Carriers - Revision history

Eric Lengyel at 22:57, 3 April 2024

2024-04-03T22:57:45Z

← Older revision		Revision as of 22:57, 3 April 2024
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	== Carrier ==		== Carrier ==

	The ''carrier'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the lowest dimensional flat object (a [[flat point]], [[line]], or [[plane]]) that contains it. The carrier of an object $$\mathbf x$$ is denoted by $$\operatorname{car}(\mathbf x)$$, and it is calculated by simply multiplying $$\mathbf x$$ by $$\mathbf e_5$$ with the [[wedge product]] to extract the round part of $$\mathbf x$$ as a flat geometry:		The ''carrier'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the lowest dimensional flat object (a [[flat point]], [[line]], or [[plane]]) that contains it. The carrier of an object $$\mathbf u$$ is denoted by $$\operatorname{car}(\mathbf u)$$, and it is calculated by simply multiplying $$\mathbf u$$ by $$\mathbf e_5$$ with the [[wedge product]] to extract the round part of $$\mathbf u$$ as a flat geometry:

	:$$\operatorname{car}(\mathbf x) = \mathbf x \wedge \mathbf e_5$$ .		:$$\operatorname{car}(\mathbf u) = \mathbf u \wedge \mathbf e_5$$ .

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

	The ''cocarrier'' of a round object is the carrier of its [[antidual]]. The cocarrier of an object $$\mathbf x$$ is denoted by $$\operatorname{ccr}(\mathbf x)$$, and it is calculated by		The ''cocarrier'' of a round object is the carrier of its [[antidual]]. The cocarrier of an object $$\mathbf u$$ is denoted by $$\operatorname{ccr}(\mathbf u)$$, and it is calculated by

	:$$\operatorname{ccr}(\mathbf x) = \mathbf x^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5$$ .		:$$\operatorname{ccr}(\mathbf u) = \mathbf u^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5$$ .

	The cocarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and cocarrier can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{~~car~~}(\mathbf x) \vee \operatorname{~~ccr~~}(\mathbf x)$$.		The cocarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and cocarrier can be used to calculate the center of an object $$\mathbf u$$ as a [[flat point]] with the formula $$\operatorname{ccr}(\mathbf u) \vee \operatorname{car}(\mathbf u)$$.

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

Eric Lengyel at 23:40, 1 December 2023

2023-12-01T23:40:44Z

Eric Lengyel: /* Anticarrier */

2023-11-17T06:49:30Z

Anticarrier

← Older revision		Revision as of 06:49, 17 November 2023
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	== Anticarrier ==		== Anticarrier ==

	The ''anticarrier'' of a round object is the carrier of its [[~~dual~~]]. The anticarrier of an object $$\mathbf x$$ is denoted by $$\operatorname{acr}(\mathbf x)$$, and it is calculated by		The ''anticarrier'' of a round object is the carrier of its [[antidual]]. The anticarrier of an object $$\mathbf x$$ is denoted by $$\operatorname{acr}(\mathbf x)$$, and it is calculated by

	:$$\operatorname{acr}(\mathbf x) = \mathbf x^\unicode["segoe ui symbol"]{~~x2605~~} \wedge \mathbf e_5$$ .		:$$\operatorname{acr}(\mathbf x) = \mathbf x^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5$$ .

	The anticarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and anticarrier can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{car}(\mathbf x) \vee \operatorname{acr}(\mathbf x)$$.		The anticarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and anticarrier can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{car}(\mathbf x) \vee \operatorname{acr}(\mathbf x)$$.

Eric Lengyel: /* Anticarrier */

2023-08-27T01:00:03Z

Anticarrier

← Older revision		Revision as of 01:00, 27 August 2023
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	The ''anticarrier'' of a round object is the carrier of its [[dual]]. The anticarrier of an object $$\mathbf x$$ is denoted by $$\operatorname{acr}(\mathbf x)$$, and it is calculated by		The ''anticarrier'' of a round object is the carrier of its [[dual]]. The anticarrier of an object $$\mathbf x$$ is denoted by $$\operatorname{acr}(\mathbf x)$$, and it is calculated by

	:$$\operatorname{acr}(\mathbf x) = \mathbf x^* \wedge \mathbf e_5$$ .		:$$\operatorname{acr}(\mathbf x) = \mathbf x^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf e_5$$ .

	The anticarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and anticarrier can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{car}(\mathbf x) \vee \operatorname{acr}(\mathbf x)$$.		The anticarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and anticarrier can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{car}(\mathbf x) \vee \operatorname{acr}(\mathbf x)$$.

