Duals - Revision history

Eric Lengyel at 07:16, 23 October 2023

2023-10-23T07:16:11Z

← Older revision		Revision as of 07:16, 23 October 2023
Line 9:		Line 9:
	:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,		:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,

	where $$\mathbb G$$ is the ~~antimetric~~ anti-exomorphism matrix. In conformal geometric algebra, it is always the case that $$\mathbf u^\unicode["segoe ui symbol"]{x2606} = -\mathbf u^\unicode["segoe ui symbol"]{x2605}$$, so the two duals differ only in sign.		where $$\mathbb G$$ is the metric anti-exomorphism matrix. In conformal geometric algebra, it is always the case that $$\mathbf u^\unicode["segoe ui symbol"]{x2606} = -\mathbf u^\unicode["segoe ui symbol"]{x2605}$$, so the two duals differ only in sign.

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

Eric Lengyel at 03:28, 23 October 2023

2023-10-23T03:28:01Z

← Older revision		Revision as of 03:28, 23 October 2023
Line 4:		Line 4:

	where $$\mathbf G$$ is the metric exomorphism matrix, and the overline is the complement operation.		where $$\mathbf G$$ is the metric exomorphism matrix, and the overline is the complement operation.

			The ''antidual'' of an object $$\mathbf u$$, denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$, is given by

			:$$\mathbf u^\unicode["segoe ui symbol"]{x2606} = \overline{\mathbb G \mathbf u}$$ ,

			where $$\mathbb G$$ is the antimetric anti-exomorphism matrix. In conformal geometric algebra, it is always the case that $$\mathbf u^\unicode["segoe ui symbol"]{x2606} = -\mathbf u^\unicode["segoe ui symbol"]{x2605}$$, so the two duals differ only in sign.

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

Eric Lengyel at 03:21, 23 October 2023

2023-10-23T03:21:16Z

← Older revision		Revision as of 03:21, 23 October 2023
Line 1:		Line 1:
	The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^\unicode["segoe ui symbol"]{x2605}$$, is given by		The ''dual'' of an object $$\mathbf u$$, denoted by $$\mathbf u^\unicode["segoe ui symbol"]{x2605}$$, is given by

	:$$\mathbf x^\unicode["segoe ui symbol"]{x2605} = \mathbf G ~~\underline{~~\mathbf x}$$ ,		:$$\mathbf u^\unicode["segoe ui symbol"]{x2605} = \overline{\mathbf G \mathbf u}$$ ,

	where $$\mathbf G$$ is the ~~extended~~ metric ~~tensor~~, and ~~$$\underline{\mathbf x}$$~~ is the ~~left~~ complement ~~of $$\mathbf x$$~~.		where $$\mathbf G$$ is the metric exomorphism matrix, and the overline is the complement operation.

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

Eric Lengyel at 00:59, 27 August 2023

2023-08-27T00:59:37Z

Eric Lengyel at 21:05, 25 August 2023

2023-08-25T21:05:55Z

Eric Lengyel: Created page with "The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^$$, is given by :$$\mathbf x^ = \mathbf{\tilde x} \mathbin{\unicode{x27D1}} {\large\unicode{x1d7d9}}$$ . The following table lists the duals for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. {| class="wikitable" ! Type !! Definition !! Dual |- | style="padding: 12px;" | Flat point | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p..."

2023-08-06T03:16:57Z

Created page with "The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^*$$, is given by :$$\mathbf x^* = \mathbf{\tilde x} \mathbin{\unicode{x27D1}} {\large\unicode{x1d7d9}}$$ . The following table lists the duals for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. {| class="wikitable" ! Type !! Definition !! Dual |- | style="padding: 12px;" | Flat point | style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p..."

New page

The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^*$$, is given by

:$$\mathbf x^* = \mathbf{\tilde x} \mathbin{\unicode{x27D1}} {\large\unicode{x1d7d9}}$$ .

