Containers
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The container of a round object (a round point, dipole, circle, or sphere) is the smallest sphere that contains it. The container of an object $$\mathbf u$$ is denoted by $$\operatorname{con}(\mathbf u)$$, and it is given by the expansion of $$\mathbf u$$ into its own carrier:
- $$\operatorname{con}(\mathbf u) = \mathbf u \wedge \operatorname{car}(\mathbf u)^\unicode["segoe ui symbol"]{x2606}$$ .
The squared radius of an object's container has the same sign as the squared radius of the object itself. That is, a real object has a real container, and an imaginary object has an imaginary container.
The following table lists the containers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.
| Type | Definition | Container |
|---|---|---|
| Round point | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ | $$\begin{split}\operatorname{con}(\mathbf a) = &-a_w^2 \mathbf e_{1234} \\ &+ a_xa_w \mathbf e_{4235} \\ &+ a_ya_w \mathbf e_{4315} \\ &+ a_za_w \mathbf e_{4125} \\ &+ (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}\end{split}$$ |
| Dipole | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ | $$\begin{split}\operatorname{con}(\mathbf d) &= (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_{1234} \\ &+ (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\,\mathbf e_{4235} \\ &+ (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\,\mathbf e_{4315} \\ &+ (d_{vy} d_{mx} - d_{vx} d_{my} - d_{vz} d_{pw})\,\mathbf e_{4125} \\ &+ (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\,\mathbf e_{3215}\end{split}$$ |
| Circle | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ | $$\begin{split}\operatorname{con}(\mathbf c) = -\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\,\mathbf e_{1234} \\ +\,&(c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\,\mathbf e_{4235} \\ +\,&(c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\,\mathbf e_{4315} \\ +\,&(c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\,\mathbf e_{4125} \\ +\,&(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\,\mathbf e_{3215}\end{split}$$ |
| Sphere | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ | $$\begin{split}\operatorname{con}(\mathbf s) &= s_u^2 \mathbf e_{1234} \\ &+ s_xs_u \mathbf e_{4235} \\ &+ s_ys_u \mathbf e_{4315} \\ &+ s_zs_u \mathbf e_{4125} \\ &+ s_ws_u \mathbf e_{3215}\end{split}$$ |