaternion.unit()
>>> q2 = Quaternion(0.1, 0.2, 0.3, 0.9, Quaternion.Format.XYZS)
>>> result = q1.cross_multiply(q2)

dot_multiply( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Perform dot multiplication with another quaternion.

Parameters:: quaternion (Quaternion) -- The quaternion to multiply with.
Returns:: The result of dot multiplication.
Return type:: Quaternion

Example

>>> q1 = Quaternion.unit()
>>> q2 = Quaternion(0.1, 0.2, 0.3, 0.9, Quaternion.Format.XYZS)
>>> result = q1.dot_multiply(q2)

dot_product( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.core.type.Real¶

Compute the dot product with another quaternion.

Parameters:: quaternion (Quaternion) -- The quaternion to compute dot product with.
Returns:: The dot product result.
Return type:: float

Example

>>> q1 = Quaternion.unit()
>>> q2 = Quaternion(0.1, 0.2, 0.3, 0.9, Quaternion.Format.XYZS)
>>> dot = q1.dot_product(q2)

static euler_angle( euler_angle: ostk::mathematics::geometry::d3::transformation::rotation::EulerAngle, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Create a quaternion from Euler angles.

Parameters:: euler_angle (EulerAngle) -- The Euler angles.
Returns:: The quaternion representing the rotation.
Return type:: Quaternion

Example

>>> ea = EulerAngle.zyx(Angle.radians(0.1), Angle.radians(0.2), Angle.radians(0.3))
>>> q = Quaternion.euler_angle(ea)

exp( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Compute the exponential of the quaternion.

Returns:: The exponential of the quaternion.
Return type:: Quaternion

Example

>>> q = Quaternion(0.1, 0.2, 0.3, 0.0, Quaternion.Format.XYZS)
>>> exp_q = q.exp()

get_scalar_part( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.core.type.Real¶

Get the scalar part of the quaternion.

Returns:: The scalar part.
Return type:: float

Example

>>> q = Quaternion(0.0, 0.0, 0.0, 1.0, Quaternion.Format.XYZS)
>>> q.get_scalar_part()  # 1.0

get_vector_part( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → numpy.ndarray[numpy.float64[3, 1]]¶

Get the vector part of the quaternion.

Returns:: The vector part.
Return type:: Vector3d

inverse( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → None¶

Invert the quaternion in-place.

Example

>>> q = Quaternion.unit()
>>> q.inverse()

is_defined( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → bool¶

Check if the quaternion is defined.

Returns:: True if the quaternion is defined, False otherwise.
Return type:: bool

is_near( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, angular_tolerance: ostk::mathematics::geometry::Angle, ) → bool¶

Check if the quaternion is near another quaternion.

Returns:: True if the quaternion is near another quaternion, False otherwise.
Return type:: bool

is_unitary( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, norm_tolerance: ostk.core.type.Real = Real.epsilon(), ) → bool¶

Check if the quaternion is unitary.

Returns:: True if the quaternion is unitary, False otherwise.
Return type:: bool

static lerp( first_quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, second_quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ratio: ostk.core.type.Real, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Linear interpolation between two quaternions.

Parameters:

first_quaternion (Quaternion) -- The first quaternion.
second_quaternion (Quaternion) -- The second quaternion.
ratio (float) -- The interpolation ratio (0.0 to 1.0).

Returns:

The interpolated quaternion.

Return type:

Quaternion

Example

>>> q1 = Quaternion.unit()
>>> q2 = Quaternion(0.1, 0.2, 0.3, 0.9, Quaternion.Format.XYZS)
>>> q_interp = Quaternion.lerp(q1, q2, 0.5)

log( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Compute the natural logarithm of the quaternion.

Returns:: The natural logarithm of the quaternion.
Return type:: Quaternion

Example

>>> q = Quaternion.unit()
>>> log_q = q.log()

static nlerp( first_quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, second_quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ratio: ostk.core.type.Real, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Normalized linear interpolation between two quaternions.

Parameters:

first_quaternion (Quaternion) -- The first quaternion.
second_quaternion (Quaternion) -- The second quaternion.
ratio (float) -- The interpolation ratio (0.0 to 1.0).

