Quaternion

class compas.geometry.Quaternion

Bases: Geometry

A quaternion is defined by 4 components, X, Y, Z, and W.

Parameters:

wfloat

The scalar (real) part of a quaternion.

xfloat

X component of the vector (complex, imaginary) part of a quaternion.

yfloat

Y component of the vector (complex, imaginary) part of a quaternion.

zfloat

Z component of the vector (complex, imaginary) part of a quaternion.

namestr, optional

The name of the transformation.

Attributes:

wfloat

The W component of the quaternion.

xfloat

The X component of the quaternion.

yfloat

The Y component of the quaternion.

zfloat

The Z component of the quaternion.

wxyzlist[float], read-only

Quaternion as a list of float in the ‘wxyz’ convention.

xyzwlist[float], read-only

Quaternion as a list of float in the ‘xyzw’ convention.

normfloat, read-only

The length (euclidean norm) of the quaternion.

is_unitbool, read-only

True if the quaternion is unit-length. False otherwise.

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]:

$$\begin{array}{r}\begin{array}{rl}ii& =jj=kk=ijk=-1\\ ij& =k,ji=-k\\ jk& =i,kj=-i\\ ki& =j,ik=-j\end{array}\end{array}$$

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.:

$$\begin{array}{r}\begin{array}{rl}w& =cos(\frac{\theta }{2})\\ x& =sin(\frac{\theta }{2})u_x\\ y& =sin(\frac{\theta }{2})u_y\\ z& =sin(\frac{\theta }{2})u_z\end{array}\end{array}$$

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]

matthew-brett/transforms3d

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.
dot	Computes the cosine of the angle between the two quaternions
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 Rotatation.
slerp	Slerp: spherical interpolation of two quaternions.
unitize	Scales the quaternion to make it unit-length.
unitized	Returns a quaternion with a unit-length.

Inherited Methods

ToString	Converts the instance to a string.
compute_aabb	Compute the axis-aligned bounding box of the geometry.
compute_obb	Compute the oriented bounding box of the geometry.
copy	Make an independent copy of the data object.
from_json	Construct an object of this type from a JSON file.
from_jsonstring	Construct an object of this type from a JSON string.
rotate	Rotate the geometry.
rotated	Returns a rotated copy of this geometry.
scale	Scale the geometry.
scaled	Returns a scaled copy of this geometry.
sha256	Compute a hash of the data for comparison during version control using the sha256 algorithm.
to_json	Convert an object to its native data representation and save it to a JSON file.
to_jsonstring	Convert an object to its native data representation and save it to a JSON string.
transform	Transform the geometry.
transformed	Returns a transformed copy of this geometry.
translate	Translate the geometry.
translated	Returns a translated copy of this geometry.
validate_data	Validate the data against the object's data schema.