Eric Lengyel: /* Anticarrier */

2023-08-25T21:31:53Z

Anticarrier

← Older revision		Revision as of 21:31, 25 August 2023
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	\| style="padding: 12px;" \| [[Round point]]		\| 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;" \| $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
	\| style="padding: 12px;" \| $$\operatorname{acr}(\mathbf a) = -a_w {\large\unicode{x1d7d9}}$$		\| style="padding: 12px;" \| $$\operatorname{acr}(\mathbf a) = a_w {\large\unicode{x1d7d9}}$$
	\|-		\|-
	\| 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;" \| $$\operatorname{acr}(\mathbf d) = -d_{vx} \mathbf e_{4235} - d_{vy} \mathbf e_{4315} - d_{vz} \mathbf e_{4125} + d_{pw} \mathbf e_{3215}$$		\| style="padding: 12px;" \| $$\operatorname{acr}(\mathbf d) = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Circle]]		\| 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;" \| $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
	\| style="padding: 12px;" \| $$\operatorname{acr}(\mathbf c) = c_{gx} \mathbf e_{415} + c_{gy} \mathbf e_{425} + c_{gz} \mathbf e_{435} + c_{vx} \mathbf e_{235} + c_{vy} \mathbf e_{315} + c_{vz} \mathbf e_{125}$$		\| style="padding: 12px;" \| $$\operatorname{acr}(\mathbf c) = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$
	\|-		\|-
	\| 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;" \| $$\operatorname{acr}(\mathbf s) = -s_x \mathbf e_{15} - s_y \mathbf e_{25} - s_z \mathbf e_{35} + s_u \mathbf e_{45}$$		\| style="padding: 12px;" \| $$\operatorname{acr}(\mathbf s) = s_x \mathbf e_{15} + s_y \mathbf e_{25} + s_z \mathbf e_{35} - s_u \mathbf e_{45}$$
	\|}		\|}

Eric Lengyel: /* Anticarrier */

2023-08-06T07:12:37Z

Anticarrier

← Older revision		Revision as of 07:12, 6 August 2023
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	== Anticarrier ==		== Anticarrier ==

	The ''anticarrier'' of a round object is the carrier of its [[dual]]. The ~~carrier~~ of an object $$\mathbf x$$ is denoted by $$\operatorname{acr}(\mathbf x)$$, and it is calculated by		The ''anticarrier'' of a round object is the carrier of its [[dual]]. The anticarrier of an object $$\mathbf x$$ is denoted by $$\operatorname{acr}(\mathbf x)$$, and it is calculated by

	:$$\operatorname{acr}(\mathbf x) = \mathbf x^* \wedge \mathbf e_5$$ .		:$$\operatorname{acr}(\mathbf x) = \mathbf x^* \wedge \mathbf e_5$$ .

Eric Lengyel: Created page with "== Carrier == The ''carrier'' of a round object (a round point, dipole, circle, or sphere) is the lowest dimensional flat object (a flat point, line, or plane) that contains it. The carrier of an object $$\mathbf x$$ is denoted by $$\operatorname{car}(\mathbf x)$$, and it is calculated by simply multiplying $$\mathbf x$$ by $$\mathbf e_5$$ with the wedge product to extract the round part of $$\mathbf x$$ as a flat geometry: :$$\operatorn..."

2023-08-06T03:16:20Z

Created page with "== Carrier == The ''carrier'' of a round object (a round point, dipole, circle, or sphere) is the lowest dimensional flat object (a flat point, line, or plane) that contains it. The carrier of an object $$\mathbf x$$ is denoted by $$\operatorname{car}(\mathbf x)$$, and it is calculated by simply multiplying $$\mathbf x$$ by $$\mathbf e_5$$ with the wedge product to extract the round part of $$\mathbf x$$ as a flat geometry: :$$\operatorn..."

New page

== Carrier ==

The ''carrier'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the lowest dimensional flat object (a [[flat point]], [[line]], or [[plane]]) that contains it. The carrier of an object $$\mathbf x$$ is denoted by $$\operatorname{car}(\mathbf x)$$, and it is calculated by simply multiplying $$\mathbf x$$ by $$\mathbf e_5$$ with the [[wedge product]] to extract the round part of $$\mathbf x$$ as a flat geometry:

:$$\operatorname{car}(\mathbf x) = \mathbf x \wedge \mathbf e_5$$ .