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

{| class="wikitable"
! Type !! Definition !! Dual
|-
| style="padding: 12px;" | [[Flat point]]
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$
| style="padding: 12px;" | $$\mathbf p^* = p_w \mathbf e_{321} - p_x \mathbf e_{235} - p_y \mathbf e_{315} - p_z \mathbf e_{125}$$
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\boldsymbol l^* = l_{vx} \mathbf e_{23} + l_{vy} \mathbf e_{31} + l_{vz} \mathbf e_{12} + l_{mx} \mathbf e_{15} + l_{my} \mathbf e_{25} + l_{mz} \mathbf e_{35}$$
|-
| style="padding: 12px;" | [[Plane]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\mathbf g^* = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3 + g_w \mathbf e_5$$
|-
| 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;" | $$\mathbf a^* = -a_w \mathbf e_{1234} + a_x \mathbf e_{4235} + a_y \mathbf e_{4315} + a_z \mathbf e_{4125} - a_u \mathbf e_{3215}$$
|-
| 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;" | $$\mathbf d^* = -d_{vx} \mathbf e_{423} - d_{vy} \mathbf e_{431} - d_{vz} \mathbf e_{412} + d_{pw} \mathbf e_{321} - d_{mx} \mathbf e_{415} - d_{my} \mathbf e_{425} - d_{mz} \mathbf e_{435} - d_{px} \mathbf e_{235} - d_{py} \mathbf e_{315} - d_{pz} \mathbf e_{125}$$
|-
| 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;" | $$\mathbf c^* = c_{gx} \mathbf e_{41} + c_{gy} \mathbf e_{42} + c_{gz} \mathbf e_{43} + c_{vx} \mathbf e_{23} + c_{vy} \mathbf e_{31} + c_{vz} \mathbf e_{12} + c_{mx} \mathbf e_{15} + c_{my} \mathbf e_{25} + c_{mz} \mathbf e_{35} - c_{gw} \mathbf e_{45}$$
|-
| 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;" | $$\mathbf s^* = -s_x \mathbf e_1 - s_y \mathbf e_2 - s_z \mathbf e_3 + s_u \mathbf e_4 + s_w \mathbf e_5$$
|}

== See Also ==

* [[Partners]]

← Older revision		Revision as of 00:59, 27 August 2023
Line 1:		Line 1:
	The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^*$$, is given by		The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^\unicode["segoe ui symbol"]{x2605}$$, is given by

	:$$\mathbf x^* = \mathbf G \underline{\mathbf x}$$ ,		:$$\mathbf x^\unicode["segoe ui symbol"]{x2605} = \mathbf G \underline{\mathbf x}$$ ,

