Translation and Reciprocal translation: Difference between pages

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


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


:$$\mathbf T = {\tau_x \mathbf e_{23} + \tau_y \mathbf e_{31} + \tau_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$
:$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$


performs a translation by twice the displacement vector $$\boldsymbol \tau = (\tau_x, \tau_y, \tau_z)$$ when used as an operator in the sandwich antiproduct. This can be interpreted as a rotation about the [[line]] at infinity perpendicular to the direction $$\boldsymbol \tau$$.
performs a perspective projection in the direction of $$\mathbf t = (t_x, t_y, t_z)$$ with the focal length given by


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


A translation by a distance $$\delta$$ perpendicular to a unitized [[plane]] $$\mathbf g$$ can be expressed as an exponential of the plane's [[attitude]] as
== Example ==


:$$\mathbf T = \exp_\unicode{x27C7}\left(\dfrac{1}{2}\delta \operatorname{att}(\mathbf g)\right) = \dfrac{\delta}{2} \operatorname{att}(\mathbf g) + {\large\unicode{x1d7d9}}$$
The left image below shows the flow field in the ''x''-''z'' plane for the translation $$\mathbf T = -\frac{1}{2} \mathbf e_{31} + {\large\unicode{x1d7d9}}$$. The right image shows the flow field in the ''x''-''z'' plane for the reciprocal translation $$\mathbf T = \frac{1}{2} \mathbf e_{42} + \mathbf 1$$. 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.


== Matrix Form ==
[[Image:Translation.svg|480px]]
 
[[Image:DualTranslation.svg|480px]]
When a translation $$\mathbf T$$ is applied to a [[point]], it is equivalent to premultiplying the point by the $$4 \times 4$$ matrix
 
:$$\begin{bmatrix}
1 & 0 & 0 & \tau_x \\
0 & 1 & 0 & \tau_y \\
0 & 0 & 1 & \tau_z \\
0 & 0 & 0 & 1 \\
\end{bmatrix}$$ .
 
== Translation to Origin ==
 
A point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ is translated to the origin by the operator
 
:$$\mathbf T = {-\dfrac{p_{x\vphantom{y}}}{2p_w} \mathbf e_{23} - \dfrac{p_y}{2p_w} \mathbf e_{31} - \dfrac{p_{z\vphantom{y}}}{2p_w} \mathbf e_{12} + \large\unicode{x1d7d9}}$$ .


== Calculation ==
== Calculation ==


The exact 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 || Translation
! Type || Reciprocal Translation
|-
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | [[Point]]


$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = (p_x + 2\tau_xp_w)\mathbf e_1 + (p_y + 2\tau_yp_w)\mathbf e_2 + (p_z + 2\tau_zp_w)\mathbf e_3 + p_w\mathbf e_4$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf p \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + (2t_xp_x + 2t_yp_y + 2t_zp_z + p_w) \mathbf e_4$$
|-
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | [[Line]]


$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$
$$\begin{split}\mathbf L =\, &v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} \\ +\, &m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + (l_{mx} + 2\tau_y l_{vz} - 2\tau_z l_{vy})\mathbf e_{23} + (l_{my} + 2\tau_z l_{vx} - 2\tau_x l_{vz})\mathbf e_{31} + (l_{mz} + 2\tau_x l_{vy} - 2\tau_y l_{vx})\mathbf e_{12}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf L \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (v_x - 2t_ym_z + 2t_zm_y)\mathbf e_{41} + (v_y - 2t_zm_x + 2t_xm_z)\mathbf e_{42} + (v_z - 2t_xm_y - 2t_ym_x)\mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$
|-
|-
| style="padding: 12px;" | [[Plane]]
| style="padding: 12px;" | [[Plane]]


$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + (g_w - 2\tau_xg_x - 2\tau_yg_y - 2\tau_zg_z) \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf f \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (f_x - 2t_xf_w) \mathbf e_{234} + (f_y - 2t_yf_w) \mathbf e_{314} + (f_z - 2t_zf_w) \mathbf e_{124} + f_w \mathbf e_{321}$$
|}
|}
== 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 ==
== See Also ==


* [[Reciprocal translation]]
* [[Translation]]
* [[Rotation]]
* [[Reciprocal rotation]]
* [[Reflection]]
* [[Reciprocal reflection]]
* [[Inversion]]
* [[Transflection]]

Revision as of 02:57, 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 $$\mathbf T = -\frac{1}{2} \mathbf e_{31} + {\large\unicode{x1d7d9}}$$. The right image shows the flow field in the x-z plane for the reciprocal translation $$\mathbf T = \frac{1}{2} \mathbf e_{42} + \mathbf 1$$. 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

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