Translation - Revision history

Eric Lengyel: /* Matrix Form */

2024-10-12T23:27:53Z

Matrix Form

← Older revision		Revision as of 23:27, 12 October 2024
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	:$$\begin{bmatrix}		:$$\begin{bmatrix}
	1 & 0 & 0 & \tau_x \\		1 & 0 & 0 & 2\tau_x \\
	0 & 1 & 0 & \tau_y \\		0 & 1 & 0 & 2\tau_y \\
	0 & 0 & 1 & \tau_z \\		0 & 0 & 1 & 2\tau_z \\
	0 & 0 & 0 & 1 \\		0 & 0 & 0 & 1 \\
	\end{bmatrix}$$ .		\end{bmatrix}$$ .

Eric Lengyel: /* See Also */

2024-08-08T07:11:29Z

Eric Lengyel: /* Exponential Form */

2024-07-14T02:26:20Z

Exponential Form

← Older revision		Revision as of 02:26, 14 July 2024
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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 ==

Eric Lengyel at 00:52, 8 February 2024

2024-02-08T00:52:02Z

← Older revision		Revision as of 00:52, 8 February 2024
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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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	:$$\begin{bmatrix}		:$$\begin{bmatrix}
	1 & 0 & 0 & ~~t_x~~ \\		1 & 0 & 0 & \tau_x \\
	0 & 1 & 0 & ~~t_y~~ \\		0 & 1 & 0 & \tau_y \\
	0 & 0 & 1 & ~~t_z~~ \\		0 & 0 & 1 & \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;" \| $$\mathbf T \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{T}}} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + (g_w - 2\tau_xg_x - 2\tau_yg_y - 2\tau_zg_z) \mathbf e_{321}$$
	\|}		\|}

Eric Lengyel: /* See Also */

2023-09-05T20:19:56Z

Eric Lengyel at 07:46, 4 August 2023

2023-08-04T07:46:14Z

← Older revision		Revision as of 07:46, 4 August 2023
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	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 $$\mathbf t = (t_x, t_y, t_z)$$ 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 $$\mathbf t$$.

	==~~= Translation to Origin =~~==		== Exponential Form ==

	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~~		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

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

			== 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

	:$$\~~mathbf T =~~ {-\~~dfrac{p_{x~~\~~vphantom{y}}}{2p_w}~~ \~~mathbf e_{23} -~~ \~~dfrac{p_y}{2p_w}~~ \~~mathbf e_{31} -~~ \~~dfrac{p_{z\vphantom{y}}}{2p_w}~~ \~~mathbf e_{12} +~~ \~~large~~\~~unicode~~{~~x1d7d9}~~}$$ .		:$$\begin{bmatrix}
			1 & 0 & 0 & t_x \\
			0 & 1 & 0 & t_y \\
			0 & 0 & 1 & t_z \\
			0 & 0 & 0 & 1 \\
			\end{bmatrix}$$ .

	==~~= Exponential Form =~~==		== Translation to Origin ==

	A ~~direction vector~~ $$\mathbf v = ~~v_x~~ \mathbf e_1 + ~~v_y~~ \mathbf e_2 + ~~v_z~~ \mathbf e_3$$ is ~~converted~~ to ~~a translation~~ operator ~~$$\mathbf T$$ through the exponential~~		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

	:$$\mathbf T = \~~overline~~{\~~exp_~~\~~unicode~~{~~x27D1~~}~~\left(~~\dfrac{1}{2}\mathbf ~~v \wedge \mathbf e_4\right)~~} = \~~exp_\unicode~~{~~x27C7}~~{\~~left(\dfrac~~{1}{2}\~~overline~~{~~\mathbf v~~} \~~vee~~ \~~mathbf e_~~{~~321~~}~~\right)~~}$$ .		:$$\mathbf T = {-\dfrac{p_{x\vphantom{y}}}{2p_w} \mathbf e_{23} - \dfrac{p_y}{2p_w} \mathbf e_{31} - \dfrac{p_{z\vphantom{y}}}{2p_w} \mathbf e_{12} + \large\unicode{x1d7d9}}$$ .

	== Calculation ==		== Calculation ==

Eric Lengyel: Created page with "NOTOC A ''translation'' is a proper isometry of Euclidean space. The specific kind of motor :$$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$ 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 $$\mathbf t$$. === Trans..."

2023-07-15T05:57:34Z

Created page with "__NOTOC__ A ''translation'' is a proper isometry of Euclidean space. The specific kind of motor :$$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$ 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 $$\mathbf t$$. === Trans..."

New page

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

The specific kind of [[motor]]

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

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 $$\mathbf t$$.

=== 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

:$$\mathbf T = {-\dfrac{p_{x\vphantom{y}}}{2p_w} \mathbf e_{23} - \dfrac{p_y}{2p_w} \mathbf e_{31} - \dfrac{p_{z\vphantom{y}}}{2p_w} \mathbf e_{12} + \large\unicode{x1d7d9}}$$ .

=== Exponential Form ===

A direction vector $$\mathbf v = v_x \mathbf e_1 + v_y \mathbf e_2 + v_z \mathbf e_3$$ is converted to a translation operator $$\mathbf T$$ through the exponential

:$$\mathbf T = \overline{\exp_\unicode{x27D1}\left(\dfrac{1}{2}\mathbf v \wedge \mathbf e_4\right)} = \exp_\unicode{x27C7}{\left(\dfrac{1}{2}\overline{\mathbf v} \vee \mathbf e_{321}\right)}$$ .

== Calculation ==

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

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

$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
| 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;" | [[Line]]

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

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

== See Also ==

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

← Older revision		Revision as of 07:11, 8 August 2024
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	== See Also ==		== See Also ==

	* [[~~Reciprocal~~ translation]]		* [[Complement translation]]
	* [[Rotation]]		* [[Rotation]]
	* [[Reflection]]		* [[Reflection]]
	* [[Inversion]]		* [[Inversion]]
	* [[Transflection]]		* [[Transflection]]

← Older revision		Revision as of 20:19, 5 September 2023
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	== See Also ==		== See Also ==

	* [[~~Dual~~ translation]]		* [[Reciprocal translation]]
	* [[Rotation]]		* [[Rotation]]
	* [[Reflection]]		* [[Reflection]]
	* [[Inversion]]		* [[Inversion]]
	* [[Transflection]]		* [[Transflection]]