Centers: Difference between revisions

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The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:
The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:


:$$\operatorname{cen}(\mathbf x) = \operatorname{car}(\mathbf x^*) \vee \mathbf x$$ .
:$$\operatorname{cen}(\mathbf x) = \operatorname{car}(\mathbf x^\unicode["segoe ui symbol"]{x2605}) \vee \mathbf x$$ .


The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.
The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

Revision as of 01:00, 27 August 2023

The center of a round object (a round point, dipole, circle, or sphere) is the round point having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the meet of $$\mathbf x$$ and its own anticarrier:

$$\operatorname{cen}(\mathbf x) = \operatorname{car}(\mathbf x^\unicode["segoe ui symbol"]{x2605}) \vee \mathbf x$$ .

The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

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

Type Definition Center
Round point $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ $$\begin{split}\operatorname{cen}(\mathbf a) = {\phantom +}\,&a_xa_w \mathbf e_1 \\ +\,&a_ya_w \mathbf e_2 \\ +\,&a_za_w \mathbf e_3 \\ +\,&a_w^2 \mathbf e_4 \\ +\,&a_wa_u \mathbf e_5\end{split}$$
Dipole $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ $$\begin{split}\operatorname{cen}(\mathbf d) = {\phantom +}\,&(d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\,\mathbf e_1 \\ +\,&(d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\,\mathbf e_2 \\ +\,&(d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\,\mathbf e_3 \\ +\,&(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_4 \\ +\,&(d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\,\mathbf e_5\end{split}$$
Circle $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ $$\begin{split}\operatorname{cen}(\mathbf c) = {\phantom +}\,&(c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\,\mathbf e_1 \\ +\,&(c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\,\mathbf e_2 \\ +\,&(c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\,\mathbf e_3 \\ +\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\,\mathbf e_4 \\ +\,&(c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\,\mathbf e_5\end{split}$$
Sphere $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ $$\begin{split}\operatorname{cen}(\mathbf s) = -\,&s_xs_u \mathbf e_1 \\ -\,&s_ys_u \mathbf e_2 \\ -\,&s_zs_u \mathbf e_3 \\ +\,&s_u^2 \mathbf e_4 \\ +\,&(s_x^2 + s_y^2 + s_z^2 - s_ws_u)\,\mathbf e_5\end{split}$$

See Also