Quaternion

class compas.geometry.Quaternion(w, x, y, z, **kwargs)[source]

Bases: compas.geometry.primitives._primitive.Primitive

Creates a Quaternion object.

Parameters

w (float) – The scalar (real) part of a quaternion.
x, y, z (float) – Components of the vector (complex, imaginary) part of a quaternion.

Attributes

wxyz (list of float, read-only) – Quaternion data listing the real part first.
xyzw (list of float, read-only) – Quaternion data listing the real part last.
norm (float, read-only) – The length of the quaternion.
is_unit (bool, read-only) – True if the quaternion has unit length.

Notes

The default convention to represent a quaternion \(q\) in this module is by four real values \(w\), \(x\), \(y\), \(z\). The first value \(w\) is the scalar (real) part, and \(x\), \(y\), \(z\) form the vector (complex, imaginary) part 1, so that:

\[q = w + xi + yj + zk\]

where \(i, j, k\) are basis components with following multiplication rules 2:

\[\begin{split}\begin{align} ii &= jj = kk = ijk = -1 \\ ij &= k, \quad ji = -k \\ jk &= i, \quad kj = -i \\ ki &= j, \quad ik = -j \end{align}\end{split}\]

Quaternions are associative but not commutative.

Quaternion as rotation.

A rotation through an angle \(\theta\) around an axis defined by a euclidean unit vector \(u = u_{x}i + u_{y}j + u_{z}k\) can be represented as a quaternion:

\[q = cos(\frac{\theta}{2}) + sin(\frac{\theta}{2}) [u_{x}i + u_{y}j + u_{z}k]\]

i.e.:

\[\begin{split}\begin{align} w &= cos(\frac{\theta}{2}) \\ x &= sin(\frac{\theta}{2}) u_{x} \\ y &= sin(\frac{\theta}{2}) u_{y} \\ z &= sin(\frac{\theta}{2}) u_{z} \end{align}\end{split}\]

For a quaternion to represent a rotation or orientation, it must be unit-length. A quaternion representing a rotation \(p\) resulting from applying a rotation \(r\) to a rotation \(q\), i.e.: \(p = rq\), is also unit-length.

References

1: http://mathworld.wolfram.com/Quaternion.html
2: http://mathworld.wolfram.com/HamiltonsRules.html
3: https://github.com/matthew-brett/transforms3d/blob/master/transforms3d/quaternions.py

Examples

>>> Q = Quaternion(1.0, 1.0, 1.0, 1.0).unitized()
>>> R = Quaternion(0.0,-0.1, 0.2,-0.3).unitized()
>>> P = R*Q
>>> P.is_unit
True

Attributes

`DATASCHEMA`	The schema of the data of this object.
`JSONSCHEMANAME`
`data`	The representation of the object as native Python data.
`is_unit`	`True` if the quaternion is unit-length or `False` if otherwise.
`norm`	The length (euclidean norm) of the quaternion.
`w`	The W component of the quaternion.
`wxyz`	Quaternion as a list of float in the 'wxyz' convention.
`x`	The X component of the quaternion.
`xyzw`	Quaternion as a list of float in the 'xyzw' convention.
`y`	The Y component of the quaternion.
`z`	The Z component of the quaternion.

Inherited Attributes

`JSONSCHEMA`	The schema of the JSON representation of the data of this object.
`dtype`	The type of the object in the form of a '2-level' import and a class name.
`guid`	The globally unique identifier of the object.
`jsondefinitions`	Reusable schema definitions.
`jsonstring`	The representation of the object data in JSON format.
`jsonvalidator`	JSON schema validator for draft 7.
`name`	The name of the object.

Methods

`canonize`	Makes the quaternion canonic.
`canonized`	Returns a quaternion in canonic form.
`conjugate`	Conjugate the quaternion.
`conjugated`	Returns a conjugate quaternion.
`from_data`	Construct a quaternion from a data dict.
`from_frame`	Creates a quaternion object from a frame.
`from_matrix`	Create a `Quaternion` from a transformation matrix.
`from_rotation`	Create a `Quaternion` from a `compas.geometry.Rotatation`.
`unitize`	Scales the quaternion to make it unit-length.
`unitized`	Returns a quaternion with a unit-length.

Inherited Methods

`copy`	Make an independent copy of the data object.
`from_json`	Construct an object from serialized data contained in a JSON file.
`from_jsonstring`	Construct an object from serialized data contained in a JSON string.
`to_data`	Convert an object to its native data representation.
`to_json`	Serialize the data representation of an object to a JSON file.
`to_jsonstring`	Serialize the data representation of an object to a JSON string.
`transform`	Transform the geometry.
`transformed`	Returns a transformed copy of this geometry.
`validate_data`	Validate the object's data against its data schema (self.DATASCHEMA).
`validate_json`	Validate the object's data against its json schema (self.JSONSCHEMA).

Quaternion

COMPAS