Rotation - Revision history

Eric Lengyel: /* See Also */

2024-08-08T07:11:37Z

Eric Lengyel: /* Calculation */

2024-04-08T05:16:19Z

Calculation

← Older revision		Revision as of 05:16, 8 April 2024
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	== Calculation ==		== Calculation ==

	The exact rotation calculations for points, lines, and planes are shown in the following table.		The exact rotation calculations for points, lines, and planes are shown in the following table, where $$\mathbf v = (R_{vx}, R_{vy}, R_{vz})$$ and $$\mathbf m = (R_{mx}, R_{my}, R_{mz})$$.

	{\| class="wikitable"		{\| class="wikitable"
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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;" \| $$\~~begin{split}~~\mathbf R \~~mathbin{~~\~~unicode~~{~~x27C7}~~} \mathbf ~~p \mathbin{\unicode{x27C7}} \smash{~~\mathbf{~~\underset{\Large\unicode{x7E}}{R}}~~} =\~~, &\left[(1 - 2R_~~{vy}~~^2 - 2R_{vz}^2)p_x~~ + 2(~~R_{vx}R_{vy} - R_{vz}~~R_{vw}~~)p_y + 2(R_{vz}R_{vx} + R_{vy}R_{vw})p_z + 2(R_{vy}R_{mz} - R_{vz}R_{my} + R_{vw}R_{mx})p_w\right]~~\mathbf ~~e_1 \\~~ +\~~, &~~\~~left[(1 - 2R_{vz}^2 - 2R_{vx}^2)p_y + 2(R_{vy}R_{vz} - R_{vx}R_{vw})p_z + 2(R_{vx}R_{vy} + R_{vz}R_{vw})p_x + 2(R_{vz}R_{mx} - R_{vx}R_{mz} + R_{vw}R_{my})p_w\right]~~\mathbf ~~e_2 \\ +\, &\left[(1 - 2R_{vx}^2 - 2R_{vy}^2)p_z + 2(R_{vz}R_{vx} - R_{vy}R_{vw})p_x + 2(R_{vy}R_{vz} + R_{vx}R_{vw})p_y + 2(R_{vx}R_{my} - R_{vy}R_{mx} + R_{vw}R_{mz}~~)p_w~~\right]\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}~~$$		\| style="padding: 12px;" \| $$\mathbf a = \mathbf v \times \mathbf p_{xyz} + p_w\mathbf m$$

			$$\mathbf p'_{xyz} = \mathbf p_{xyz} + 2(R_{vw}\mathbf a + \mathbf v \times \mathbf a)$$

			$$p'_w = p_w$$
	\|-		\|-
	\| 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;" \| $$\~~begin{split}~~\mathbf ~~R \mathbin{~~\~~unicode{x27C7}}~~ \boldsymbol ~~l \mathbin~~{\~~unicode{x27C7}~~} \~~smash{~~\mathbf{\~~underset{~~\~~Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2R_{vy}^2 - 2R_{vz}^2)~~l_{~~vx} + 2(R_{vx}R_{vy} - R_{vz}R_{vw})l_{vy} + 2(R_{vz}R_{vx} + R_{vy}R_{vw})l_{vz}\right]~~\mathbf ~~e_{41~~} \\ +\~~, &~~\~~left[(1 - 2R_{vz}^2 - 2R_{vx}^2)~~l_{vy} ~~+ 2(R_~~{vy}~~R_{vz} - R_{vx}R_{vw})~~l_{vz} + 2(~~R_{vx}R_{vy} + R_{vz}~~R_{vw~~})l_{vx~~}\~~right]~~\mathbf ~~e_{42}~~ \\ +\~~, &\left[(1 - 2R_~~{~~vx}^2 - 2R_{vy}^2)l_{vz} + 2(R_{vz}R_{vx} - R_{vy}R_{vw})l_{vx} + 2(R_{vy}R_{vz} + R_{vx}R_{vw})l_{vy}\right]~~\mathbf ~~e_{43~~} \~~\ +\, &\left[-4(R_{vy}R_{my} + R_{vz}R_{mz})~~l_{~~vx} + 2(R_{vy}R_{mx} + R_{vx}R_{my} - R_{vw}R_{mz})l_{vy} + 2(R_{vz}R_{mx} + R_{vx}R_{mz} + R_{vw}R_{my})l_{vz} + (1 - 2R_{vy}^2 - 2R_{vz}^2)l_{mx~~} + 2(~~R_{vx}R_{vy} - R_{vz}~~R_{vw}~~)l_{my} + 2~~(~~R_{vz}R_{vx} + R_{vy}R_{vw})l_{mz}\right]~~\mathbf ~~e_{23} \\~~ +\~~, &\left[-4(R_{vz}R_{mz} + R_{vx}R_{mx}~~)~~l_{vy}~~ + 2(R_{vz}R_{my} + R_{vy}R_{mz} - R_{vw}R_{mx})l_{vz} + 2(R_{vx}R_{my} + R_{vy}R_{mx} + R_{vw}R_{mz})l_{vx} + (1 - 2R_{vz}^2 - 2R_{vx}^2)l_{my} + 2(R_{vy}R_{vz} - R_{vx}R_{vw})l_{mz} + 2(R_{vx}R_{vy} + R_{vz}R_{vw})l_{mx}\right]\mathbf ~~e_{31}~~ \\ +\~~, &\left[-4(R_{vx}R_{mx} + R_{vy}R_{my}~~)~~l_{vz}~~ + 2(R_{vx}R_{mz} + R_{vz}R_{mx} - R_{vw}R_{my})l_{vx} + 2(R_{vy}R_{mz} + R_{vz}R_{my} + R_{vw}R_{mx})l_{vy} + (1 - 2R_{vx}^2 - 2R_{vy}^2)l_{mz} + 2(R_{vz}R_{vx} - R_{vy}R_{vw})l_{mx} + 2(R_{vy}R_{vz} + R_{vx}R_{vw})l_{my}\~~right]~~\mathbf ~~e_{12}\end{split}~~$$		\| style="padding: 12px;" \| $$\mathbf a = \mathbf v \times \boldsymbol l_{\mathbf v}$$

