Bulk and weight - Revision history

Eric Lengyel at 01:16, 8 July 2024

2024-07-08T01:16:10Z

Eric Lengyel at 23:32, 13 April 2024

2024-04-13T23:32:00Z

← Older revision		Revision as of 23:32, 13 April 2024
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	\| style="padding: 12px;" \| $$\mathbf F_\unicode{x25CB} = F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412}$$		\| style="padding: 12px;" \| $$\mathbf F_\unicode{x25CB} = F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412}$$
	\|}		\|}

			== In the Book ==

			* Bulk and weight are introduced in Section 2.8.3.

	== See Also ==		== See Also ==
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	* [[Complements]]		* [[Complements]]
	* [[Duals]]		* [[Duals]]

	~~== In the Book ==~~

	* Bulk and weight are introduced in Section 2.8.3.

Eric Lengyel at 05:08, 13 April 2024

2024-04-13T05:08:37Z

← Older revision		Revision as of 05:08, 13 April 2024
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	* [[Complements]]		* [[Complements]]
	* [[Duals]]		* [[Duals]]

			== In the Book ==

			* Bulk and weight are introduced in Section 2.8.3.

Eric Lengyel at 01:54, 13 April 2024

2024-04-13T01:54:30Z

← Older revision		Revision as of 01:54, 13 April 2024
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	:$$\mathbf u_\unicode{x25CF} = \mathbf G \mathbf u$$,		:$$\mathbf u_\unicode{x25CF} = \mathbf G \mathbf u$$,

	where $$\mathbf G$$ is the metric exomorphism matrix. The bulk consists of the components of $$\mathbf u$$ that do not have the projective basis vector $$\mathbf e_4$$ as a factor.		where $$\mathbf G$$ is the [[metric exomorphism matrix]]. The bulk consists of the components of $$\mathbf u$$ that do not have the projective basis vector $$\mathbf e_4$$ as a factor.

	The weight is denoted by $$\mathbf u_\unicode{x25CB}$$, and it is defined as		The weight is denoted by $$\mathbf u_\unicode{x25CB}$$, and it is defined as
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	:$$\mathbf u_\unicode{x25CB} = \mathbb G \mathbf u$$,		:$$\mathbf u_\unicode{x25CB} = \mathbb G \mathbf u$$,

	where $$\mathbb G$$ is the metric ~~anti-exomorphism~~ matrix. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.		where $$\mathbb G$$ is the [[metric antiexomorphism matrix]]. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.

	The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.		The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.

Eric Lengyel: /* See Also */

2024-04-12T06:27:02Z

Eric Lengyel at 07:15, 23 October 2023

2023-10-23T07:15:52Z

← Older revision		Revision as of 07:15, 23 October 2023
Line 11:		Line 11:
	:$$\mathbf u_\unicode{x25CB} = \mathbb G \mathbf u$$,		:$$\mathbf u_\unicode{x25CB} = \mathbb G \mathbf u$$,

	where $$\mathbb G$$ is the ~~antimetric~~ anti-exomorphism matrix. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.		where $$\mathbb G$$ is the metric anti-exomorphism matrix. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.

	The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.		The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.

Eric Lengyel at 07:15, 23 October 2023

2023-10-23T07:15:32Z

← Older revision		Revision as of 07:15, 23 October 2023
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	The ~~components of an element~~ of a rigid geometric algebra ~~can be divided~~ into two groups called the ''bulk'' and the ''weight'' ~~of the element~~. The bulk of an element $$\mathbf x$$ is denoted by $$\mathbf x_\unicode{x25CF}$$, and it consists of the components of $$\mathbf x$$ that do not have the projective basis vector as a factor. The weight is denoted by $$\mathbf x_\unicode{x25CB}$$, and it ~~consists of the components that do have the projective basis vector~~ as ~~a factor. In the 4D rigid geometric algebra~~ $$\~~mathcal G_~~{~~3,0,1~~}$$, the ~~bulk is thus all~~ components ~~that do not contain the factor~~ $$\mathbf ~~e_4~~$$~~, and the weight is all components~~ that do ~~contain~~ the ~~factor~~ $$\mathbf e_4$$.		The degenerate metric of rigid geometric algebra naturally divides the components of every quantity into two groups called the ''bulk'' and the ''weight''.

			The bulk of an element $$\mathbf u$$ is denoted by $$\mathbf u_\unicode{x25CF}$$, and it is defined as

			:$$\mathbf u_\unicode{x25CF} = \mathbf G \mathbf u$$,

			where $$\mathbf G$$ is the metric exomorphism matrix. The bulk consists of the components of $$\mathbf u$$ that do not have the projective basis vector $$\mathbf e_4$$ as a factor.

			The weight is denoted by $$\mathbf u_\unicode{x25CB}$$, and it is defined as

			:$$\mathbf u_\unicode{x25CB} = \mathbb G \mathbf u$$,

			where $$\mathbb G$$ is the antimetric anti-exomorphism matrix. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.

	The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.		The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.

