Dipole - Revision history

Eric Lengyel at 04:30, 31 July 2024

2024-07-31T04:30:04Z

← Older revision		Revision as of 04:30, 31 July 2024
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	Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as		Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as

	:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .		:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{\mathbf p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .

	The various properties of a dipole are summarized in the following table.		The various properties of a dipole are summarized in the following table.
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	The radius of a dipole $$\mathbf d$$ is given by		The radius of a dipole $$\mathbf d$$ is given by

	:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .		:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .

	== Contained Points ==		== Contained Points ==

Eric Lengyel: /* Contained Points */

2024-04-21T20:45:05Z

Contained Points

← Older revision		Revision as of 20:45, 21 April 2024
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	The [[attitude]] of a dipole $$\mathbf d$$ is given by		The [[attitude]] of a dipole $$\mathbf d$$ is given by

	:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \~~underline~~{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .		:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \overline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .

	The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as		The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as

Eric Lengyel at 23:45, 1 December 2023

2023-12-01T23:45:06Z

Eric Lengyel: /* Carrier and Anticarrier */

2023-11-17T07:22:14Z

Carrier and Anticarrier

← Older revision		Revision as of 07:22, 17 November 2023
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	The carrier and anticarrier meet at the flat center of the dipole, which is given by the [[flat point]]		The carrier and anticarrier meet at the flat center of the dipole, which is given by the [[flat point]]

	:$$\operatorname{car}(\mathbf d) \vee \operatorname{acr}(\mathbf d) = (d_{vx} d_{pw} - d_{vz} d_{my} + d_{vy} d_{mz})\mathbf e_{15} + (d_{vy} d_{pw} - d_{vx} d_{mz} + d_{vz} d_{mx})\mathbf e_{25} + (d_{vz} d_{pw} - d_{vy} d_{mx} + d_{vx} d_{my})\mathbf e_{35} + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .		:$$\operatorname{car}(\mathbf d) \vee \operatorname{acr}(\mathbf d) = (d_{vz} d_{my} - d_{vx} d_{pw} - d_{vy} d_{mz})\mathbf e_{15} + (d_{vx} d_{mz} - d_{vy} d_{pw} - d_{vz} d_{mx})\mathbf e_{25} + (d_{vy} d_{mx} - d_{vz} d_{pw} - d_{vx} d_{my})\mathbf e_{35} - (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

	== Center and Container ==		== Center and Container ==

Eric Lengyel: /* Center and Container */

2023-11-17T07:16:25Z

Center and Container

← Older revision		Revision as of 07:16, 17 November 2023
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	The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by		The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

	:$$\operatorname{cen}(\mathbf d) = -\operatorname{~~car~~}(\mathbf d^*) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .		:$$\operatorname{cen}(\mathbf d) = \operatorname{acr}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

	The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by		The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by

	:$$\operatorname{con}(\mathbf d) = \operatorname{car}(\mathbf d)^* \~~wedge \mathbf d~~ = -(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_{4235} + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_{4315} + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_{4125} - (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .		:$$\operatorname{con}(\mathbf d) = \mathbf d \wedge \operatorname{car}(\mathbf d)^\unicode["segoe ui symbol"]{x2606} = (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\mathbf e_{4235} + (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\mathbf e_{4315} + (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\mathbf e_{4125} + (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .

	A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as		A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as

Eric Lengyel: /* Carrier and Anticarrier */

2023-11-17T07:13:43Z

Carrier and Anticarrier

← Older revision		Revision as of 07:13, 17 November 2023
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	The [[anticarrier]] of a dipole $$\mathbf d$$ is the [[plane]]		The [[anticarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

	:$$\operatorname{acr}(\mathbf d) = \mathbf d^* \wedge \mathbf e_5 = -d_{vx} \mathbf e_{4235} - d_{vy} \mathbf e_{4315} - d_{vz} \mathbf e_{4125} + d_{pw} \mathbf e_{3215}$$ .		:$$\operatorname{acr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .

	The carrier and anticarrier meet at the flat center of the dipole, which is given by the [[flat point]]		The carrier and anticarrier meet at the flat center of the dipole, which is given by the [[flat point]]

Eric Lengyel: Created page with "NOTOC In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form :$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ . If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ com..."

2023-08-06T03:14:49Z

Created page with "__NOTOC__ In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form :$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ . If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ com..."

New page

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''dipole'' $$\mathbf d$$ is a bivector with ten components having the general form

:$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ .

If the $$vx$$, $$vy$$, $$vz$$, $$mx$$, $$my$$, and $$mz$$ components are all zero, then the dipole contains the [[point at infinity]], and it is thus a [[flat point]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a line direction $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a dipole can be formulated as

:$$\mathbf d = n_x \mathbf e_{41} + n_y \mathbf e_{42} + n_z \mathbf e_{43} + (p_yn_z - p_zn_y) \mathbf e_{23} + (p_zn_x - p_xn_z) \mathbf e_{31} + (p_xn_y - p_yn_x) \mathbf e_{12} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + \mathbf e_{45}) - \dfrac{p^2 + r^2}{2}(n_x \mathbf e_{15} + n_y \mathbf e_{25} + n_z \mathbf e_{35})$$ .

The various properties of a dipole are summarized in the following table.

[[Image:dipole.svg|800px]]

== Constraints ==

A valid dipole $$\mathbf d$$ must satisfy the following constraints, where $$\mathbf v = (d_{vx}, d_{vy}, d_{vz})$$, $$\mathbf m = (d_{mx}, d_{my}, d_{mz})$$, and $$\mathbf p = (d_{px}, d_{py}, d_{pz})$$ are treated as ordinary 3D vectors.