			$$\mathbf b = \mathbf v \times \boldsymbol l_{\mathbf m}$$

			$$\mathbf c = \mathbf m \times \boldsymbol l_{\mathbf v}$$

			$$\boldsymbol l'_{\mathbf v} = \boldsymbol l_{\mathbf v} + 2(R_{vw}\mathbf a + \mathbf v \times \mathbf a)$$

			$$\boldsymbol l'_{\mathbf m} = \boldsymbol l_{\mathbf m} + 2(R_{vw}(\mathbf b + \mathbf c) + \mathbf v \times (\mathbf b + \mathbf c) + \mathbf m \times \mathbf a)$$
	\|-		\|-
	\| 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;" \| $$\~~begin{split}~~\mathbf R \~~mathbin{~~\~~unicode~~{~~x27C7}~~} \mathbf g ~~\mathbin~~{~~\unicode{x27C7~~}~~} \smash{~~\mathbf{~~\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2R_{vy}^2 - 2R_{vz~~}~~^2)g_x~~ + 2(~~R_{vx}R_{vy} - R_{vz}~~R_{vw}~~)g_y + 2(R_{vz}R_{vx}~~ + ~~R_{vy}R_{vw})g_z\right]~~\mathbf ~~e_{423}~~ \\ ~~+\, &\left[(1 - 2R_{vz}^2 - 2R_{vx}^2~~)~~g_y~~ + 2(~~R_{vy}R_{vz} - R_{vx}R_{vw})g_z + 2(R_{vx}R_{vy} + R_{vz}R_{vw})g_x~~\~~right]~~\mathbf e_{~~431~~} \\ ~~+\, &\left[(1~~ - ~~2R_{vx}^2 - 2R_{vy}^2)g_z + 2(R_{vz}R_{vx} - R_{vy}~~R_{vw}~~)g_x + 2~~(~~R_{vy}R_{vz} + R_{vx}R_{vw})g_y\right]~~\mathbf ~~e_{412}~~ \\ ~~+\, &\left[2(R_{vy}R_{mz} - R_{vz}R_~~{~~my} - R_{vw}R_{mx~~})~~g_x + 2(R_{vz}R_{mx} - R_{vx}R_{mz} - R_{vw}R_{my})g_y + 2(R_{vx}R_{my} - R_{vy}R_{mx} - R_{vw}R_{mz})g_z + g_w\right~~]~~\mathbf e_{321}\end{split}~~$$		\| style="padding: 12px;" \| $$\mathbf a = \mathbf v \times \mathbf g_{xyz}$$

			$$\mathbf g'_{xyz} = \mathbf g_{xyz} + 2(R_{vw}\mathbf a + \mathbf v \times \mathbf a)$$

			$$g'_w = g_w + 2[(\mathbf m \times \mathbf g_{xyz}) \cdot \mathbf v - R_{vw}(\mathbf m \cdot \mathbf g_{xyz})]$$
	\|}		\|}

