Carriers
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 | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ | $$\operatorname{ccr}(\mathbf s) = s_x \mathbf e_{15} + s_y \mathbf e_{25} + s_z \mathbf e_{35} - s_u \mathbf e_{45}$$ |