Containers: Difference between revisions

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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 $$\mathbf x$$ is denoted by $$\operatorname{con}(\mathbf x)$$, and it is given by the [[connect]] of $$\mathbf x$$ with its own [[carrier]]:
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]]:


:$$\operatorname{con}(\mathbf x) = -\operatorname{car}(\mathbf x)^* \wedge \mathbf x$$ .
:$$\operatorname{con}(\mathbf u) = \mathbf u \wedge \operatorname{car}(\mathbf u)^\unicode["segoe ui symbol"]{x2606}$$ .


(The negative sign is not strictly necessary, but is included so the function always produces a sphere having a negative $$\mathbf e_{1234}$$ component.) 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 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 $$\mathcal G_{4,1}$$.
The following table lists the containers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.
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| 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;" | $$\begin{split}\operatorname{con}(\mathbf d) =
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf d)
&-(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_{1234} \\
&= (d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_{1234} \\
&+ (d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\,\mathbf e_{4235} \\
&+ (d_{vz} d_{my} - d_{vy} d_{mz} - d_{vx} d_{pw})\,\mathbf e_{4235} \\
&+ (d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\,\mathbf e_{4315} \\
&+ (d_{vx} d_{mz} - d_{vz} d_{mx} - d_{vy} d_{pw})\,\mathbf e_{4315} \\
&+ (d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\,\mathbf e_{4125} \\
&+ (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}$$
&+ (d_{mx}^2 + d_{my}^2 + d_{mz}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})\,\mathbf e_{3215}\end{split}$$
|-
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | [[Circle]]
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| 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;" | $$\begin{split}\operatorname{con}(\mathbf s) =
| style="padding: 12px;" | $$\begin{split}\operatorname{con}(\mathbf s)
&-s_u^2 \mathbf e_{1234} \\
&= s_u^2 \mathbf e_{1234} \\
&- s_xs_u \mathbf e_{4235} \\
&+ s_xs_u \mathbf e_{4235} \\
&- s_ys_u \mathbf e_{4315} \\
&+ s_ys_u \mathbf e_{4315} \\
&- s_zs_u \mathbf e_{4125} \\
&+ s_zs_u \mathbf e_{4125} \\
&- s_ws_u \mathbf e_{3215}\end{split}$$
&+ s_ws_u \mathbf e_{3215}\end{split}$$
|}
|}



Latest revision as of 22:59, 3 April 2024

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 $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ $$\begin{split}\operatorname{con}(\mathbf s) &= s_u^2 \mathbf e_{1234} \\ &+ s_xs_u \mathbf e_{4235} \\ &+ s_ys_u \mathbf e_{4315} \\ &+ s_zs_u \mathbf e_{4125} \\ &+ s_ws_u \mathbf e_{3215}\end{split}$$

See Also