Motor - Revision history

Eric Lengyel: /* Square Root */

2024-07-08T01:19:38Z

Square Root

Eric Lengyel: /* Norm */

2024-07-08T01:18:10Z

Norm

← Older revision		Revision as of 01:18, 8 July 2024
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	The [[bulk norm]] of a motor $$\mathbf Q$$ is given by		The [[bulk norm]] of a motor $$\mathbf Q$$ is given by

	:$$\left\Vert\mathbf Q\right\Vert_\unicode{x25CF} = \sqrt{\mathbf Q \mathbin{\unicode{~~x25CF~~}} \mathbf{\tilde Q}} = \sqrt{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}$$ ,		:$$\left\Vert\mathbf Q\right\Vert_\unicode["segoe ui symbol"]{x25CF} = \sqrt{\mathbf Q \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf{\tilde Q}} = \sqrt{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}$$ ,

	and its [[weight norm]] is given by		and its [[weight norm]] is given by

	:$$\left\Vert\mathbf Q\right\Vert_\unicode{x25CB} = \sqrt{\mathbf Q \mathbin{\unicode{~~x25CB~~}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}\vphantom{\mathbf{\tilde Q}}} = {\large\unicode{x1D7D9}}\sqrt{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}$$ .		:$$\left\Vert\mathbf Q\right\Vert_\unicode["segoe ui symbol"]{x25CB} = \sqrt{\mathbf Q \mathbin{\unicode["segoe ui symbol"]{x2218}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}\vphantom{\mathbf{\tilde Q}}} = {\large\unicode{x1D7D9}}\sqrt{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}$$ .

	The [[geometric norm]] of a motor $$\mathbf Q$$ is thus		The [[geometric norm]] of a motor $$\mathbf Q$$ is thus

Eric Lengyel at 23:51, 13 April 2024

2024-04-13T23:51:11Z

← Older revision		Revision as of 23:51, 13 April 2024
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	$$g'_w = g_w + 2[(\mathbf m \times \mathbf g_{xyz} + Q_{mw}\mathbf g_{xyz}) \cdot \mathbf v - Q_{vw}(\mathbf m \cdot \mathbf g_{xyz})]$$		$$g'_w = g_w + 2[(\mathbf m \times \mathbf g_{xyz} + Q_{mw}\mathbf g_{xyz}) \cdot \mathbf v - Q_{vw}(\mathbf m \cdot \mathbf g_{xyz})]$$
	\|}		\|}

			== In the Book ==

			* General motion operators (motors) are discussed in Section 3.6.

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

Eric Lengyel: /* Motor Transformations */

2024-04-08T05:01:32Z

Motor Transformations

← Older revision		Revision as of 05:01, 8 April 2024
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	== Motor Transformations ==		== Motor Transformations ==

	[[Point \| Points]], [[Line \| lines]], and [[Plane \| planes]] are transformed by a [[unitized]] motor $$\mathbf Q$$ as shown in the following table.		[[Point \| Points]], [[Line \| lines]], and [[Plane \| planes]] are transformed by a [[unitized]] motor $$\mathbf Q$$ as shown in the following table, where $$\mathbf v = (Q_{vx}, Q_{vy}, Q_{vz})$$ and $$\mathbf m = (Q_{mx}, Q_{my}, Q_{mz})$$.

	{\| class="wikitable"		{\| class="wikitable"
Line 251:		Line 251:

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

			$$\mathbf p'_{xyz} = \mathbf p_{xyz} + 2(Q_{vw}\mathbf a + \mathbf v \times \mathbf a - Q_{mw}p_w\mathbf v)$$

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

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

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

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

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

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

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

Eric Lengyel at 01:01, 9 February 2024

2024-02-09T01:01:44Z

← Older revision		Revision as of 01:01, 9 February 2024
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	:$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ .		:$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ .

	To ~~possess~~ the [[geometric ~~property~~]], the components of $$\mathbf Q$$ must satisfy the equation		To satisfy the [[geometric constraint]], the components of $$\mathbf Q$$ must satisfy the equation

	:$$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$ .		:$$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$ .

Eric Lengyel at 05:36, 6 August 2023

2023-08-06T05:36:24Z

← Older revision		Revision as of 05:36, 6 August 2023
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	[[Image:proper_isom.svg\|400px\|thumb\|right\|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol L$$ and a displacement along the same line.]]		[[Image:proper_isom.svg\|400px\|thumb\|right\|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol l$$ and a displacement along the same line.]]
	A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of [[rotations]] and [[translations]]. The name motor is a portmanteau of ''motion operator'' or ''moment vector''. Motors are equivalent to the set of ''dual quaternions'' used in conventional theories, and the functionality is properly generalized in rigid geometric algebra. Motors cannot perform improper isometries that include an odd number of [[reflections]]; those are instead performed by operators called [[flectors]].		A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of [[rotations]] and [[translations]]. The name motor is a portmanteau of ''motion operator'' or ''moment vector''. Motors are equivalent to the set of ''dual quaternions'' used in conventional theories, and the functionality is properly generalized in rigid geometric algebra. Motors cannot perform improper isometries that include an odd number of [[reflections]]; those are instead performed by operators called [[flectors]].