Eric Lengyel at 23:37, 7 February 2024

2024-02-07T23:37:56Z

← Older revision		Revision as of 23:37, 7 February 2024
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	:$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ ,		:$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ ,

	performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathbf R$$ differs from a general [[motor]] only in that it is always the case that $$R_{mw} = 0$$. The line $$\boldsymbol l$$ and its ~~weight~~ [[~~complement~~]] $$~~\underline{~~\boldsymbol ~~l_\smash{~~\unicode{~~x25CB}}~~}$$ are invariant under this operation. The line $$~~\underline{~~\boldsymbol ~~l_\smash{~~\unicode{~~x25CB}}~~}$$ lies in the [[horizon]] in directions perpendicular to the direction of $$\boldsymbol l$$.		performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathbf R$$ differs from a general [[motor]] only in that it is always the case that $$R_{mw} = 0$$. The line $$\boldsymbol l$$ and its [[weight dual]] $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606}$$ are invariant under this operation. The line $$\boldsymbol l^\unicode["segoe ui symbol"]{x2606}$$ lies in the [[horizon]] in directions perpendicular to the direction of $$\boldsymbol l$$.

	== Calculation ==		== Calculation ==

Eric Lengyel: /* See Also */

2023-09-05T20:19:45Z

Eric Lengyel: Created page with "A ''rotation'' is a proper isometry of Euclidean space. For a unitized line $$\boldsymbol l$$, the specific kind of motor :$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ , performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathb..."

2023-07-15T05:57:00Z

Created page with "A ''rotation'' is a proper isometry of Euclidean space. For a unitized line $$\boldsymbol l$$, the specific kind of motor :$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ , performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathb..."

New page

A ''rotation'' is a proper isometry of Euclidean space.

For a [[Unitization | unitized]] [[line]] $$\boldsymbol l$$, the specific kind of [[motor]]

:$$\mathbf R = \boldsymbol l\sin\phi + {\large\unicode{x1d7d9}}\cos\phi$$ ,

performs a rotation by twice the angle $$\phi$$ about the line $$\boldsymbol l$$ with the sandwich product $$\mathbf R \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$. The operator $$\mathbf R$$ differs from a general [[motor]] only in that it is always the case that $$R_{mw} = 0$$. The line $$\boldsymbol l$$ and its weight [[complement]] $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}}$$ are invariant under this operation. The line $$\underline{\boldsymbol l_\smash{\unicode{x25CB}}}$$ lies in the [[horizon]] in directions perpendicular to the direction of $$\boldsymbol l$$.

