Complement translation: Difference between revisions

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__NOTOC__
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A ''dual translation'' is a proper isometry of dual Euclidean space.
A ''reciprocal translation'' is a proper isometry of reciprocal Euclidean space.


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


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


The left image below shows the flow field in the ''x''-''z'' plane for the translation $$\mathbf T = -\frac{1}{2} \mathbf e_{12} + {\large\unicode{x1d7d9}}$$. The right image shows the flow field in the ''x''-''z'' plane for the dual translation $$\mathbf T = \frac{1}{2} \mathbf e_{43} + \mathbf 1$$. The yellow line is fixed as a whole, but points on it move to other locations on the line. All points with $$z = 0$$, represented by the blue plane, are fixed. The white plane at $$z = -1$$ represents the division between regions where the signs of projected $$z$$ coordinates are positive and negative.
The left image below shows the flow field in the ''x''-''z'' plane for the translation $$\mathbf T = -\frac{1}{2} \mathbf e_{12} + {\large\unicode{x1d7d9}}$$. The right image shows the flow field in the ''x''-''z'' plane for the reciprocal translation $$\mathbf T = \frac{1}{2} \mathbf e_{43} + \mathbf 1$$. The yellow line is fixed as a whole, but points on it move to other locations on the line. All points with $$z = 0$$, represented by the blue plane, are fixed. The white plane at $$z = -1$$ represents the division between regions where the signs of projected $$z$$ coordinates are positive and negative.


[[Image:Translation.svg|480px]]
[[Image:Translation.svg|480px]]
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== Calculation ==
== Calculation ==


The exact dual translation calculations for points, lines, and planes are shown in the following table.
The exact reciprocal translation calculations for points, lines, and planes are shown in the following table.


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


== Dual Translation to Horizon ==
== Reciprocal Translation to Horizon ==


A plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ is dual translated to the horizon by the operator
A plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ is reciprocal translated to the horizon by the operator


:$$\mathbf T = \dfrac{f_{x\vphantom{y}}}{2f_w} \mathbf e_{41} + \dfrac{f_y}{2f_w} \mathbf e_{42} + \dfrac{f_{z\vphantom{y}}}{2f_w} \mathbf e_{32} + \mathbf 1$$ .
:$$\mathbf T = \dfrac{f_{x\vphantom{y}}}{2f_w} \mathbf e_{41} + \dfrac{f_y}{2f_w} \mathbf e_{42} + \dfrac{f_{z\vphantom{y}}}{2f_w} \mathbf e_{43} + \mathbf 1$$ .


== See Also ==
== See Also ==


* [[Translation]]
* [[Translation]]
* [[Dual rotation]]
* [[Reciprocal rotation]]
* [[Dual reflection]]
* [[Reciprocal reflection]]

Revision as of 03:00, 28 February 2024

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

The specific kind of reciprocal motor

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

performs a perspective projection in the direction of $$\mathbf t = (t_x, t_y, t_z)$$ with the focal length given by

$$g = \dfrac{1}{2\Vert \mathbf t \Vert}$$ .

Example

The left image below shows the flow field in the x-z plane for the translation $$\mathbf T = -\frac{1}{2} \mathbf e_{12} + {\large\unicode{x1d7d9}}$$. The right image shows the flow field in the x-z plane for the reciprocal translation $$\mathbf T = \frac{1}{2} \mathbf e_{43} + \mathbf 1$$. The yellow line is fixed as a whole, but points on it move to other locations on the line. All points with $$z = 0$$, represented by the blue plane, are fixed. The white plane at $$z = -1$$ represents the division between regions where the signs of projected $$z$$ coordinates are positive and negative.

Calculation

The exact reciprocal translation calculations for points, lines, and planes are shown in the following table.

Type Reciprocal 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

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

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

Reciprocal Translation to Horizon

A plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ is reciprocal translated to the horizon by the operator

$$\mathbf T = \dfrac{f_{x\vphantom{y}}}{2f_w} \mathbf e_{41} + \dfrac{f_y}{2f_w} \mathbf e_{42} + \dfrac{f_{z\vphantom{y}}}{2f_w} \mathbf e_{43} + \mathbf 1$$ .

See Also