Translation: Difference between revisions

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


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


performs a translation by twice the displacement vector $$\mathbf t = (t_x, t_y, t_z)$$ 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 $$\mathbf t$$.
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 $$\boldsymbol \tau$$.


== Exponential Form ==
== Exponential Form ==
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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
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


:$$\mathbf T = \exp_\unicode{x27C7}\left(\dfrac{1}{2}\delta \operatorname{att}(\mathbf g)\right) = \dfrac{\delta}{2} \operatorname{att}(\mathbf g) + {\large\unicode{x1d7d9}}$$
:$$\mathbf T = \exp_\unicode{x27C7}\left(\dfrac{\delta}{2} \operatorname{att}(\mathbf g)\right) = \dfrac{\delta}{2} \operatorname{att}(\mathbf g) + {\large\unicode{x1d7d9}}$$


== Matrix Form ==
== Matrix Form ==
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:$$\begin{bmatrix}
:$$\begin{bmatrix}
1 & 0 & 0 & t_x \\
1 & 0 & 0 & 2\tau_x \\
0 & 1 & 0 & t_y \\
0 & 1 & 0 & 2\tau_y \\
0 & 0 & 1 & t_z \\
0 & 0 & 1 & 2\tau_z \\
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 \\
\end{bmatrix}$$ .
\end{bmatrix}$$ .
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$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = (p_x + 2t_xp_w)\mathbf e_1 + (p_y + 2t_yp_w)\mathbf e_2 + (p_z + 2t_zp_w)\mathbf e_3 + p_w\mathbf e_4$$
| 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;" | [[Line]]
| style="padding: 12px;" | [[Line]]


$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$
$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + (l_{mx} + 2t_y l_{vz} - 2t_z l_{vy})\mathbf e_{23} + (l_{my} + 2t_z l_{vx} - 2t_x l_{vz})\mathbf e_{31} + (l_{mz} + 2t_x l_{vy} - 2t_y l_{vx})\mathbf e_{12}$$
| 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;" | [[Plane]]
| style="padding: 12px;" | [[Plane]]


$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + (g_w - 2t_xg_x - 2t_yg_y - 2t_zg_z) \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + (g_w - 2\tau_xg_x - 2\tau_yg_y - 2\tau_zg_z) \mathbf e_{321}$$
|}
|}


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


* [[Dual translation]]
* [[Complement translation]]
* [[Rotation]]
* [[Rotation]]
* [[Reflection]]
* [[Reflection]]
* [[Inversion]]
* [[Inversion]]
* [[Transflection]]
* [[Transflection]]

Latest revision as of 23:27, 12 October 2024

A translation is a proper isometry of Euclidean space.

The specific kind of motor

$$\mathbf T = {\tau_x \mathbf e_{23} + \tau_y \mathbf e_{31} + \tau_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$

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 $$\boldsymbol \tau$$.

Exponential Form

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

$$\mathbf T = \exp_\unicode{x27C7}\left(\dfrac{\delta}{2} \operatorname{att}(\mathbf g)\right) = \dfrac{\delta}{2} \operatorname{att}(\mathbf g) + {\large\unicode{x1d7d9}}$$

Matrix Form

When a translation $$\mathbf T$$ is applied to a point, it is equivalent to premultiplying the point by the $$4 \times 4$$ matrix

$$\begin{bmatrix} 1 & 0 & 0 & 2\tau_x \\ 0 & 1 & 0 & 2\tau_y \\ 0 & 0 & 1 & 2\tau_z \\ 0 & 0 & 0 & 1 \\ \end{bmatrix}$$ .

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

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

Type Translation
Point

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

$$\mathbf T \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = (p_x + 2\tau_xp_w)\mathbf e_1 + (p_y + 2\tau_yp_w)\mathbf e_2 + (p_z + 2\tau_zp_w)\mathbf e_3 + p_w\mathbf e_4$$
Line

$$\begin{split}\boldsymbol l =\, &l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} \\ +\, &l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}\end{split}$$

$$\mathbf T \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + (l_{mx} + 2\tau_y l_{vz} - 2\tau_z l_{vy})\mathbf e_{23} + (l_{my} + 2\tau_z l_{vx} - 2\tau_x l_{vz})\mathbf e_{31} + (l_{mz} + 2\tau_x l_{vy} - 2\tau_y l_{vx})\mathbf e_{12}$$
Plane

$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$

$$\mathbf T \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + (g_w - 2\tau_xg_x - 2\tau_yg_y - 2\tau_zg_z) \mathbf e_{321}$$

See Also

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