== Calculation ==

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

{| class="wikitable"
! Type || Transformation
|-
| 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;" | $$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2R_{vy}^2 - 2R_{vz}^2)p_x + 2(R_{vx}R_{vy} - R_{vz}R_{vw})p_y + 2(R_{vz}R_{vx} + R_{vy}R_{vw})p_z + 2(R_{vy}R_{mz} - R_{vz}R_{my} + R_{vw}R_{mx})p_w\right]\mathbf e_1 \\ +\, &\left[(1 - 2R_{vz}^2 - 2R_{vx}^2)p_y + 2(R_{vy}R_{vz} - R_{vx}R_{vw})p_z + 2(R_{vx}R_{vy} + R_{vz}R_{vw})p_x + 2(R_{vz}R_{mx} - R_{vx}R_{mz} + R_{vw}R_{my})p_w\right]\mathbf e_2 \\ +\, &\left[(1 - 2R_{vx}^2 - 2R_{vy}^2)p_z + 2(R_{vz}R_{vx} - R_{vy}R_{vw})p_x + 2(R_{vy}R_{vz} + R_{vx}R_{vw})p_y + 2(R_{vx}R_{my} - R_{vy}R_{mx} + R_{vw}R_{mz})p_w\right]\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}$$
|-
| 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;" | $$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2R_{vy}^2 - 2R_{vz}^2)l_{vx} + 2(R_{vx}R_{vy} - R_{vz}R_{vw})l_{vy} + 2(R_{vz}R_{vx} + R_{vy}R_{vw})l_{vz}\right]\mathbf e_{41} \\ +\, &\left[(1 - 2R_{vz}^2 - 2R_{vx}^2)l_{vy} + 2(R_{vy}R_{vz} - R_{vx}R_{vw})l_{vz} + 2(R_{vx}R_{vy} + R_{vz}R_{vw})l_{vx}\right]\mathbf e_{42} \\ +\, &\left[(1 - 2R_{vx}^2 - 2R_{vy}^2)l_{vz} + 2(R_{vz}R_{vx} - R_{vy}R_{vw})l_{vx} + 2(R_{vy}R_{vz} + R_{vx}R_{vw})l_{vy}\right]\mathbf e_{43} \\ +\, &\left[-4(R_{vy}R_{my} + R_{vz}R_{mz})l_{vx} + 2(R_{vy}R_{mx} + R_{vx}R_{my} - R_{vw}R_{mz})l_{vy} + 2(R_{vz}R_{mx} + R_{vx}R_{mz} + R_{vw}R_{my})l_{vz} + (1 - 2R_{vy}^2 - 2R_{vz}^2)l_{mx} + 2(R_{vx}R_{vy} - R_{vz}R_{vw})l_{my} + 2(R_{vz}R_{vx} + R_{vy}R_{vw})l_{mz}\right]\mathbf e_{23} \\ +\, &\left[-4(R_{vz}R_{mz} + R_{vx}R_{mx})l_{vy} + 2(R_{vz}R_{my} + R_{vy}R_{mz} - R_{vw}R_{mx})l_{vz} + 2(R_{vx}R_{my} + R_{vy}R_{mx} + R_{vw}R_{mz})l_{vx} + (1 - 2R_{vz}^2 - 2R_{vx}^2)l_{my} + 2(R_{vy}R_{vz} - R_{vx}R_{vw})l_{mz} + 2(R_{vx}R_{vy} + R_{vz}R_{vw})l_{mx}\right]\mathbf e_{31} \\ +\, &\left[-4(R_{vx}R_{mx} + R_{vy}R_{my})l_{vz} + 2(R_{vx}R_{mz} + R_{vz}R_{mx} - R_{vw}R_{my})l_{vx} + 2(R_{vy}R_{mz} + R_{vz}R_{my} + R_{vw}R_{mx})l_{vy} + (1 - 2R_{vx}^2 - 2R_{vy}^2)l_{mz} + 2(R_{vz}R_{vx} - R_{vy}R_{vw})l_{mx} + 2(R_{vy}R_{vz} + R_{vx}R_{vw})l_{my}\right]\mathbf e_{12}\end{split}$$
|-
| 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;" | $$\begin{split}\mathbf R \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{R}}} =\, &\left[(1 - 2R_{vy}^2 - 2R_{vz}^2)g_x + 2(R_{vx}R_{vy} - R_{vz}R_{vw})g_y + 2(R_{vz}R_{vx} + R_{vy}R_{vw})g_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2R_{vz}^2 - 2R_{vx}^2)g_y + 2(R_{vy}R_{vz} - R_{vx}R_{vw})g_z + 2(R_{vx}R_{vy} + R_{vz}R_{vw})g_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2R_{vx}^2 - 2R_{vy}^2)g_z + 2(R_{vz}R_{vx} - R_{vy}R_{vw})g_x + 2(R_{vy}R_{vz} + R_{vx}R_{vw})g_y\right]\mathbf e_{412} \\ +\, &\left[2(R_{vy}R_{mz} - R_{vz}R_{my} - R_{vw}R_{mx})g_x + 2(R_{vz}R_{mx} - R_{vx}R_{mz} - R_{vw}R_{my})g_y + 2(R_{vx}R_{my} - R_{vy}R_{mx} - R_{vw}R_{mz})g_z + g_w\right]\mathbf e_{321}\end{split}$$
|}

== See Also ==

* [[Dual rotation]]
* [[Translation]]
* [[Reflection]]
* [[Inversion]]
* [[Transflection]]

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

	* [[~~Reciprocal~~ rotation]]		* [[Complement rotation]]
	* [[Translation]]		* [[Translation]]
	* [[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~~ rotation]]		* [[Reciprocal rotation]]
	* [[Translation]]		* [[Translation]]
	* [[Reflection]]		* [[Reflection]]
	* [[Inversion]]		* [[Inversion]]
	* [[Transflection]]		* [[Transflection]]