:$$\mathbf p \times \mathbf v - d_{pw}\mathbf m = \mathbf 0$$
:$$\mathbf p \cdot \mathbf m = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.

== Carrier and Anticarrier ==

The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]

:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .

The [[anticarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

:$$\operatorname{acr}(\mathbf d) = \mathbf d^* \wedge \mathbf e_5 = -d_{vx} \mathbf e_{4235} - d_{vy} \mathbf e_{4315} - d_{vz} \mathbf e_{4125} + d_{pw} \mathbf e_{3215}$$ .

The carrier and anticarrier meet at the flat center of the dipole, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf d) \vee \operatorname{acr}(\mathbf d) = (d_{vx} d_{pw} - d_{vz} d_{my} + d_{vy} d_{mz})\mathbf e_{15} + (d_{vy} d_{pw} - d_{vx} d_{mz} + d_{vz} d_{mx})\mathbf e_{25} + (d_{vz} d_{pw} - d_{vy} d_{mx} + d_{vx} d_{my})\mathbf e_{35} + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{cen}(\mathbf d) = -\operatorname{car}(\mathbf d^*) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by

:$$\operatorname{con}(\mathbf d) = \operatorname{car}(\mathbf d)^* \wedge \mathbf d = -(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{1234} + (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_{4235} + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_{4315} + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_{4125} - (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\mathbf e_{3215}$$ .

A unitized dipole is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf d = \operatorname{car}(\mathbf d) \vee \operatorname{con}(\mathbf d)$$ .

== Norms ==

The radius of a dipole $$\mathbf d$$ is given by

:$$\operatorname{rad}(\mathbf d) = \dfrac{\left\Vert\mathbf d\right\Vert_R}{\left\Vert\mathbf d\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - 2(d_{px} d_{vx} + d_{py} d_{vy} + d_{pz} d_{vz})}{d_{vx}^2 + d_{vy}^2 + d_{vz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a dipole $$\mathbf d$$ is given by

:$$\operatorname{att}(\mathbf d) = \mathbf d \vee \underline{\mathbf e_4} = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ .

The set of round points contained by a dipole $$\mathbf d$$ can be expressed parametrically in terms of the center and attitude as

:$$\mathbf p(\alpha) = \operatorname{cen}(\mathbf d) + \alpha \operatorname{att}(\mathbf d)$$ .

That is, $$\mathbf d \wedge \mathbf p(\alpha) = 0$$ for all real numbers $$\alpha$$. In particular, the two points on the surface of a dipole are given by the parameter $$\alpha_R = \pm \left\Vert\mathbf d\right\Vert_R$$ (the weighted radius). When $$\mathbf d$$ is a real dipole, this is precisely where the radius of $$\mathbf p(\alpha)$$ is zero. For smaller absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is real, and for larger absolute values of $$\alpha$$, the round point $$\mathbf p(\alpha)$$ is imaginary. When $$\mathbf d$$ is an imaginary dipole, the round point $$\mathbf p(\alpha)$$ is always imaginary, and it has an absolute radius at least as large as the dipole itself.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Circle]]
* [[Sphere]]

← Older revision		Revision as of 23:45, 1 December 2023
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	The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.		The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf p$$ and $$\mathbf v$$.

	== Carrier and ~~Anticarrier~~ ==		== Carrier and Cocarrier ==

	The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]		The [[carrier]] of a dipole $$\mathbf d$$ is the [[line]]
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	:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .		:$$\operatorname{car}(\mathbf d) = \mathbf d \wedge \mathbf e_5 = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$ .

	The [[~~anticarrier~~]] of a dipole $$\mathbf d$$ is the [[plane]]		The [[cocarrier]] of a dipole $$\mathbf d$$ is the [[plane]]

	:$$\operatorname{~~acr~~}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .		:$$\operatorname{ccr}(\mathbf d) = \mathbf d^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$ .

	The carrier and ~~anticarrier~~ meet at the flat center of the dipole, which is given by the [[flat point]]		The carrier and cocarrier meet at the flat center of the dipole, which is given by the [[flat point]]

	:$$\operatorname{car}(\mathbf d) \vee \operatorname{~~acr~~}(\mathbf d) = (d_{vz} d_{my} - d_{vx} d_{pw} - d_{vy} d_{mz})\mathbf e_{15} + (d_{vx} d_{mz} - d_{vy} d_{pw} - d_{vz} d_{mx})\mathbf e_{25} + (d_{vy} d_{mx} - d_{vz} d_{pw} - d_{vx} d_{my})\mathbf e_{35} - (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .		:$$\operatorname{car}(\mathbf d) \vee \operatorname{ccr}(\mathbf d) = (d_{vz} d_{my} - d_{vx} d_{pw} - d_{vy} d_{mz})\mathbf e_{15} + (d_{vx} d_{mz} - d_{vy} d_{pw} - d_{vz} d_{mx})\mathbf e_{25} + (d_{vy} d_{mx} - d_{vz} d_{pw} - d_{vx} d_{my})\mathbf e_{35} - (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_{45}$$ .

	== Center and Container ==		== Center and Container ==
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	The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by		The round [[center]] of a dipole $$\mathbf d$$ is the [[round point]] having the same center and radius as $$\mathbf d$$, and it is given by

	:$$\operatorname{cen}(\mathbf d) = \operatorname{~~acr~~}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .		:$$\operatorname{cen}(\mathbf d) = \operatorname{ccr}(\mathbf d) \vee \mathbf d = (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\mathbf e_1 + (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\mathbf e_2 + (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\mathbf e_3 + (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\mathbf e_4 + (d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\mathbf e_5$$ .

	The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by		The [[container]] of a dipole $$\mathbf d$$ is the [[sphere]] having the same center and radius as $$\mathbf d$$, and it is given by