Returns:

The normalized interpolated quaternion.

Return type:

Quaternion

Example

>>> q1 = Quaternion.unit()
>>> q2 = Quaternion(0.1, 0.2, 0.3, 0.9, Quaternion.Format.XYZS)
>>> q_interp = Quaternion.nlerp(q1, q2, 0.5)

norm( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.core.type.Real¶

Compute the norm (magnitude) of the quaternion.

Returns:: The norm of the quaternion.
Return type:: float

Example

>>> q = Quaternion(1.0, 2.0, 3.0, 4.0, Quaternion.Format.XYZS)
>>> magnitude = q.norm()

normalize( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → None¶

Normalize the quaternion in-place.

Example

>>> q = Quaternion(1.0, 1.0, 1.0, 1.0, Quaternion.Format.XYZS)
>>> q.normalize()

static parse( string: ostk.core.type.String, format: ostk::mathematics::geometry::d3::transformation::rotation::Quaternion::Format, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Parse a quaternion from a string.

Parameters:

string (str) -- The string representation of the quaternion.
format (Quaternion.Format) -- The format of the quaternion.

Returns:

The parsed quaternion.

Return type:

Quaternion

Example

>>> q = Quaternion.parse("[0.0, 0.0, 0.0, 1.0]", Quaternion.Format.XYZS)

pow( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, value: ostk.core.type.Real, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Raise the quaternion to a power.

Parameters:: value (float) -- The exponent.
Returns:: The quaternion raised to the power.
Return type:: Quaternion

Example

>>> q = Quaternion.unit()
>>> powered = q.pow(2.0)

rectify( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → None¶

Rectify the quaternion in-place (ensure positive scalar part).

Example

>>> q = Quaternion(0.0, 0.0, 0.0, -1.0, Quaternion.Format.XYZS)
>>> q.rectify()

rotate_vector( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, vector: numpy.ndarray[numpy.float64[3, 1]], norm_tolerance: ostk.core.type.Real = Real.epsilon(), ) → numpy.ndarray[numpy.float64[3, 1]]¶

Rotate a vector using this quaternion.

Parameters:

vector (numpy.array) -- The 3D vector to rotate.
norm_tolerance (float, optional) -- Tolerance for normalization check.

Returns:

The rotated vector.

Return type:

Vector3d

Example

>>> q = Quaternion.unit()
>>> vector = numpy.array([1.0, 0.0, 0.0])
>>> rotated = q.rotate_vector(vector)

static rotation_matrix( rotation_matrix: ostk::mathematics::geometry::d3::transformation::rotation::RotationMatrix, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Create a quaternion from a rotation matrix.

Parameters:: rotation_matrix (RotationMatrix) -- The rotation matrix.
Returns:: The quaternion representing the rotation.
Return type:: Quaternion

Example

>>> rm = RotationMatrix.identity()
>>> q = Quaternion.rotation_matrix(rm)

static rotation_vector( rotation_vector: ostk::mathematics::geometry::d3::transformation::rotation::RotationVector, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Create a quaternion from a rotation vector.

Parameters:: rotation_vector (RotationVector) -- The rotation vector.
Returns:: The quaternion representing the rotation.
Return type:: Quaternion

Example

>>> rv = RotationVector([0.1, 0.2, 0.3], Angle.radians(1.0))
>>> q = Quaternion.rotation_vector(rv)

s( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.core.type.Real¶

Get the s-coordinate of the quaternion.

Returns:: The s-coordinate.
Return type:: float

static shortest_rotation( first_vector: numpy.ndarray[numpy.float64[3, 1]], second_vector: numpy.ndarray[numpy.float64[3, 1]], ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Create a quaternion representing the shortest rotation between two vectors.

Parameters:

first_vector (numpy.array) -- The first vector.
second_vector (numpy.array) -- The second vector.

Returns:

The quaternion representing the shortest rotation.

Return type:

Quaternion

Example

>>> v1 = numpy.array([1.0, 0.0, 0.0])
>>> v2 = numpy.array([0.0, 1.0, 0.0])
>>> q = Quaternion.shortest_rotation(v1, v2)

static slerp( first_quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, second_quaternion: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ratio: ostk.core.type.Real, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Spherical linear interpolation between two quaternions.