Eric Lengyel at 05:36, 6 August 2023

2023-08-06T05:36:16Z

← Older revision		Revision as of 05:36, 6 August 2023
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	[[Image:proper_isom.svg\|400px\|thumb\|right\|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\~~mathbf~~ L$$ and a displacement along the same line.]]		[[Image:proper_isom.svg\|400px\|thumb\|right\|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\boldsymbol L$$ and a displacement along the same line.]]
	A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of [[rotations]] and [[translations]]. The name motor is a portmanteau of ''motion operator'' or ''moment vector''. Motors are equivalent to the set of ''dual quaternions'' used in conventional theories, and the functionality is properly generalized in rigid geometric algebra. Motors cannot perform improper isometries that include an odd number of [[reflections]]; those are instead performed by operators called [[flectors]].		A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of [[rotations]] and [[translations]]. The name motor is a portmanteau of ''motion operator'' or ''moment vector''. Motors are equivalent to the set of ''dual quaternions'' used in conventional theories, and the functionality is properly generalized in rigid geometric algebra. Motors cannot perform improper isometries that include an odd number of [[reflections]]; those are instead performed by operators called [[flectors]].

Eric Lengyel at 05:26, 6 August 2023

2023-08-06T05:26:32Z

← Older revision		Revision as of 05:26, 6 August 2023
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	:$$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$ .		:$$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$ .

	Motors are capable of representing any general screw motion consisting of a rotation about a line combined with a displacement along the same line. By ~~Chaslesâ€™~~ theorem, all proper rigid motions in 3D space are screw motions.		Motors are capable of representing any general screw motion consisting of a rotation about a line combined with a displacement along the same line. By Chasles' theorem, all proper rigid motions in 3D space are screw motions.

	An element $$\mathbf x$$ is transformed by a motor $$\mathbf Q$$ through the operation $$\mathbf x' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$, where $$\unicode{x27C7}$$ is the [[geometric antiproduct]].		An element $$\mathbf x$$ is transformed by a motor $$\mathbf Q$$ through the operation $$\mathbf x' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$, where $$\unicode{x27C7}$$ is the [[geometric antiproduct]].

Eric Lengyel: Created page with "'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\mathbf L$$ and a displacement along the same line. A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of rotations and translations. The name motor is a portmanteau of ''motion operator'' or ''moment vector..."

2023-07-15T05:27:08Z

Created page with "400px|thumb|right|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\mathbf L$$ and a displacement along the same line. A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of rotations and translations. The name motor is a portmanteau of ''motion operator'' or ''moment vector..."

New page

[[Image:proper_isom.svg|400px|thumb|right|'''Figure 1.''' A motor represents a proper Euclidean isometry, which can always be regarded as a rotation about a line $$\mathbf L$$ and a displacement along the same line.]]
A ''motor'' is an operator that performs a proper isometry in Euclidean space. Such isometries encompass all possible combinations of any number of [[rotations]] and [[translations]]. The name motor is a portmanteau of ''motion operator'' or ''moment vector''. Motors are equivalent to the set of ''dual quaternions'' used in conventional theories, and the functionality is properly generalized in rigid geometric algebra. Motors cannot perform improper isometries that include an odd number of [[reflections]]; those are instead performed by operators called [[flectors]].

In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, a motor $$\mathbf Q$$ has the general form

:$$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$ .

To possess the [[geometric property]], the components of $$\mathbf Q$$ must satisfy the equation

:$$Q_{vx} Q_{mx} + Q_{vy} Q_{my} + Q_{vz} Q_{mz} + Q_{vw} Q_{mw} = 0$$ .

Motors are capable of representing any general screw motion consisting of a rotation about a line combined with a displacement along the same line. By Chaslesâ€™ theorem, all proper rigid motions in 3D space are screw motions.

An element $$\mathbf x$$ is transformed by a motor $$\mathbf Q$$ through the operation $$\mathbf x' = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf x \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}$$, where $$\unicode{x27C7}$$ is the [[geometric antiproduct]].

The set of all motors, sometimes denoted by $$\unicode{x1D544}$$, forms a subgroup of index 2 in a geometric algebra. Its coset is the set of [[flectors]].

== Simple Motors ==

If $$Q_{mw} = 0$$ (i.e., the scalar part $$Q_{\mathbf 1}$$ is zero), then the motor $$\mathbf Q$$ is called a ''simple motor''. Every simple motor represents either a pure [[translation]] or a pure [[rotation]] about a line without any displacement along that line.

In the case of a pure translation, the motor $$\mathbf T$$ is given by

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

and this performs a translation by twice the displacement vector $$(t_x, t_y, t_z)$$.

In the case of a pure rotation, the motor $$\mathbf R$$ is given by

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

and this performs a rotation by twice the angle $$\phi$$ about the [[line]] $$\boldsymbol l$$.

== Motors Built from Geometry ==

Motors having specific geometric constructions can be built from [[Point | points]], [[Line | lines]], and [[Plane | planes]] as shown in the following table.