	where $$\mathbf G$$ is the extended metric tensor, and $$\underline{\mathbf x}$$ is the left complement of $$\mathbf x$$.		where $$\mathbf G$$ is the extended metric tensor, and $$\underline{\mathbf x}$$ is the left complement of $$\mathbf x$$.
Line 12:		Line 12:
	\| style="padding: 12px;" \| [[Flat point]]		\| style="padding: 12px;" \| [[Flat point]]
	\| style="padding: 12px;" \| $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$		\| style="padding: 12px;" \| $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$
	\| style="padding: 12px;" \| $$\mathbf p^* = -p_w \mathbf e_{321} + p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125}$$		\| style="padding: 12px;" \| $$\mathbf p^\unicode["segoe ui symbol"]{x2605} = -p_w \mathbf e_{321} + p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Line]]		\| style="padding: 12px;" \| [[Line]]
	\| style="padding: 12px;" \| $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$		\| style="padding: 12px;" \| $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$
	\| style="padding: 12px;" \| $$\boldsymbol l^* = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12} - l_{mx} \mathbf e_{15} - l_{my} \mathbf e_{25} - l_{mz} \mathbf e_{35}$$		\| style="padding: 12px;" \| $$\boldsymbol l^\unicode["segoe ui symbol"]{x2605} = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12} - l_{mx} \mathbf e_{15} - l_{my} \mathbf e_{25} - l_{mz} \mathbf e_{35}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Plane]]		\| style="padding: 12px;" \| [[Plane]]
	\| style="padding: 12px;" \| $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$		\| style="padding: 12px;" \| $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$
	\| style="padding: 12px;" \| $$\mathbf g^* = g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3 - g_w \mathbf e_5$$		\| style="padding: 12px;" \| $$\mathbf g^\unicode["segoe ui symbol"]{x2605} = g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3 - g_w \mathbf e_5$$
	\|-		\|-
	\| 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;" \| $$\mathbf a^* = a_w \mathbf e_{1234} - a_x \mathbf e_{4235} - a_y \mathbf e_{4315} - a_z \mathbf e_{4125} + a_u \mathbf e_{3215}$$		\| style="padding: 12px;" \| $$\mathbf a^\unicode["segoe ui symbol"]{x2605} = a_w \mathbf e_{1234} - a_x \mathbf e_{4235} - a_y \mathbf e_{4315} - a_z \mathbf e_{4125} + a_u \mathbf e_{3215}$$
	\|-		\|-
	\| 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;" \| $$\mathbf d^* = d_{vx} \mathbf e_{423} + d_{vy} \mathbf e_{431} + d_{vz} \mathbf e_{412} - d_{pw} \mathbf e_{321} + d_{mx} \mathbf e_{415} + d_{my} \mathbf e_{425} + d_{mz} \mathbf e_{435} + d_{px} \mathbf e_{235} + d_{py} \mathbf e_{315} + d_{pz} \mathbf e_{125}$$		\| style="padding: 12px;" \| $$\mathbf d^\unicode["segoe ui symbol"]{x2605} = d_{vx} \mathbf e_{423} + d_{vy} \mathbf e_{431} + d_{vz} \mathbf e_{412} - d_{pw} \mathbf e_{321} + d_{mx} \mathbf e_{415} + d_{my} \mathbf e_{425} + d_{mz} \mathbf e_{435} + d_{px} \mathbf e_{235} + d_{py} \mathbf e_{315} + d_{pz} \mathbf e_{125}$$
	\|-		\|-
	\| 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;" \| $$\mathbf c^* = -c_{gx} \mathbf e_{41} - c_{gy} \mathbf e_{42} - c_{gz} \mathbf e_{43} - c_{vx} \mathbf e_{23} - c_{vy} \mathbf e_{31} - c_{vz} \mathbf e_{12} - c_{mx} \mathbf e_{15} - c_{my} \mathbf e_{25} - c_{mz} \mathbf e_{35} + c_{gw} \mathbf e_{45}$$		\| style="padding: 12px;" \| $$\mathbf c^\unicode["segoe ui symbol"]{x2605} = -c_{gx} \mathbf e_{41} - c_{gy} \mathbf e_{42} - c_{gz} \mathbf e_{43} - c_{vx} \mathbf e_{23} - c_{vy} \mathbf e_{31} - c_{vz} \mathbf e_{12} - c_{mx} \mathbf e_{15} - c_{my} \mathbf e_{25} - c_{mz} \mathbf e_{35} + c_{gw} \mathbf e_{45}$$
	\|-		\|-
	\| 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;" \| $$\mathbf s^* = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 - s_u \mathbf e_4 - s_w \mathbf e_5$$		\| style="padding: 12px;" \| $$\mathbf s^\unicode["segoe ui symbol"]{x2605} = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 - s_u \mathbf e_4 - s_w \mathbf e_5$$
	\|}		\|}

← Older revision		Revision as of 21:05, 25 August 2023
Line 1:		Line 1:
	The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^*$$, is given by		The ''dual'' of an object $$\mathbf x$$, denoted by $$\mathbf x^*$$, is given by

	:$$\mathbf x^* = \mathbf{\~~tilde~~ x} \~~mathbin~~{\~~unicode{x27D1~~}~~} {~~\~~large\unicode{x1d7d9}}~~$$ .		:$$\mathbf x^* = \mathbf G \underline{\mathbf x}$$ ,

			where $$\mathbf G$$ is the extended metric tensor, and $$\underline{\mathbf x}$$ is the left complement of $$\mathbf x$$.

	The following table lists the duals for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.		The following table lists the duals for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.
Line 10:		Line 12:
	\| style="padding: 12px;" \| [[Flat point]]		\| style="padding: 12px;" \| [[Flat point]]
	\| style="padding: 12px;" \| $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$		\| style="padding: 12px;" \| $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$
	\| style="padding: 12px;" \| $$\mathbf p^* = p_w \mathbf e_{321} - p_x \mathbf e_{235} - p_y \mathbf e_{315} - p_z \mathbf e_{125}$$		\| style="padding: 12px;" \| $$\mathbf p^* = -p_w \mathbf e_{321} + p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Line]]		\| style="padding: 12px;" \| [[Line]]
	\| style="padding: 12px;" \| $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$		\| style="padding: 12px;" \| $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$
	\| style="padding: 12px;" \| $$\boldsymbol l^* = l_{vx} \mathbf e_{23} + l_{vy} \mathbf e_{31} + l_{vz} \mathbf e_{12} + l_{mx} \mathbf e_{15} + l_{my} \mathbf e_{25} + l_{mz} \mathbf e_{35}$$		\| style="padding: 12px;" \| $$\boldsymbol l^* = -l_{vx} \mathbf e_{23} - l_{vy} \mathbf e_{31} - l_{vz} \mathbf e_{12} - l_{mx} \mathbf e_{15} - l_{my} \mathbf e_{25} - l_{mz} \mathbf e_{35}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Plane]]		\| style="padding: 12px;" \| [[Plane]]
	\| style="padding: 12px;" \| $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$		\| style="padding: 12px;" \| $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$
	\| style="padding: 12px;" \| $$\mathbf g^* = -g_x \mathbf e_1 - g_y \mathbf e_2 - g_z \mathbf e_3 + g_w \mathbf e_5$$		\| style="padding: 12px;" \| $$\mathbf g^* = g_x \mathbf e_1 + g_y \mathbf e_2 + g_z \mathbf e_3 - g_w \mathbf e_5$$
	\|-		\|-
	\| 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;" \| $$\mathbf a^* = -a_w \mathbf e_{1234} + a_x \mathbf e_{4235} + a_y \mathbf e_{4315} + a_z \mathbf e_{4125} - a_u \mathbf e_{3215}$$		\| style="padding: 12px;" \| $$\mathbf a^* = a_w \mathbf e_{1234} - a_x \mathbf e_{4235} - a_y \mathbf e_{4315} - a_z \mathbf e_{4125} + a_u \mathbf e_{3215}$$
	\|-		\|-
	\| 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;" \| $$\mathbf d^* = -d_{vx} \mathbf e_{423} - d_{vy} \mathbf e_{431} - d_{vz} \mathbf e_{412} + d_{pw} \mathbf e_{321} - d_{mx} \mathbf e_{415} - d_{my} \mathbf e_{425} - d_{mz} \mathbf e_{435} - d_{px} \mathbf e_{235} - d_{py} \mathbf e_{315} - d_{pz} \mathbf e_{125}$$		\| style="padding: 12px;" \| $$\mathbf d^* = d_{vx} \mathbf e_{423} + d_{vy} \mathbf e_{431} + d_{vz} \mathbf e_{412} - d_{pw} \mathbf e_{321} + d_{mx} \mathbf e_{415} + d_{my} \mathbf e_{425} + d_{mz} \mathbf e_{435} + d_{px} \mathbf e_{235} + d_{py} \mathbf e_{315} + d_{pz} \mathbf e_{125}$$
	\|-		\|-
	\| 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;" \| $$\mathbf c^* = c_{gx} \mathbf e_{41} + c_{gy} \mathbf e_{42} + c_{gz} \mathbf e_{43} + c_{vx} \mathbf e_{23} + c_{vy} \mathbf e_{31} + c_{vz} \mathbf e_{12} + c_{mx} \mathbf e_{15} + c_{my} \mathbf e_{25} + c_{mz} \mathbf e_{35} - c_{gw} \mathbf e_{45}$$		\| style="padding: 12px;" \| $$\mathbf c^* = -c_{gx} \mathbf e_{41} - c_{gy} \mathbf e_{42} - c_{gz} \mathbf e_{43} - c_{vx} \mathbf e_{23} - c_{vy} \mathbf e_{31} - c_{vz} \mathbf e_{12} - c_{mx} \mathbf e_{15} - c_{my} \mathbf e_{25} - c_{mz} \mathbf e_{35} + c_{gw} \mathbf e_{45}$$
	\|-		\|-
	\| 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;" \| $$\mathbf s^* = -s_x \mathbf e_1 - s_y \mathbf e_2 - s_z \mathbf e_3 + s_u \mathbf e_4 + s_w \mathbf e_5$$		\| style="padding: 12px;" \| $$\mathbf s^* = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 - s_u \mathbf e_4 - s_w \mathbf e_5$$
	\|}		\|}