Eric Lengyel at 05:33, 15 July 2023

2023-07-15T05:33:25Z

← Older revision		Revision as of 05:33, 15 July 2023
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			The components of an element of a rigid geometric algebra can be divided into two groups called the ''bulk'' and the ''weight'' of the element. The bulk of an element $$\mathbf x$$ is denoted by $$\mathbf x_\unicode{x25CF}$$, and it consists of the components of $$\mathbf x$$ that do not have the projective basis vector as a factor. The weight is denoted by $$\mathbf x_\unicode{x25CB}$$, and it consists of the components that do have the projective basis vector as a factor. In the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$, the bulk is thus all components that do not contain the factor $$\mathbf e_4$$, and the weight is all components that do contain the factor $$\mathbf e_4$$.

	The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.		The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.

Eric Lengyel: Created page with "The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon. An element is unitized when the magnitude of its weight is one. The following table lists the bulk and weight for the main types in the 4D rigid geometric algebra $$\mathcal..."

2023-07-15T05:31:09Z

Created page with "The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon. An element is unitized when the magnitude of its weight is one. The following table lists the bulk and weight for the main types in the 4D rigid geometric algebra $$\mathcal..."

New page

The bulk generally contains information about the position of an element relative to the origin, and the weight generally contains information about the attitude and orientation of an element. An object with zero bulk contains the origin. An object with zero weight is contained by the horizon.

An element is [[unitized]] when the magnitude of its weight is one.

The following table lists the bulk and weight for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

{| class="wikitable"
! Type !! Definition !! Bulk !! Weight
|-
| style="padding: 12px;" | [[Magnitude]]
| style="padding: 12px;" | $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
| style="padding: 12px;" | $$\mathbf z_\unicode{x25CF} = x \mathbf 1$$
| style="padding: 12px;" | $$\mathbf z_\unicode{x25CB} = y {\large\unicode{x1d7d9}}$$
|-
| style="padding: 12px;" | [[Point]]
| style="padding: 12px;" | $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
| style="padding: 12px;" | $$\mathbf p_\unicode{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
| style="padding: 12px;" | $$\mathbf p_\unicode{x25CB} = p_w \mathbf e_4$$
|-
| style="padding: 12px;" | [[Line]]
| style="padding: 12px;" | $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
| style="padding: 12px;" | $$\boldsymbol l_\unicode{x25CF} = l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
| style="padding: 12px;" | $$\boldsymbol l_\unicode{x25CB} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}$$
|-
| style="padding: 12px;" | [[Plane]]
| style="padding: 12px;" | $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf g_\unicode{x25CF} = g_w \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf g_\unicode{x25CB} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}$$
|-
| style="padding: 12px;" | [[Motor]]
| style="padding: 12px;" | $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
| style="padding: 12px;" | $$\mathbf Q_\unicode{x25CF} = Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
| style="padding: 12px;" | $$\mathbf Q_\unicode{x25CB} = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}}$$
|-
| style="padding: 12px;" | [[Flector]]
| style="padding: 12px;" | $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf F_\unicode{x25CF} = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{gw} \mathbf e_{321}$$
| style="padding: 12px;" | $$\mathbf F_\unicode{x25CB} = F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412}$$
|}

== See Also ==

* [[Attitude]]
* [[Geometric norm]]
* [[Unitization]]
* [[Complements]]

← Older revision		Revision as of 01:16, 8 July 2024
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	The degenerate metric of rigid geometric algebra naturally divides the components of every quantity into two groups called the ''bulk'' and the ''weight''.		The degenerate metric of rigid geometric algebra naturally divides the components of every quantity into two groups called the ''bulk'' and the ''weight''.

	The bulk of an element $$\mathbf u$$ is denoted by $$\mathbf u_\unicode{x25CF}$$, and it is defined as		The bulk of an element $$\mathbf u$$ is denoted by $$\mathbf u_\unicode["segoe ui symbol"]{x25CF}$$, and it is defined as

	:$$\mathbf u_\unicode{x25CF} = \mathbf G \mathbf u$$,		:$$\mathbf u_\unicode["segoe ui symbol"]{x25CF} = \mathbf G \mathbf u$$,

	where $$\mathbf G$$ is the [[metric exomorphism matrix]]. The bulk consists of the components of $$\mathbf u$$ that do not have the projective basis vector $$\mathbf e_4$$ as a factor.		where $$\mathbf G$$ is the [[metric exomorphism matrix]]. The bulk consists of the components of $$\mathbf u$$ that do not have the projective basis vector $$\mathbf e_4$$ as a factor.