{| class="wikitable"
! Motor|| Description
|-
| style="padding: 12px;" | $$\begin{split}\mathbf h \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{g}}} =\, &(g_yh_z - g_zh_y)\mathbf e_{41} + (g_zh_x - g_xh_z)\mathbf e_{42} + (g_xh_y - g_yh_x)\mathbf e_{43} \\ +\, &(h_xg_w - g_xh_w)\mathbf e_{23} + (h_yg_w - g_yh_w)\mathbf e_{31} + (h_zg_w - g_zh_w)\mathbf e_{12} \\ +\, &(g_xh_x + g_yh_y + g_zh_z)\smash{\large\unicode{x1d7d9}}\end{split}$$
| style="padding: 12px;" | Rotation about the line where planes $$\mathbf g$$ and $$\mathbf h$$ intersect by twice the angle between them in the direction from $$\mathbf g$$ to $$\mathbf h$$.

$$\mathbf g = g_x \mathbf e_{234} + g_y \mathbf e_{314} + g_z \mathbf e_{124} + g_w \mathbf e_{321}$$

$$\mathbf h = h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$
|-
| style="padding: 12px;" | $$\begin{split}\boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{k}}}
=\, &(l_{vz} k_{vy} - l_{vy} k_{vz})\mathbf e_{41} + (l_{vx} k_{vz} - l_{vz} k_{vx})\mathbf e_{42} + (l_{vy} k_{vx} - l_{vx} k_{vy})\mathbf e_{43} \\
+\, &(l_{vz} k_{my} - l_{vy} k_{mz} + l_{mz} k_{vy} - l_{my} k_{vz})\mathbf e_{23} + (l_{vx} k_{mz} - l_{vz} k_{mx} + l_{mx} k_{vz} - l_{mz} k_{vx})\mathbf e_{31} + (l_{vy} k_{mx} - l_{vx} k_{my} + l_{my} k_{vx} - l_{mx} k_{vy})\mathbf e_{12} \\
+\, &(l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} + l_{mx} k_{vx} + l_{my} k_{vy} + l_{mz} k_{vz}) \\
+\, &(l_{vx} k_{vx} + l_{vy} k_{vy} + l_{vz} k_{vz})\smash{\large\unicode{x1d7d9}}\end{split}$$
| style="padding: 12px;" | Rotation about the line containing the closest points on lines $$\mathbf k$$ and $$\boldsymbol l$$ by twice the angle between the directions $$(l_{vx}, l_{vy}, l_{vz})$$ and $$(k_{vx}, k_{vy}, k_{vz})$$, and translation by twice the distance between the lines in the direction from $$\mathbf k$$ to $$\boldsymbol l$$.

$$\mathbf k = k_{vx} \mathbf e_{41} + k_{vy} \mathbf e_{42} + k_{vz} \mathbf e_{43} + k_{mx} \mathbf e_{23} + k_{my} \mathbf e_{31} + k_{mz} \mathbf e_{12}$$

$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
|-
| style="padding: 12px;" | $$\mathbf q \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{p}}} = (q_xp_w - p_xq_w)\mathbf e_{23} + (q_yp_w - p_yq_w)\mathbf e_{31} + (q_zp_w - p_zq_w)\mathbf e_{12} + (p_wq_w)\smash{\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | Translation by twice the distance between points $$\mathbf p$$ and $$\mathbf q$$ in the direction from $$\mathbf p$$ to $$\mathbf q$$.

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

$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
|}

== Norm ==

The [[bulk norm]] of a motor $$\mathbf Q$$ is given by

:$$\left\Vert\mathbf Q\right\Vert_\unicode{x25CF} = \sqrt{\mathbf Q \mathbin{\unicode{x25CF}} \mathbf{\tilde Q}} = \sqrt{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}$$ ,

and its [[weight norm]] is given by

:$$\left\Vert\mathbf Q\right\Vert_\unicode{x25CB} = \sqrt{\mathbf Q \mathbin{\unicode{x25CB}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}}\vphantom{\mathbf{\tilde Q}}} = {\large\unicode{x1D7D9}}\sqrt{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}$$ .

The [[geometric norm]] of a motor $$\mathbf Q$$ is thus

:$$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{\dfrac{Q_{mx}^2 + Q_{my}^2 + Q_{mz}^2 + Q_{mw}^2}{Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2}}$$ ,

and this is equal to half the distance that the origin is moved by the operator.

A motor is [[unitized]] when $$Q_{vx}^2 + Q_{vy}^2 + Q_{vz}^2 + Q_{vw}^2 = 1$$.

== Exponential Form ==

A motor $$\mathbf Q$$ can be expressed as the exponential of a [[unitized]] [[line]] $$\boldsymbol l$$ multiplied by $$d + \phi{\large\unicode{x1D7D9}}$$, where $$\phi$$ is half the angle of rotation about the line $$\boldsymbol l$$, and $$d$$ is half the displacement distance along the line $$\boldsymbol l$$. The exponential form can be written as

:$$\mathbf Q = \exp_\unicode{x27C7}((d + \phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l) = \cos_\unicode{x27C7}(d + \phi{\large\unicode{x1D7D9}}) + \sin_\unicode{x27C7}(d + \phi{\large\unicode{x1D7D9}}) \mathbin{\unicode{x27C7}} \boldsymbol l$$ .

