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

Methods

`canonize`()	Makes the quaternion canonic.
`canonized`()	Returns a quaternion in canonic form.
`conjugate`()	Conjugate the quaternion.
`conjugated`()	Returns a conjugate quaternion.
`copy`([cls])	Make an independent copy of the data object.
`from_data`(data)	Construct a quaternion from a data dict.
`from_frame`(frame)	Creates a quaternion object from a frame.
`from_json`(filepath)	Construct an object from serialized data contained in a JSON file.
`from_jsonstring`(string)	Construct an object from serialized data contained in a JSON string.
`from_matrix`(M)	Create a `Quaternion` from a transformation matrix.
`from_rotation`(R)	Create a `Quaternion` from a `compas.geometry.Rotatation`.
`to_data`()	Convert an object to its native data representation.
`to_json`(filepath[, pretty])	Serialize the data representation of an object to a JSON file.
`to_jsonstring`([pretty])	Serialize the data representation of an object to a JSON string.
`transform`(transformation)	Transform the geometry.
`transformed`(transformation)	Returns a transformed copy of this geometry.
`unitize`()	Scales the quaternion to make it unit-length.
`unitized`()	Returns a quaternion with a unit-length.
`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