Homogeneous Points

The class PointHomogeneous is used to represent points in n-dimensional space.

A 3D point with coordinates \((x, y, z)\) in Eucledian space \(\mathbb{R}^3\) can be represented as a 4D point in homogeneous space \(\mathbb{R}^4\) as \((w, x, y, z)\).

3D points can be embedded into dual quaternion space in the following way:

\[\mathbf{q} = (w, x, y, z) \in \mathbb{R}^4 \rightarrow \mathbf{q'} = (w, 0, 0, 0, 0, x, y, z) \in SE(3)\]

where \(SE(3)\) is the special Euclidean group of rigid body transformations in 3D space.

How dual quaternions act on points can be found in DQ Action on a Point.

Normalized Lines

The class NormalizedLine is used to represent lines.

A line in 3D space can be represented using Plücker coordinates. These are six-dimensional vectors \(\mathbf{l}=(g_0, g_1, g_2, g_3, g_4, g_5)\). The line’s direction is \(\mathbf{g}=(g_0,g_1,g_2)\), its moment vector \(\overline{\mathbf{g}} = (g_3,g_4,g_5)\) is obtained as

\[\overline{\mathbf{g}} = \mathbf{q} \times \mathbf{g} = - \mathbf{g} \times \mathbf{q}\]

where \(\mathbf{q}\) is an arbitrary point on the line. Plücker coordinates fulfill the Plücker condition

\[\mathbf{g}^T\cdot\overline{\mathbf{g}} = 0\]

Normalized line then corresponds to Plücker coordinates with the condition, that

\[\mathbf{||g||} = 1\]

i.e. the direction vector is normalized. More on Plücker coordinates can be found in Pottmann and Wallner[1].

Lines can be embedded into dual quaternion space in the following way:

\[\mathbf{l} = (g_0, g_1, g_2, g_3, g_4, g_5) \in \mathbb{R}^6 \rightarrow \mathbf{l'} = (0, g_0, g_1, g_2, 0, -g_3, -g_4, -g_5) \in SE(3)\]

How dual quaternions act on lines can be found in DQ Action on a Line.

Dual quaternions that correspond to lines can be interpreted as the half-turn transformations, i.e. transformations that rotate the origin by 180 degrees around the line. The following code snippet plots the half-turn transformation corresponding to a line in 3D space.

from rational_linkages import Plotter, DualQuaternion, NormalizedLine, PointHomogeneous
from copy import deepcopy

pt1 = PointHomogeneous([1, 0.3, 1, 0])  # w = 1, x = 0.3, y = 1, z = 0
pt2 = PointHomogeneous([1, 0.3, 1, 1])  # w = 1, x = 0.3, y = 1, z = 1
l = NormalizedLine.from_two_points(pt1, pt2)

dq1 = DualQuaternion(l.line2dq_array())

dq2 = deepcopy(
    dq1)  # alter the rotation part of the dual quaternion to get some other transformation
dq2[0] = 2

p = Plotter(arrows_length=0.5, backend='matplotlib')
p.plot(l)
p.plot(pt1, label='pt1', color='red')
p.plot(pt2, label='pt2', color='red')

p.plot(DualQuaternion(), label='origin')
p.plot(dq1, label='half-turn')
p.plot(dq2, label='some-rotation')
p.show()

We can additionally see that if we alter the zero element of dq1 to get dq2, we get a transformation that corresponds to some rotation around the line, but not a half-turn transformation anymore.

Half-turn transformation corresponding to a line in 3D space

References: