Difference between revisions of "Dual translation"

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m (Eric Lengyel moved page Antitranslation to Dual translation without leaving a redirect)
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An ''antitranslation'' is a proper isometry of dual Euclidean space.
A ''dual translation'' is a proper isometry of dual Euclidean space.


The specific kind of [[antimotor]]
The specific kind of [[dual motor]]


:$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + 1$$
:$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + 1$$


performs a perspective projection. The exact antitranslation calculations for points, lines, and planes are shown in the following table.
performs a perspective projection. The exact dual translation calculations for points, lines, and planes are shown in the following table.


{| class="wikitable"
{| class="wikitable"
! Type || Antitranslation
! Type || Dual Translation
|-
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | [[Point]]
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== See Also ==
== See Also ==


* [[Antirotation]]
* [[Dual rotation]]
* [[Antireflection]]
* [[Dual reflection]]

Revision as of 05:22, 19 June 2022

A dual translation is a proper isometry of dual Euclidean space.

The specific kind of dual motor

$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + 1$$

performs a perspective projection. The exact dual translation calculations for points, lines, and planes are shown in the following table.

Type Dual Translation
Point

$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$

$$\mathbf T \mathbin{\unicode{x27D1}} \mathbf p \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + (2t_xp_x + 2t_yp_y + 2t_zp_z + p_w) \mathbf e_4$$
Line

$$\begin{split}\mathbf L =\, &v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} \\ +\, &m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}\end{split}$$

$$\mathbf T \mathbin{\unicode{x27D1}} \mathbf L \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (v_x - 2t_ym_z + 2t_zm_y)\mathbf e_{41} + (v_y - 2t_zm_x + 2t_xm_z)\mathbf e_{42} + (v_z - 2t_xm_y - 2t_ym_x)\mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$
Plane

$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$

$$\mathbf T \mathbin{\unicode{x27D1}} \mathbf f \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (f_x - 2t_xf_w) \mathbf e_{234} + (f_y - 2t_yf_w) \mathbf e_{314} + (f_z - 2t_zf_w) \mathbf e_{124} + f_w \mathbf e_{321}$$

See Also