Carriers: Difference between revisions

From Conformal Geometric Algebra
Jump to navigation Jump to search
No edit summary
 
(One intermediate revision by the same user not shown)
Line 1: Line 1:
== 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 $$\mathcal G_{4,1}$$.
Line 27: Line 27:
|}
|}


== 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 $$\mathbf u$$ is denoted by $$\operatorname{ccr}(\mathbf u)$$, and it is calculated by


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


The 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 $$\mathbf u$$ 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 $$\mathcal G_{4,1}$$.


{| 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;" | $$\operatorname{ccr}(\mathbf s) = s_x \mathbf e_{15} + s_y \mathbf e_{25} + s_z \mathbf e_{35} - s_u \mathbf e_{45}$$
|}
|}



Latest revision as of 22:57, 3 April 2024

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 $$\mathbf u$$ 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:

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

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

Type Definition Carrier
Round point $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ $$\operatorname{car}(\mathbf a) = a_x \mathbf e_{15} + a_y \mathbf e_{25} + a_z \mathbf e_{35} + a_w \mathbf e_{45}$$
Dipole $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ $$\operatorname{car}(\mathbf d) = d_{vx} \mathbf e_{415} + d_{vy} \mathbf e_{425} + d_{vz} \mathbf e_{435} + d_{mx} \mathbf e_{235} + d_{my} \mathbf e_{315} + d_{mz} \mathbf e_{125}$$
Circle $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ $$\operatorname{car}(\mathbf c) = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$
Sphere $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ $$\operatorname{car}(\mathbf s) = s_u {\large\unicode{x1d7d9}}$$

Cocarrier

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

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

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 $$\mathbf u$$ 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 $$\mathcal G_{4,1}$$.

Type Definition Cocarrier
Round point $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ $$\operatorname{ccr}(\mathbf a) = a_w {\large\unicode{x1d7d9}}$$
Dipole $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ $$\operatorname{ccr}(\mathbf d) = d_{vx} \mathbf e_{4235} + d_{vy} \mathbf e_{4315} + d_{vz} \mathbf e_{4125} - d_{pw} \mathbf e_{3215}$$
Circle $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ $$\operatorname{ccr}(\mathbf c) = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$
Sphere $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ $$\operatorname{ccr}(\mathbf s) = s_x \mathbf e_{15} + s_y \mathbf e_{25} + s_z \mathbf e_{35} - s_u \mathbf e_{45}$$

See Also