Containers - Revision history

Eric Lengyel at 22:59, 3 April 2024

2024-04-03T22:59:40Z

← Older revision		Revision as of 22:59, 3 April 2024
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	The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[expansion]] of $$\mathbf x$$ into its own [[carrier]]:		The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf u$$ is denoted by $$\operatorname{con}(\mathbf u)$$, and it is given by the [[expansion]] of $$\mathbf u$$ into its own [[carrier]]:

	:$$\operatorname{con}(\mathbf x) = \mathbf x \wedge \operatorname{car}(\mathbf x)^\unicode["segoe ui symbol"]{x2606}$$ .		:$$\operatorname{con}(\mathbf u) = \mathbf u \wedge \operatorname{car}(\mathbf u)^\unicode["segoe ui symbol"]{x2606}$$ .

	The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.		The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.

Eric Lengyel at 06:54, 17 November 2023

2023-11-17T06:54:38Z

← Older revision		Revision as of 06:54, 17 November 2023
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	The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[~~connect~~]] of $$\mathbf x$$ ~~with~~ its own [[carrier]]:		The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[expansion]] of $$\mathbf x$$ into its own [[carrier]]:

	:$$\operatorname{con}(\mathbf x) = -\operatorname{car}(\mathbf x)^\unicode["segoe ui symbol"]{~~x2605~~} ~~\wedge \mathbf x~~$$ .		:$$\operatorname{con}(\mathbf x) = \mathbf x \wedge \operatorname{car}(\mathbf x)^\unicode["segoe ui symbol"]{x2606}$$ .

	~~(The negative sign is not strictly necessary, but is included so the function always produces a sphere having a negative $$\mathbf e_{1234}$$ component.)~~ The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.		The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.

	The following table lists the containers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.		The following table lists the containers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.
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	\| style="padding: 12px;" \| [[Dipole]]		\| style="padding: 12px;" \| [[Dipole]]
	\| style="padding: 12px;" \| $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$		\| style="padding: 12px;" \| $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
	\| style="padding: 12px;" \| $$\begin{split}\operatorname{con}(\mathbf d) =		\| style="padding: 12px;" \| $$\begin{split}\operatorname{con}(\mathbf d)
	&-(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_{1234} \\		&= (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_{1234} \\
	&+ (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\,\mathbf e_{4235} \\		&+ (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\,\mathbf e_{4235} \\
	&+ (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\,\mathbf e_{4315} \\		&+ (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\,\mathbf e_{4315} \\
	&+ (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\,\mathbf e_{4125} \\		&+ (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\,\mathbf e_{4125} \\
	&- (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\,\mathbf e_{3215}\end{split}$$		&+ (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\,\mathbf e_{3215}\end{split}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Circle]]		\| style="padding: 12px;" \| [[Circle]]
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	\| style="padding: 12px;" \| [[Sphere]]		\| style="padding: 12px;" \| [[Sphere]]
	\| style="padding: 12px;" \| $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$		\| style="padding: 12px;" \| $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
	\| style="padding: 12px;" \| $$\begin{split}\operatorname{con}(\mathbf s) =		\| style="padding: 12px;" \| $$\begin{split}\operatorname{con}(\mathbf s)
	&-s_u^2 \mathbf e_{1234} \\		&= s_u^2 \mathbf e_{1234} \\
	&- s_xs_u \mathbf e_{4235} \\		&+ s_xs_u \mathbf e_{4235} \\
	&- s_ys_u \mathbf e_{4315} \\		&+ s_ys_u \mathbf e_{4315} \\
	&- s_zs_u \mathbf e_{4125} \\		&+ s_zs_u \mathbf e_{4125} \\
	&- s_ws_u \mathbf e_{3215}\end{split}$$		&+ s_ws_u \mathbf e_{3215}\end{split}$$
	\|}		\|}

Eric Lengyel at 01:00, 27 August 2023

2023-08-27T01:00:28Z

← Older revision		Revision as of 01:00, 27 August 2023
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	The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[connect]] of $$\mathbf x$$ with its own [[carrier]]:		The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[connect]] of $$\mathbf x$$ with its own [[carrier]]:

	:$$\operatorname{con}(\mathbf x) = -\operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ .		:$$\operatorname{con}(\mathbf x) = -\operatorname{car}(\mathbf x)^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf x$$ .

	(The negative sign is not strictly necessary, but is included so the function always produces a sphere having a negative $$\mathbf e_{1234}$$ component.) The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.		(The negative sign is not strictly necessary, but is included so the function always produces a sphere having a negative $$\mathbf e_{1234}$$ component.) The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.

Eric Lengyel at 21:47, 25 August 2023

2023-08-25T21:47:41Z

← Older revision		Revision as of 21:47, 25 August 2023
Line 1:		Line 1:
	The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[connect]] of $$\mathbf x$$ with its own [[carrier]]:		The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[connect]] of $$\mathbf x$$ with its own [[carrier]]:

	:$$\operatorname{con}(\mathbf x) = \operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ .		:$$\operatorname{con}(\mathbf x) = -\operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ .

	The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.		(The negative sign is not strictly necessary, but is included so the function always produces a sphere having a negative $$\mathbf e_{1234}$$ component.) The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.

	The following table lists the containers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.		The following table lists the containers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

Eric Lengyel: Created page with "The ''container'' of a round object (a round point, dipole, circle, or sphere) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the connect of $$\mathbf x$$ with its own carrier: :$$\operatorname{con}(\mathbf x) = \operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ . The squared radius of an object's container has the same sign as the squared radi..."

2023-08-06T03:15:48Z

Created page with "The ''container'' of a round object (a round point, dipole, circle, or sphere) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the connect of $$\mathbf x$$ with its own carrier: :$$\operatorname{con}(\mathbf x) = \operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ . The squared radius of an object's container has the same sign as the squared radi..."

New page

The ''container'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the smallest sphere that contains it. The container of an object $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[connect]] of $$\mathbf x$$ with its own [[carrier]]:

:$$\operatorname{con}(\mathbf x) = \operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ .

The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.

The following table lists the containers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Container
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf a) =
&-a_w^2 \mathbf e_{1234} \\
&+ a_xa_w \mathbf e_{4235} \\
&+ a_ya_w \mathbf e_{4315} \\
&+ a_za_w \mathbf e_{4125} \\
&+ (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}\end{split}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf d) =
&-(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_{1234} \\
&+ (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\,\mathbf e_{4235} \\
&+ (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\,\mathbf e_{4315} \\
&+ (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\,\mathbf e_{4125} \\
&- (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\,\mathbf e_{3215}\end{split}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf c) =
-\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\,\mathbf e_{1234} \\
+\,&(c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\,\mathbf e_{4235} \\
+\,&(c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\,\mathbf e_{4315} \\
+\,&(c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\,\mathbf e_{4125} \\
+\,&(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\,\mathbf e_{3215}\end{split}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf s) =
&-s_u^2 \mathbf e_{1234} \\
&- s_xs_u \mathbf e_{4235} \\
&- s_ys_u \mathbf e_{4315} \\
&- s_zs_u \mathbf e_{4125} \\
&- s_ws_u \mathbf e_{3215}\end{split}$$
|}

== See Also ==

* [[Centers]]
* [[Carriers]]
* [[Partners]]
* [[Attitude]]