This expands to

:$$\mathbf Q = \boldsymbol l\sin\phi + (d \mathbin{\unicode{x27C7}} \boldsymbol l)\cos\phi - d\sin\phi + {\large\unicode{x1D7D9}}\cos\phi$$ ,

and replacing the [[line]] $$\boldsymbol l$$ with its components gives us

:$$\mathbf Q = (l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12})\sin\phi + d(l_{vx} \mathbf e_{23} + l_{vy} \mathbf e_{31} + l_{vz} \mathbf e_{12})\cos\phi - d\sin\phi + {\large\unicode{x1D7D9}}\cos\phi$$ .

A motor in exponential form is always [[unitized]], and its [[geometric norm]] can be written as

:$$\widehat{\left\Vert\mathbf Q\right\Vert} = \sqrt{d^2 + (l_{mx}^2 + l_{my}^2 + l_{mz}^2)\sin^2\phi}$$ .

The quantity $$d + \phi{\large\unicode{x1D7D9}}$$, as a homogeneous magnitude, is the ''pitch'' of the screw transformation performed by the motor, which is the amount of translation along the screw axis per radian of rotation.

== Logarithm ==

Given an arbitrary [[unitized]] motor $$\mathbf Q$$ with $$|Q_{vw}| < 1$$, the values of $$\boldsymbol l$$, $$d$$, and $$\phi$$ can be determined from the components of $$\mathbf Q$$, essentially taking the logarithm of a motor. First, the [[scalar]] and [[antiscalar]] terms satisfy the equations

:$$Q_{mw} = -d\sin\phi$$
:$$Q_{vw} = \cos\phi$$ .

Assuming that $$\phi > 0$$, we define $$s = \sin\phi = \sqrt{1 - Q_{vw}^2}$$. Then,

:$$d = -\dfrac{Q_{mw}}{s}$$
:$$\phi = \tan^{-1}\left(\dfrac{s}{Q_{vw}}\right)$$ .

If $$Q_{vw} = 0$$, then we assign $$\phi = \pi/2$$. If $$Q_{vw} < 0$$, then we can add $$\pi$$ to $$\phi$$ to make the angle positive, but this is not required.

The components of $$\boldsymbol l$$ are then given by

:$$(l_{vx}, l_{vy}, l_{vz}) = \dfrac{1}{s}(Q_{vx}, Q_{vy}, Q_{vz})$$

:$$(l_{mx}, l_{my}, l_{mz}) = \dfrac{1}{s}\left[(Q_{mx}, Q_{my}, Q_{mz}) - \dfrac{dQ_{vw}}{s}(Q_{vx}, Q_{vy}, Q_{vz})\right] = \dfrac{1}{s}\left[(Q_{mx}, Q_{my}, Q_{mz}) + \dfrac{Q_{vw} Q_{mw}}{s^2}(Q_{vx}, Q_{vy}, Q_{vz})\right]$$ .

In the special case that $$Q_{vw} = \pm 1$$, the motor must be a pure [[translation]] with $$\phi = 0$$ and $$(Q_{vx}, Q_{vy}, Q_{vz}) = (0, 0, 0)$$.

== Square Root ==

The square root of a [[quaternion]] can be calculated by summing with the identity and renormalizing. If we attempt to do the same thing with a motor $$\mathbf Q$$, we find that it works only under certain conditions. However, such a calculation does give us a starting point from which we can make a correction for the general case.

Using the exponential form of $$\mathbf Q$$, we first examine the [[weight norm]] of $$\mathbf Q + {\large\unicode{x1D7D9}}$$ and find that

:$$\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB} = \sqrt{\sin^2\phi + (\cos\phi + 1)^2} = \sqrt{\vphantom{\sin^2\phi}2 + 2Q_\smash{\large\unicode{x1D7D9}}}$$ ,

where $$Q_{\large\unicode{x1D7D9}} = \cos\phi$$. Applying the trigonometric identity $$\cos^2(\phi/2) = (1 + \cos\phi)/2$$, we can rewrite this as

:$$\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB} = 2\cos(\phi/2)$$ .

Applying several more trigonometric identities, we now observe

:$$\dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{2\cos(\phi/2)} = \boldsymbol l\sin(\phi/2) + \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\cos(\phi/2) - \dfrac{d}{2}\sin(\phi/2) + {\large\unicode{x1D7D9}}\cos(\phi/2) - \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\sin(\phi/2)\tan(\phi/2) - \dfrac{d}{2}\sin(\phi/2)$$ .

The first four terms are exactly the square root of $$\mathbf Q$$ because the distance $$d$$ and angle $$\phi$$ have both been halved. But there are two additional terms that we need to eliminate. It just so happens that

:$$\dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB}} \mathbin{\unicode{x27C7}} \dfrac{d}{2}\tan(\phi/2) = \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\sin(\phi/2)\tan(\phi/2) + \dfrac{d}{2}\sin(\phi/2)$$ ,

which means that

:$$\sqrt{\mathbf Q} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\sqrt{2 + 2Q_\smash{\large\unicode{x1D7D9}}}} \mathbin{\unicode{x27C7}} \left({\large\unicode{x1D7D9}} + \dfrac{d}{2}\tan(\phi/2)\right)$$ .