	The weight is denoted by $$\mathbf u_\unicode{x25CB}$$, and it is defined as		The weight is denoted by $$\mathbf u_\unicode["segoe ui symbol"]{x25CB}$$, and it is defined as

	:$$\mathbf u_\unicode{x25CB} = \mathbb G \mathbf u$$,		:$$\mathbf u_\unicode["segoe ui symbol"]{x25CB} = \mathbb G \mathbf u$$,

	where $$\mathbb G$$ is the [[metric antiexomorphism matrix]]. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.		where $$\mathbb G$$ is the [[metric antiexomorphism matrix]]. The weight consists of the components of $$\mathbf u$$ that do have the projective basis vector $$\mathbf e_4$$ as a factor.
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	\| style="padding: 12px;" \| [[Magnitude]]		\| style="padding: 12px;" \| [[Magnitude]]
	\| style="padding: 12px;" \| $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$		\| style="padding: 12px;" \| $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$
	\| style="padding: 12px;" \| $$\mathbf z_\unicode{x25CF} = x \mathbf 1$$		\| style="padding: 12px;" \| $$\mathbf z_\unicode["segoe ui symbol"]{x25CF} = x \mathbf 1$$
	\| style="padding: 12px;" \| $$\mathbf z_\unicode{x25CB} = y {\large\unicode{x1d7d9}}$$		\| style="padding: 12px;" \| $$\mathbf z_\unicode["segoe ui symbol"]{x25CB} = y {\large\unicode{x1d7d9}}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Point]]		\| style="padding: 12px;" \| [[Point]]
	\| style="padding: 12px;" \| $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$		\| style="padding: 12px;" \| $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
	\| style="padding: 12px;" \| $$\mathbf p_\unicode{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$		\| style="padding: 12px;" \| $$\mathbf p_\unicode["segoe ui symbol"]{x25CF} = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3$$
	\| style="padding: 12px;" \| $$\mathbf p_\unicode{x25CB} = p_w \mathbf e_4$$		\| style="padding: 12px;" \| $$\mathbf p_\unicode["segoe ui symbol"]{x25CB} = p_w \mathbf e_4$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Line]]		\| style="padding: 12px;" \| [[Line]]
	\| style="padding: 12px;" \| $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$		\| style="padding: 12px;" \| $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
	\| style="padding: 12px;" \| $$\boldsymbol l_\unicode{x25CF} = l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$		\| style="padding: 12px;" \| $$\boldsymbol l_\unicode["segoe ui symbol"]{x25CF} = l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
	\| style="padding: 12px;" \| $$\boldsymbol l_\unicode{x25CB} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}$$		\| style="padding: 12px;" \| $$\boldsymbol l_\unicode["segoe ui symbol"]{x25CB} = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Plane]]		\| style="padding: 12px;" \| [[Plane]]
	\| style="padding: 12px;" \| $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$		\| style="padding: 12px;" \| $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\mathbf g_\unicode{x25CF} = g_w \mathbf e_{321}$$		\| style="padding: 12px;" \| $$\mathbf g_\unicode["segoe ui symbol"]{x25CF} = g_w \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\mathbf g_\unicode{x25CB} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}$$		\| style="padding: 12px;" \| $$\mathbf g_\unicode["segoe ui symbol"]{x25CB} = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Motor]]		\| style="padding: 12px;" \| [[Motor]]
	\| style="padding: 12px;" \| $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$		\| style="padding: 12px;" \| $$\mathbf Q = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}} + Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
	\| style="padding: 12px;" \| $$\mathbf Q_\unicode{x25CF} = Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$		\| style="padding: 12px;" \| $$\mathbf Q_\unicode["segoe ui symbol"]{x25CF} = Q_{mx} \mathbf e_{23} + Q_{my} \mathbf e_{31} + Q_{mz} \mathbf e_{12} + Q_{mw} \mathbf 1$$
	\| style="padding: 12px;" \| $$\mathbf Q_\unicode{x25CB} = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}}$$		\| style="padding: 12px;" \| $$\mathbf Q_\unicode["segoe ui symbol"]{x25CB} = Q_{vx} \mathbf e_{41} + Q_{vy} \mathbf e_{42} + Q_{vz} \mathbf e_{43} + Q_{vw} {\large\unicode{x1d7d9}}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Flector]]		\| style="padding: 12px;" \| [[Flector]]
	\| style="padding: 12px;" \| $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$		\| style="padding: 12px;" \| $$\mathbf F = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412} + F_{gw} \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\mathbf F_\unicode{x25CF} = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{gw} \mathbf e_{321}$$		\| style="padding: 12px;" \| $$\mathbf F_\unicode["segoe ui symbol"]{x25CF} = F_{px} \mathbf e_1 + F_{py} \mathbf e_2 + F_{pz} \mathbf e_3 + F_{gw} \mathbf e_{321}$$
	\| style="padding: 12px;" \| $$\mathbf F_\unicode{x25CB} = F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412}$$		\| style="padding: 12px;" \| $$\mathbf F_\unicode["segoe ui symbol"]{x25CB} = F_{pw} \mathbf e_4 + F_{gx} \mathbf e_{423} + F_{gy} \mathbf e_{431} + F_{gz} \mathbf e_{412}$$
	\|}		\|}

← Older revision		Revision as of 06:27, 12 April 2024
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	* [[Unitization]]		* [[Unitization]]
	* [[Complements]]		* [[Complements]]
			* [[Duals]]