Relativistic screw - Revision history

Eric Lengyel at 06:33, 8 February 2025

2025-02-08T06:33:35Z

← Older revision		Revision as of 06:33, 8 February 2025
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	A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by		A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by

	:$$\mathbf Q(\tau) = \exp_\unicode{x27C7}\left[\dfrac{1}{2}\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \~~frac~~{1}{2}\gamma c\tau\,\mathbf e_{321}\right]$$ ,		:$$\mathbf Q(\tau) = \exp_\unicode{x27C7}\left[\dfrac{1}{2}\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \dfrac{1}{2}\gamma c\tau\,\mathbf e_{321}\right]$$ ,

	where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to		where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to

Eric Lengyel at 08:55, 26 December 2024

2024-12-26T08:55:00Z

← Older revision		Revision as of 08:55, 26 December 2024
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	A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by		A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by

	:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,		:$$\mathbf Q(\tau) = \exp_\unicode{x27C7}\left[\dfrac{1}{2}\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \frac{1}{2}\gamma c\tau\,\mathbf e_{321}\right]$$ ,

	where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to		where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to

	:$$\mathbf Q(2\tau) = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) ~~+ {\large~~\~~unicode~~{~~x1D7D9~~}}~~\cos(~~\gamma\tau\~~dot\phi) -~~ [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \~~gamma c~~\tau\,\mathbf e_{321}$$.		:$$\mathbf Q(\tau) = \boldsymbol l\sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) + {\large\unicode{x1D7D9}}\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \left(\dfrac{1}{2}\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0\right)\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \dfrac{1}{2}\gamma\tau\dot\delta \mathbf e_0 \sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) - \dfrac{1}{2}\gamma c\tau\left[\boldsymbol l \sin\left(\dfrac{1}{2}\gamma\tau\dot\phi\right) + {\large\unicode{x1D7D9}}\cos\left(\dfrac{1}{2}\gamma\tau\dot\phi\right)\right] \vee \mathbf e_{321}$$.

			This operator has 12 components and can be written generically as

			:$$\mathbf Q(\tau) = q_x\,\mathbf e_{410} + q_y\,\mathbf e_{420} + q_z\,\mathbf e_{430} + q_w\,\unicode{x1D7D9} + m_x\,\mathbf e_{230} + m_y\,\mathbf e_{310} + m_z\,\mathbf e_{120} + m_w\,\mathbf e_0 + s_x\,\mathbf e_1 + s_y\,\mathbf e_2 + s_z\,\mathbf e_3 + s_w\,e_{321}$$ .

	The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product		The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product
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	The bulk [[norm]] of a unitized relativistic screw operator is given by		The bulk [[norm]] of a unitized relativistic screw operator is given by

	:$$\left\Vert\mathbf Q(2\tau)\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - ~~Q_{mx}~~^2 - ~~Q_{my}~~^2 - ~~Q_{mz}~~^2 - ~~Q_{mw}~~^2}$$ ,		:$$\left\Vert\mathbf Q(\tau)\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \dfrac{1}{2}\sqrt{\gamma^2c^2\tau^2 - m_x^2 - m_y^2 - m_z^2 - m_w^2}$$ ,

	and it corresponds to the distance that the origin has moved after time $$\tau$$. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).		and it corresponds to the distance that the origin has moved after time $$\tau$$. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).

Eric Lengyel at 01:54, 21 December 2024

2024-12-21T01:54:57Z

← Older revision		Revision as of 01:54, 21 December 2024
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	A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by		A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by

	:$$\mathbf Q(\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,		:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,

	where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The ~~half~~ rate of rotation about the line is specified by $$\dot\phi$$, and the ~~half~~ rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to		where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to

	:$$\mathbf Q(\tau) = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi) - [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \gamma c\tau\,\mathbf e_{321}$$.		:$$\mathbf Q(2\tau) = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi) - [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \gamma c\tau\,\mathbf e_{321}$$.

	The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product		The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product
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	The bulk [[norm]] of a unitized relativistic screw operator is given by		The bulk [[norm]] of a unitized relativistic screw operator is given by

	:$$\left\Vert\mathbf Q\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - Q_{mx}^2 - Q_{my}^2 - Q_{mz}^2 - Q_{mw}^2}$$ ,		:$$\left\Vert\mathbf Q(2\tau)\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - Q_{mx}^2 - Q_{my}^2 - Q_{mz}^2 - Q_{mw}^2}$$ ,

	and it corresponds to the distance that the origin is moved. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).		and it corresponds to the distance that the origin has moved after time $$\tau$$. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).