The scalar component of $$\mathbf Q$$ can be rewritten as $$Q_{\mathbf 1} = -d\sin\phi = -2d\sin(\phi/2)\cos(\phi/2) $$. Dividing this by $$2 + 2Q_{\large\unicode{x1D7D9}} = 4\cos^2(\phi/2)$$ yields

:$$\dfrac{Q_\mathbf 1}{2 + 2Q_{\large\unicode{x1D7D9}}} = -\dfrac{d}{2}\tan(\phi/2)$$ .

The square root of a general motor $$\mathbf Q$$ is thus given by

:$$\sqrt{\mathbf Q} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\sqrt{2 + 2Q_\smash{\large\unicode{x1D7D9}}}} \mathbin{\unicode{x27C7}} \left({\large\unicode{x1D7D9}} - \dfrac{Q_\mathbf 1}{2 + 2Q_{\large\unicode{x1D7D9}}}\right)$$ .

If $$\mathbf Q$$ is a simple motor, then $$Q_{\mathbf 1} = 0$$, and this reduces to

:$$\sqrt{\mathbf Q} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\sqrt{2 + 2Q_\smash{\large\unicode{x1D7D9}}}} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB}}$$ .

== Factorization ==

Any [[unitized]] motor $$\mathbf Q$$ can be factored into the product of a motor $$\mathbf R = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1D7D9}}$$ corresponding to a pure [[rotation]] about a line through the origin and a motor $$\mathbf T = {t_x \mathbf e_{23} + t_y \mathbf e_{31} + t_z \mathbf e_{12} + \large\unicode{x1d7d9}}$$ corresponding to a pure [[translation]] by calculating

:$$\mathbf T = \mathbf Q \mathbin{\unicode{x27C7}} \mathbf{\underset{\Large\unicode{x7E}}{R}}$$ .

The motor $$\mathbf T$$ is then given by

:$$\mathbf T = (Q_{vy} Q_{mz} - Q_{vz} Q_{my} + Q_{mx} Q_{vw} - Q_{vx} Q_{mw})\mathbf e_{23} + (Q_{vz} Q_{mx} - Q_{vx} Q_{mz} + Q_{my} Q_{vw} - Q_{vy} Q_{mw})\mathbf e_{31} + (Q_{vx} Q_{my} - Q_{vy} Q_{mx} + Q_{mz} Q_{vw} - Q_{vz} Q_{mw})\mathbf e_{12} + {\large\unicode{x1d7d9}}$$ .

The original motor $$\mathbf Q$$ can now be expressed as

:$$\mathbf Q = \mathbf T \mathbin{\unicode{x27C7}} \mathbf R$$ ,

where both $$\mathbf R$$ and $$\mathbf T$$ are [[unitized]] simple motors.

== Conversion from Motor to Matrix ==

Given a specific [[Unitization | unitized]] motor $$\mathbf Q$$, define the matrices

:$$\mathbf A = \begin{bmatrix}1 - 2(Q_{vy}^2 + Q_{vz}^2) & 2Q_{vx} Q_{vy} & 2Q_{vz} Q_{vx} & 2(Q_{vy} Q_{mz} - Q_{vz} Q_{my}) \\ 2Q_{vx} Q_{vy} & 1 - 2(Q_{vz}^2 + Q_{vx}^2) & 2Q_{vy} Q_{vz} & 2(Q_{vz} Q_{mx} - Q_{vx} Q_{mz}) \\ 2Q_{vz} Q_{vx} & 2Q_{vy} Q_{vz} & 1 - 2(Q_{vx}^2 + Q_{vy}^2) & 2(Q_{vx} Q_{my} - Q_{vy} Q_{mx}) \\ 0 & 0 & 0 & 1\end{bmatrix}$$

and

:$$\mathbf B = \begin{bmatrix}0 & -2Q_{vz} Q_{vw} & 2Q_{vy} Q_{vw} & 2(Q_{vw} Q_{mx} - Q_{vx} Q_{mw}) \\ 2Q_{vz} Q_{vw} & 0 & -2Q_{vx} Q_{vw} & 2(Q_{vw} Q_{my} - Q_{vy} Q_{mw}) \\ -2Q_{vy} Q_{vw} & 2Q_{vx} Q_{vw} & 0 & 2(Q_{vw} Q_{mz} - Q_{vz} Q_{mw}) \\ 0 & 0 & 0 & 0\end{bmatrix}$$ .

Then the corresponding 4×4 matrix $$\mathbf M$$ that transforms a [[point]] $$\mathbf p$$, regarded as a column matrix, as $$\mathbf p' = \mathbf{Mp}$$ is given by

:$$\mathbf M = \mathbf A + \mathbf B$$ .

