Carriers and Exterior products: Difference between pages

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(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...")
 
(Created page with "The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct. == Exterior Product == The following Cayley table shows the exterior products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$. 1440px == Exterior Anti...")
 
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== Carrier ==
The ''exterior product'' is the fundamental product of Grassmann Algebra, and it forms part of the [[geometric product]] in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct.


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:
== Exterior Product ==


:$$\operatorname{car}(\mathbf x) = \mathbf x \wedge \mathbf e_5$$ .
The following Cayley table shows the exterior products between all pairs of basis elements 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 $$\mathcal G_{4,1}$$.


{| class="wikitable"
[[Image:WedgeProduct.svg|1440px]]
! 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 ==
== Exterior Antiproduct ==


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 following Cayley table shows the exterior antiproducts between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.


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


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 $$\operatorname{car}(\mathbf x) \vee \operatorname{acr}(\mathbf x)$$.
[[Image:AntiwedgeProduct.svg|1440px]]


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


{| class="wikitable"
We can express the product and antiproduct in terms of each other through an analog of De Morgan's laws as follows.
! Type !! Definition !! Anticarrier
 
|-
:$$\overline{\mathbf a \wedge \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \vee \overline{\mathbf b}$$
| style="padding: 12px;" | [[Round point]]
 
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
:$$\overline{\mathbf a \vee \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \wedge \overline{\mathbf b}$$
| style="padding: 12px;" | $$\operatorname{acr}(\mathbf a) = -a_w {\large\unicode{x1d7d9}}$$
 
|-
:$$\underline{\mathbf a \wedge \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \vee \underline{\mathbf b}$$
| style="padding: 12px;" | [[Dipole]]
 
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
:$$\underline{\mathbf a \vee \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \wedge \underline{\mathbf b}$$
| 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 ==
== See Also ==


* [[Attitude]]
* [[Geometric products]]
* [[Centers]]
* [[Dot products]]
* [[Containers]]
* [[Duals]]
* [[Partners]]

Revision as of 03:18, 6 August 2023

The exterior product is the fundamental product of Grassmann Algebra, and it forms part of the geometric product in geometric algebra. There are two products with symmetric properties called the exterior product and exterior antiproduct.

Exterior Product

The following Cayley table shows the exterior products between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.


Exterior Antiproduct

The following Cayley table shows the exterior antiproducts between all pairs of basis elements in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.


De Morgan Laws

We can express the product and antiproduct in terms of each other through an analog of De Morgan's laws as follows.

$$\overline{\mathbf a \wedge \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \vee \overline{\mathbf b}$$
$$\overline{\mathbf a \vee \mathbf b} = \overline{\mathbf{a\vphantom{b}}} \wedge \overline{\mathbf b}$$
$$\underline{\mathbf a \wedge \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \vee \underline{\mathbf b}$$
$$\underline{\mathbf a \vee \mathbf b} = \underline{\mathbf{a\vphantom{b}}} \wedge \underline{\mathbf b}$$

See Also

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