Complement translation - Revision history

Eric Lengyel: Eric Lengyel moved page Reciprocal translation to Complement translation without leaving a redirect

2024-08-08T07:10:32Z

Eric Lengyel moved page Reciprocal translation to Complement translation without leaving a redirect

← Older revision	Revision as of 07:10, 8 August 2024
(No difference)

Eric Lengyel: Undo revision 299 by Eric Lengyel (talk)

2024-02-28T03:00:01Z

Undo revision 299 by Eric Lengyel (talk)

← Older revision		Revision as of 03:00, 28 February 2024
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	== Example ==		== Example ==

	The left image below shows the flow field in the ''x''-''z'' plane for the translation $$\mathbf T = -\frac{1}{2} \mathbf e_{31} + {\large\unicode{x1d7d9}}$$. 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.		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]]

Eric Lengyel: /* Example */

2024-02-28T02:57:17Z

Example

← Older revision		Revision as of 02:57, 28 February 2024
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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 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.		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.

	[[Image:Translation.svg\|480px]]		[[Image:Translation.svg\|480px]]

Eric Lengyel: /* Reciprocal Translation to Horizon */

2023-10-01T00:31:45Z

Reciprocal Translation to Horizon

← Older revision		Revision as of 00:31, 1 October 2023
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	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		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 ==

Eric Lengyel at 00:27, 26 August 2023

2023-08-26T00:27:16Z

← Older revision		Revision as of 00:27, 26 August 2023
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	__NOTOC__		__NOTOC__
	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]]

	:$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$		:$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$
Line 12:		Line 12:
	== 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

	:$$\mathbf T = \dfrac{f_{x\vphantom{y}}}{2f_w} \mathbf e_{41} + \dfrac{f_y}{2f_w} \mathbf e_{42} + \dfrac{f_{z\vphantom{y}}}{2f_w} \mathbf e_{32} + \mathbf 1$$ .		:$$\mathbf T = \dfrac{f_{x\vphantom{y}}}{2f_w} \mathbf e_{41} + \dfrac{f_y}{2f_w} \mathbf e_{42} + \dfrac{f_{z\vphantom{y}}}{2f_w} \mathbf e_{32} + \mathbf 1$$ .
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	* [[Translation]]		* [[Translation]]
	* [[~~Dual~~ rotation]]		* [[Reciprocal rotation]]
	* [[~~Dual~~ reflection]]		* [[Reciprocal reflection]]

Eric Lengyel: Eric Lengyel moved page Dual translation to Reciprocal translation without leaving a redirect

2023-08-26T00:26:01Z

Eric Lengyel moved page Dual translation to Reciprocal translation without leaving a redirect

← Older revision	Revision as of 00:26, 26 August 2023
(No difference)

Eric Lengyel: Created page with "NOTOC A ''dual translation'' is a proper isometry of dual Euclidean space. The specific kind of dual motor :$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$ 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 $..."

2023-07-15T05:59:04Z

Created page with "__NOTOC__ A ''dual translation'' is a proper isometry of dual Euclidean space. The specific kind of dual motor :$$\mathbf T = t_x \mathbf e_{41} + t_y \mathbf e_{42} + t_z \mathbf e_{43} + \mathbf 1$$ 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 $..."

New page

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

The specific kind of [[dual motor]]

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

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 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.

[[Image:Translation.svg|480px]]
[[Image:DualTranslation.svg|480px]]

== Calculation ==

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

{| class="wikitable"
! Type || Dual 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;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf p \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + (2t_xp_x + 2t_yp_y + 2t_zp_z + p_w) \mathbf e_4$$
|-
| style="padding: 12px;" | [[Line]]

$$\begin{split}\mathbf L =\, &v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} \\ +\, &m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf L \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (v_x - 2t_ym_z + 2t_zm_y)\mathbf e_{41} + (v_y - 2t_zm_x + 2t_xm_z)\mathbf e_{42} + (v_z - 2t_xm_y - 2t_ym_x)\mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$
|-
| style="padding: 12px;" | [[Plane]]

$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf T \mathbin{\unicode{x27D1}} \mathbf f \mathbin{\unicode{x27D1}} \mathbf{\tilde T} = (f_x - 2t_xf_w) \mathbf e_{234} + (f_y - 2t_yf_w) \mathbf e_{314} + (f_z - 2t_zf_w) \mathbf e_{124} + f_w \mathbf e_{321}$$
|}

== Dual 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

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

== See Also ==

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