The inverse of $$\mathbf M$$, which transforms a [[plane]] $$\mathbf g$$, regarded as a row matrix, as $$\mathbf g' = \mathbf{gM^{-1}}$$ is given by

:$$\mathbf M^{-1} = \mathbf A - \mathbf B$$ .

== Conversion from Matrix to Motor ==

Let $$\mathbf M$$ be an orthogonal 4×4 matrix with determinant +1 having the form

:$$\mathbf M = \begin{bmatrix} M_{00} & M_{01} & M_{02} & M_{03} \\ M_{10} & M_{11} & M_{12} & M_{13} \\ M_{20} & M_{21} & M_{22} & M_{23} \\ 0 & 0 & 0 & 1\end{bmatrix}$$ .

Then, by equating the entries of $$\mathbf M$$ to the entries of $$\mathbf A + \mathbf B$$ from above, we have the following four relationships based on the diagonal entries of $$\mathbf M$$:

:$$M_{00} - M_{11} - M_{22} + 1 = 4Q_{vx}^2$$

:$$M_{11} - M_{22} - M_{00} + 1 = 4Q_{vy}^2$$

:$$M_{22} - M_{00} - M_{11} + 1 = 4Q_{vz}^2$$

:$$M_{00} + M_{11} + M_{22} + 1 = 4(1 - Q_{vx}^2 - Q_{vy}^2 - Q_{vz}^2) = 4Q_{vw}^2$$

And we have the following six relationships based on the off-diagonal entries of $$\mathbf M$$:

:$$M_{21} + M_{12} = 4Q_{vy} Q_{vz}$$

:$$M_{02} + M_{20} = 4Q_{vz} Q_{vx}$$

:$$M_{10} + M_{01} = 4Q_{vx} Q_{vy}$$

:$$M_{21} - M_{12} = 4Q_{vx} Q_{vw}$$

:$$M_{02} - M_{20} = 4Q_{vy} Q_{vw}$$

:$$M_{10} - M_{01} = 4Q_{vz} Q_{vw}$$

If $$M_{00} + M_{11} + M_{22} \geq 0$$, then we calculate

:$$Q_{vw} = \pm \dfrac{1}{2}\sqrt{M_{00} + M_{11} + M_{22} + 1}$$ ,

where either sign can be chosen. In this case, we know $$|Q_{vw}|$$ is at least $$1/2$$, so we can safely divide by $$4Q_{vw}$$ in the last three off-diagonal relationships above to solve for $$Q_{vx}$$, $$Q_{vy}$$, and $$Q_{vz}$$. Otherwise, if $$M_{00} + M_{11} + M_{22} < 0$$, then we select one of the first three diagonal relationships based on the largest diagonal entry $$M_{00}$$, $$M_{11}$$, or $$M_{22}$$. After calculating $$Q_{vx}$$, $$Q_{vy}$$, or $$Q_{vz}$$, we plug its value into two of the first three off-diagonal relationships to solve for the other two values of $$Q_{vx}$$, $$Q_{vy}$$, and $$Q_{vz}$$. Finally, we plug it into one of the last three off-diagonal relationships to solve for $$Q_{vw}$$.

Setting $$t_x = M_{03}$$, $$t_y = M_{13}$$, and $$t_z = M_{23}$$, the values of $$Q_{mx}$$, $$Q_{my}$$, $$Q_{mz}$$, and $$Q_{mw}$$ are given by

$$Q_{mx} = \dfrac{1}{2}(Q_{vw} t_x + Q_{vz} t_y - Q_{vy} t_z)$$

$$Q_{my} = \dfrac{1}{2}(Q_{vw} t_y + Q_{vx} t_z - Q_{vz} t_x)$$

$$Q_{mz} = \dfrac{1}{2}(Q_{vw} t_z + Q_{vy} t_x - Q_{vx} t_y)$$

$$Q_{mw} = -\dfrac{1}{2}(Q_{vx} t_x + Q_{vy} t_y + Q_{vz} t_z)$$ .