Parameters:

first_quaternion (Quaternion) -- The first quaternion.
second_quaternion (Quaternion) -- The second quaternion.
ratio (float) -- The interpolation ratio (0.0 to 1.0).

Returns:

The spherically interpolated quaternion.

Return type:

Quaternion

Example

>>> q1 = Quaternion.unit()
>>> q2 = Quaternion(0.1, 0.2, 0.3, 0.9, Quaternion.Format.XYZS)
>>> q_interp = Quaternion.slerp(q1, q2, 0.5)

to_conjugate( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Get the conjugate of the quaternion.

Returns:: The conjugate quaternion.
Return type:: Quaternion

Example

>>> q = Quaternion(1.0, 2.0, 3.0, 4.0, Quaternion.Format.XYZS)
>>> conjugate = q.to_conjugate()

to_inverse( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Get the inverse of the quaternion.

Returns:: The inverse quaternion.
Return type:: Quaternion

Example

>>> q = Quaternion.unit()
>>> inverse = q.to_inverse()

to_normalized( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Get a normalized copy of the quaternion.

Returns:: The normalized quaternion.
Return type:: Quaternion

Example

>>> q = Quaternion(1.0, 1.0, 1.0, 1.0, Quaternion.Format.XYZS)
>>> normalized = q.to_normalized()

to_string(*args, **kwargs)¶

Overloaded function.

to_string(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion) -> ostk.core.type.String
Convert the quaternion to string representation.
Returns:
str: String representation of the quaternion.

Example:
>>> q = Quaternion.unit() >>> q.to_string()
to_string(self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, format: ostk::mathematics::geometry::d3::transformation::rotation::Quaternion::Format) -> ostk.core.type.String
Convert the quaternion to string representation with specified format.
Args:
format (Quaternion.Format): The format for string representation.

Returns:
str: String representation of the quaternion.

Example:
>>> q = Quaternion.unit() >>> q.to_string(Quaternion.Format.SXYZ)

to_vector( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, format: ostk::mathematics::geometry::d3::transformation::rotation::Quaternion::Format, ) → numpy.ndarray[numpy.float64[4, 1]]¶

Convert the quaternion to a 4D vector.

Parameters:: format (Quaternion.Format) -- The format for the vector components.
Returns:: The quaternion as a 4D vector.
Return type:: Vector4d

Example

>>> q = Quaternion.unit()
>>> vector = q.to_vector(Quaternion.Format.XYZS)

static undefined() → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Create an undefined quaternion.

Returns:: An undefined quaternion.
Return type:: Quaternion

Example

>>> q = Quaternion.undefined()
>>> q.is_defined()  # False

static unit() → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Create a unit quaternion (identity rotation).

Returns:: A unit quaternion.
Return type:: Quaternion

Example

>>> q = Quaternion.unit()
>>> q.is_unitary()  # True

x( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.core.type.Real¶

Get the x-coordinate of the quaternion.

Returns:: The x-coordinate.
Return type:: float

static xyzs( first_component: ostk.core.type.Real, second_component: ostk.core.type.Real, third_component: ostk.core.type.Real, fourth_component: ostk.core.type.Real, ) → ostk.mathematics.geometry.d3.transformation.rotation.Quaternion¶

Create a quaternion in XYZS format.

Parameters:

first_component (float) -- The x component.
second_component (float) -- The y component.
third_component (float) -- The z component.
fourth_component (float) -- The s component.

Returns:

The quaternion in XYZS format.

Return type:

Quaternion

Example

>>> q = Quaternion.xyzs(0.0, 0.0, 0.0, 1.0)

y( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.core.type.Real¶

Get the y-coordinate of the quaternion.

Returns:: The y-coordinate.
Return type:: float

z( self: ostk.mathematics.geometry.d3.transformation.rotation.Quaternion, ) → ostk.core.type.Real¶

Get the z-coordinate of the quaternion.

Returns:: The z-coordinate.
Return type:: float