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

	* [[Translation]]		* [[Translation]]

Eric Lengyel at 01:52, 21 December 2024

2024-12-21T01:52:25Z

← Older revision		Revision as of 01:52, 21 December 2024
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	A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by		A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by

	:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,		:$$\mathbf Q(\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,

	where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to		where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The half rate of rotation about the line is specified by $$\dot\phi$$, and the half rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to

	:$$\mathbf Q = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi) - [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \gamma c\tau\,\mathbf e_{321}$$.		:$$\mathbf Q(\tau) = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi) - [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \gamma c\tau\,\mathbf e_{321}$$.

	The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product		The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product

Eric Lengyel at 08:20, 9 December 2024

2024-12-09T08:20:52Z

← Older revision		Revision as of 08:20, 9 December 2024
Line 3:		Line 3:
	:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,		:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,

	where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$.		where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$. The exponential expands to

			:$$\mathbf Q = \boldsymbol l\sin(\gamma\tau\dot\phi) - (\gamma\tau\dot\delta \boldsymbol l^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_0)\cos(\gamma\tau\dot\phi) - \gamma\tau\dot\delta \mathbf e_0 \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi) - [\boldsymbol l \sin(\gamma\tau\dot\phi) + {\large\unicode{x1D7D9}}\cos(\gamma\tau\dot\phi)] \vee \gamma c\tau\,\mathbf e_{321}$$.

	The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product		The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product

Eric Lengyel at 06:36, 9 December 2024

2024-12-09T06:36:47Z

← Older revision		Revision as of 06:36, 9 December 2024
Line 1:		Line 1:
	A relativistic screw $$\mathbf Q$$ about a line $$\boldsymbol l$$ is given by		A relativistic screw $$\mathbf Q$$ about a unitized line $$\boldsymbol l$$ is given by

	:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,		:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,

Eric Lengyel at 10:08, 24 November 2024

2024-11-24T10:08:40Z

← Older revision		Revision as of 10:08, 24 November 2024
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	:$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.		:$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.

			The bulk [[norm]] of a unitized relativistic screw operator is given by

			:$$\left\Vert\mathbf Q\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{c^2t^2 - Q_{mx}^2 - Q_{my}^2 - Q_{mz}^2 - Q_{mw}^2}$$ ,

			and it corresponds to the distance that the origin is moved. This must be real for any motion that's physically possible (i.e., without exceeding the speed of light).

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

	* [[Translation]]		* [[Translation]]

Eric Lengyel at 10:03, 24 November 2024

2024-11-24T10:03:05Z

← Older revision		Revision as of 10:03, 24 November 2024
Line 1:		Line 1:
	A relativistic screw $$\mathbf Q$$ is given by		A relativistic screw $$\mathbf Q$$ about a line $$\boldsymbol l$$ is given by

	:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta\,\mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,		:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta \mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,

	where $$c$$ is the speed of light and $$\gamma = dt/d\tau$$.		where $$c$$ is the speed of light, $$\tau$$ is proper time, and $$\gamma = dt/d\tau$$. The rate of rotation about the line is specified by $$\dot\phi$$, and the rate of translation along the line is specified by $$\dot\delta$$.

	The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product		The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product

Eric Lengyel: Created page with "A relativistic screw $$\mathbf Q$$ is given by :$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta\,\mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ , where $$c$$ is the speed of light and $$\gamma = dt/d\tau$$. The operator $$\mathbf Q$$ transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product :$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}}..."

2024-11-24T09:56:56Z

Created page with "A relativistic screw $$\mathbf Q$$ is given by :$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta\,\mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ , where $$c$$ is the speed of light and $$\gamma = dt/d\tau$$. The operator $$\mathbf Q$$ transforms a position $$\mathbf r$$ (or any other quantity) through the sandwich product :$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}}..."

New page

A relativistic screw $$\mathbf Q$$ is given by

:$$\mathbf Q(2\tau) = \exp_\unicode{x27C7}[\gamma\tau(\dot\delta\,\mathbf e_0 + \dot\phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l - \gamma c\tau\,\mathbf e_{321}]$$ ,

where $$c$$ is the speed of light and $$\gamma = dt/d\tau$$.

The operator $$\mathbf Q$$ transforms a [[position]] $$\mathbf r$$ (or any other quantity) through the sandwich product

:$$\mathbf r' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf r \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$.

== See Also ==

* [[Translation]]