== Motor Transformations ==

[[Point | Points]], [[Line | lines]], and [[Plane | planes]] are transformed by a [[unitized]] motor $$\mathbf Q$$ as 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;" | $$\begin{split}\mathbf Q \mathbin{\unicode{x27C7}} \mathbf p \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}} =\, &\left[(1 - 2Q_{vy}^2 - 2Q_{vz}^2)p_x + 2(Q_{vx}Q_{vy} - Q_{vz}Q_{vw})p_y + 2(Q_{vz}Q_{vx} + Q_{vy}Q_{vw})p_z + 2(Q_{vy}Q_{mz} - Q_{vz}Q_{my} + Q_{vw}Q_{mx} - Q_{vx}Q_{mw})p_w\right]\mathbf e_1 \\ +\, &\left[(1 - 2Q_{vz}^2 -2Q_{vx}^2))p_y + 2(Q_{vy}Q_{vz} - Q_{vx}Q_{vw})p_z + 2(Q_{vx}Q_{vy} + Q_{vz}Q_{vw})p_x + 2(Q_{vz}Q_{mx} - Q_{vx}Q_{mz} + Q_{vw}Q_{my} - Q_{vy}Q_{mw})p_w\right]\mathbf e_2 \\ +\, &\left[(1 - 2Q_{vx}^2 - 2Q_{vy}^2))p_z + 2(Q_{vz}Q_{vx} - Q_{vy}Q_{vw})p_x + 2(Q_{vy}Q_{vz} + Q_{vx}Q_{vw})p_y + 2(Q_{vx}Q_{my} - Q_{vy}Q_{mx} + Q_{vw}Q_{mz} - Q_{vz}Q_{mw})p_w\right]\mathbf e_3 \\ +\, &p_w\mathbf e_4\end{split}$$
|-
| 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;" | $$\begin{split}\mathbf Q \mathbin{\unicode{x27C7}} \boldsymbol l \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}} =\, &\left[(1 - 2Q_{vy}^2 - 2Q_{vz}^2)l_{vx} + 2(Q_{vx}Q_{vy} - Q_{vz}Q_{vw})l_{vy} + 2(Q_{vz}Q_{vx} + Q_{vy}Q_{vw})l_{vz}\right]\mathbf e_{41} \\ +\, &\left[(1 - 2Q_{vz}^2 - 2Q_{vx}^2)l_{vy} + 2(Q_{vy}Q_{vz} - Q_{vx}Q_{vw})l_{vz} + 2(Q_{vx}Q_{vy} + Q_{vz}Q_{vw})l_{vx}\right]\mathbf e_{42} \\ +\, &\left[(1 - 2Q_{vx}^2 - 2Q_{vy}^2)l_{vz} + 2(Q_{vz}Q_{vx} - Q_{vy}Q_{vw})l_{vx} + 2(Q_{vy}Q_{vz} + Q_{vx}Q_{vw})l_{vy}\right]\mathbf e_{43} \\ +\, &\left[-4(Q_{vy}Q_{my} + Q_{vz}Q_{mz})l_{vx} + 2(Q_{vy}Q_{mx} + Q_{vx}Q_{my} - Q_{vz}Q_{mw} - Q_{vw}Q_{mz})l_{vy} + 2(Q_{vz}Q_{mx} + Q_{vx}Q_{mz} + Q_{vy}Q_{mw} + Q_{vw}Q_{my})l_{vz} + (1 - 2Q_{vy}^2 - 2Q_{vz}^2)l_{mx} + 2(Q_{vx}Q_{vy} - Q_{vz}Q_{vw})l_{my} + 2(Q_{vz}Q_{vx} + Q_{vy}Q_{vw})l_{mz}\right]\mathbf e_{23} \\ +\, &\left[-4(Q_{vz}Q_{mz} + Q_{vx}Q_{mx})l_{vy} + 2(Q_{vz}Q_{my} + Q_{vy}Q_{mz} - Q_{vx}Q_{mw} - Q_{vw}Q_{mx})l_{vz} + 2(Q_{vx}Q_{my} + Q_{vy}Q_{mx} + Q_{vz}Q_{mw} + Q_{vw}Q_{mz})l_{vx} + (1 - 2Q_{vz}^2 - 2Q_{vx}^2)l_{my} + 2(Q_{vy}Q_{vz} - Q_{vx}Q_{vw})l_{mz} + 2(Q_{vx}Q_{vy} + Q_{vz}Q_{vw})l_{mx}\right]\mathbf e_{31} \\ +\, &\left[-4(Q_{vx}Q_{mx} + Q_{vy}Q_{my})l_{vz} + 2(Q_{vx}Q_{mz} + Q_{vz}Q_{mx} - Q_{vy}Q_{mw} - Q_{vw}Q_{my})l_{vx} + 2(Q_{vy}Q_{mz} + Q_{vz}Q_{my} + Q_{vx}Q_{mw} + Q_{vw}Q_{mx})l_{vy} + (1 - 2Q_{vx}^2 - 2Q_{vy}^2)l_{mz} + 2(Q_{vz}Q_{vx} - Q_{vy}Q_{vw})l_{mx} + 2(Q_{vy}Q_{vz} + 2Q_{vx}Q_{vw})l_{my}\right]\mathbf e_{12}\end{split}$$
|-
| 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;" | $$\begin{split}\mathbf Q \mathbin{\unicode{x27C7}} \mathbf g \mathbin{\unicode{x27C7}} \smash{\mathbf{\underset{\Large\unicode{x7E}}{Q}}} =\, &\left[(1 - 2Q_{vy}^2 - 2Q_{vz}^2)g_x + 2(Q_{vx}Q_{vy} - Q_{vz}Q_{vw})g_y + 2(Q_{vz}Q_{vx} + Q_{vy}Q_{vw})g_z\right]\mathbf e_{423} \\ +\, &\left[(1 - 2Q_{vz}^2 - 2Q_{vx}^2)g_y + 2(Q_{vy}Q_{vz} - Q_{vx}Q_{vw})g_z + 2(Q_{vx}Q_{vy} + Q_{vz}Q_{vw})g_x\right]\mathbf e_{431} \\ +\, &\left[(1 - 2Q_{vx}^2 - 2Q_{vy}^2)g_z + 2(Q_{vz}Q_{vx} - Q_{vy}Q_{vw})g_x + 2(Q_{vy}Q_{vz} + Q_{vx}Q_{vw})g_y\right]\mathbf e_{412} \\ +\, &\left[2(Q_{vy}Q_{mz} - Q_{vz}Q_{my} + Q_{vx}Q_{mw} - Q_{vw}Q_{mx})g_x + 2(Q_{vz}Q_{mx} - Q_{vx}Q_{mz} + Q_{vy}Q_{mw} - Q_{vw}Q_{my})g_y + 2(Q_{vx}Q_{my} - Q_{vy}Q_{mx} + Q_{vz}Q_{mw} - Q_{vw}Q_{mz})g_z + g_w\right]\mathbf e_{321}\end{split}$$
|}