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

{| class="wikitable"
! Type !! Definition !! Carrier
|-
| 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;" | $$\operatorname{car}(\mathbf a) = a_x \mathbf e_{15} + a_y \mathbf e_{25} + a_z \mathbf e_{35} + a_w \mathbf e_{45}$$
|-
| 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;" | $$\operatorname{car}(\mathbf d) = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$
|-
| 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;" | $$\operatorname{car}(\mathbf c) = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$
|-
| 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;" | $$\operatorname{car}(\mathbf s) = s_u {\large\unicode{x1d7d9}}$$
|}

== Anticarrier ==

The ''anticarrier'' of a round object is the carrier of its [[dual]]. The carrier of an object $$\mathbf x$$ is denoted by $$\operatorname{acr}(\mathbf x)$$, and it is calculated by

:$$\operatorname{acr}(\mathbf x) = \mathbf x^* \wedge \mathbf e_5$$ .

The anticarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and anticarrier can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{car}(\mathbf x) \vee \operatorname{acr}(\mathbf x)$$.

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

{| class="wikitable"
! Type !! Definition !! Anticarrier
|-
| 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;" | $$\operatorname{acr}(\mathbf a) = -a_w {\large\unicode{x1d7d9}}$$
|-
| 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;" | $$\operatorname{acr}(\mathbf d) = -d_{vx} \mathbf e_{4235} - d_{vy} \mathbf e_{4315} - d_{vz} \mathbf e_{4125} + d_{pw} \mathbf e_{3215}$$
|-
| 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;" | $$\operatorname{acr}(\mathbf c) = c_{gx} \mathbf e_{415} + c_{gy} \mathbf e_{425} + c_{gz} \mathbf e_{435} + c_{vx} \mathbf e_{235} + c_{vy} \mathbf e_{315} + c_{vz} \mathbf e_{125}$$
|-
| 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;" | $$\operatorname{acr}(\mathbf s) = -s_x \mathbf e_{15} - s_y \mathbf e_{25} - s_z \mathbf e_{35} + s_u \mathbf e_{45}$$
|}

== See Also ==

* [[Attitude]]
* [[Centers]]
* [[Containers]]
* [[Partners]]

← Older revision		Revision as of 23:40, 1 December 2023
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	\|}		\|}

	== ~~Anticarrier~~ ==		== Cocarrier ==

	The ''~~anticarrier~~'' of a round object is the carrier of its [[antidual]]. The ~~anticarrier~~ of an object $$\mathbf x$$ is denoted by $$\operatorname{~~acr~~}(\mathbf x)$$, and it is calculated by		The ''cocarrier'' of a round object is the carrier of its [[antidual]]. The cocarrier of an object $$\mathbf x$$ is denoted by $$\operatorname{ccr}(\mathbf x)$$, and it is calculated by

	:$$\operatorname{~~acr~~}(\mathbf x) = \mathbf x^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5$$ .		:$$\operatorname{ccr}(\mathbf x) = \mathbf x^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5$$ .

	The ~~anticarrier~~ is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and ~~anticarrier~~ can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{car}(\mathbf x) \vee \operatorname{~~acr~~}(\mathbf x)$$.		The cocarrier is perpendicular to the carrier, and it contains the center of the object. Thus, the meet of the carrier and cocarrier can be used to calculate the center of an object $$\mathbf x$$ as a [[flat point]] with the formula $$\operatorname{car}(\mathbf x) \vee \operatorname{ccr}(\mathbf x)$$.

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

	{\| class="wikitable"		{\| class="wikitable"
	! Type !! Definition !! ~~Anticarrier~~		! Type !! Definition !! Cocarrier
	\|-		\|-
	\| style="padding: 12px;" \| [[Round point]]		\| 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;" \| $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
	\| style="padding: 12px;" \| $$\operatorname{~~acr~~}(\mathbf a) = a_w {\large\unicode{x1d7d9}}$$		\| style="padding: 12px;" \| $$\operatorname{ccr}(\mathbf a) = a_w {\large\unicode{x1d7d9}}$$
	\|-		\|-
	\| 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;" \| $$\operatorname{~~acr~~}(\mathbf d) = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$		\| style="padding: 12px;" \| $$\operatorname{ccr}(\mathbf d) = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Circle]]		\| 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;" \| $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
	\| style="padding: 12px;" \| $$\operatorname{~~acr~~}(\mathbf c) = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$		\| style="padding: 12px;" \| $$\operatorname{ccr}(\mathbf c) = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$
	\|-		\|-
	\| 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;" \| $$\operatorname{~~acr~~}(\mathbf s) = s_x \mathbf e_{15} + s_y \mathbf e_{25} + s_z \mathbf e_{35} - s_u \mathbf e_{45}$$		\| style="padding: 12px;" \| $$\operatorname{ccr}(\mathbf s) = s_x \mathbf e_{15} + s_y \mathbf e_{25} + s_z \mathbf e_{35} - s_u \mathbf e_{45}$$
	\|}		\|}