Centers

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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 cocarrier:

$$\operatorname{cen}(\mathbf x) = \operatorname{ccr}(\mathbf x) \vee \mathbf x$$ .

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