== See Also ==

* [[Flector]]
* [[Translation]]
* [[Rotation]]

← Older revision		Revision as of 01:19, 8 July 2024
Line 131:		Line 131:
	Using the exponential form of $$\mathbf Q$$, we first examine the [[weight norm]] of $$\mathbf Q + {\large\unicode{x1D7D9}}$$ and find that		Using the exponential form of $$\mathbf Q$$, we first examine the [[weight norm]] of $$\mathbf Q + {\large\unicode{x1D7D9}}$$ and find that

	:$$\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB} = \sqrt{\sin^2\phi + (\cos\phi + 1)^2} = \sqrt{\vphantom{\sin^2\phi}2 + 2Q_\smash{\large\unicode{x1D7D9}}}$$ ,		:$$\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode["segoe ui symbol"]{x25CB} = \sqrt{\sin^2\phi + (\cos\phi + 1)^2} = \sqrt{\vphantom{\sin^2\phi}2 + 2Q_\smash{\large\unicode{x1D7D9}}}$$ ,

	where $$Q_{\large\unicode{x1D7D9}} = \cos\phi$$. Applying the trigonometric identity $$\cos^2(\phi/2) = (1 + \cos\phi)/2$$, we can rewrite this as		where $$Q_{\large\unicode{x1D7D9}} = \cos\phi$$. Applying the trigonometric identity $$\cos^2(\phi/2) = (1 + \cos\phi)/2$$, we can rewrite this as

	:$$\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB} = 2\cos(\phi/2)$$ .		:$$\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode["segoe ui symbol"]{x25CB} = 2\cos(\phi/2)$$ .

	Applying several more trigonometric identities, we now observe		Applying several more trigonometric identities, we now observe

	:$$\dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB}} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{2\cos(\phi/2)} = \boldsymbol l\sin(\phi/2) + \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\cos(\phi/2) - \dfrac{d}{2}\sin(\phi/2) + {\large\unicode{x1D7D9}}\cos(\phi/2) - \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\sin(\phi/2)\tan(\phi/2) - \dfrac{d}{2}\sin(\phi/2)$$ .		:$$\dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{2\cos(\phi/2)} = \boldsymbol l\sin(\phi/2) + \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\cos(\phi/2) - \dfrac{d}{2}\sin(\phi/2) + {\large\unicode{x1D7D9}}\cos(\phi/2) - \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\sin(\phi/2)\tan(\phi/2) - \dfrac{d}{2}\sin(\phi/2)$$ .

	The first four terms are exactly the square root of $$\mathbf Q$$ because the distance $$d$$ and angle $$\phi$$ have both been halved. But there are two additional terms that we need to eliminate. It just so happens that		The first four terms are exactly the square root of $$\mathbf Q$$ because the distance $$d$$ and angle $$\phi$$ have both been halved. But there are two additional terms that we need to eliminate. It just so happens that

	:$$\dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB}} \mathbin{\unicode{x27C7}} \dfrac{d}{2}\tan(\phi/2) = \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\sin(\phi/2)\tan(\phi/2) + \dfrac{d}{2}\sin(\phi/2)$$ ,		:$$\dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode["segoe ui symbol"]{x25CB}} \mathbin{\unicode{x27C7}} \dfrac{d}{2}\tan(\phi/2) = \left(\dfrac{d}{2} \mathbin{\unicode{x27C7}} \boldsymbol l\right)\sin(\phi/2)\tan(\phi/2) + \dfrac{d}{2}\sin(\phi/2)$$ ,

	which means that		which means that
Line 159:		Line 159:
	If $$\mathbf Q$$ is a simple motor, then $$Q_{\mathbf 1} = 0$$, and this reduces to		If $$\mathbf Q$$ is a simple motor, then $$Q_{\mathbf 1} = 0$$, and this reduces to

	:$$\sqrt{\mathbf Q} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\sqrt{2 + 2Q_\smash{\large\unicode{x1D7D9}}}} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode{x25CB}}$$ .		:$$\sqrt{\mathbf Q} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\sqrt{2 + 2Q_\smash{\large\unicode{x1D7D9}}}} = \dfrac{\mathbf Q + {\large\unicode{x1D7D9}}}{\left\Vert\mathbf Q + \smash{\large\unicode{x1D7D9}}\right\Vert_\unicode["segoe ui symbol"]{x25CB}}$$ .

	== Factorization ==		== Factorization ==