Partners
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The partner of a round object (a round point, dipole, circle, or sphere) is the round object having the same center, same carrier, and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf u$$ is denoted by $$\operatorname{par}(\mathbf u)$$, and it is given by the meet of the container of $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and the carrier of $$\mathbf u$$:
- $$\operatorname{par}(\mathbf u) = (-1)^{\operatorname{gr}(\mathbf u) + 1}\operatorname{con}(\mathbf u^\unicode["segoe ui symbol"]{x2606}) \vee \operatorname{car}(\mathbf u)$$ .
The dot product between a round object and its partner is always zero. They are orthogonal:
- $$\mathbf u \mathbin{\unicode{x25CF}} \operatorname{par}(\mathbf u) = 0$$ .
The following table lists the partners for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.
| Type | Definition | Partner |
|---|---|---|
| Round point | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ | $$\begin{split}\operatorname{par}(\mathbf a) &= a_xa_w^2 \mathbf e_1 + a_ya_w^2 \mathbf e_2 + a_za_w^2 \mathbf e_3 + a_w^3 \mathbf e_4 + (a_x^2 + a_y^2 + a_z^2 - a_wa_u)\,a_w \mathbf e_5\end{split}$$ |
| Dipole | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ | $$\begin{split}\operatorname{par}(\mathbf d) =\,&(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)(d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{pw} \mathbf e_{45}) \\ +\,&(d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})(d_{vx} \mathbf e_{15} + d_{vy} \mathbf e_{25} + d_{vz} \mathbf e_{35}) \\ +\,&(d_{mz} d_{vy} - d_{my} d_{vz})\,d_{pw}\mathbf e_{15} + (d_{mx} d_{vz} - d_{mz} d_{vx})\,d_{pw}\mathbf e_{25} + (d_{my} d_{vx} - d_{mx} d_{vy})\,d_{pw}\mathbf e_{35}\end{split}$$ |
| Circle | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ | $$\begin{split}\operatorname{par}(\mathbf c) =\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)(c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435}) \\ +\,&(c_{gw}^2 - c_{vx}^2 - c_{vy}^2 - c_{vz}^2 - c_{gx} c_{mx} - c_{gy} c_{my} - c_{gz} c_{mz})(c_{gx} \mathbf e_{235} + c_{gy} \mathbf e_{315} + c_{gz} \mathbf e_{125}) \\ +\,&(c_{vy} c_{gz} - c_{vz} c_{gy})\,c_{gw}\mathbf e_{235} + (c_{vz} c_{gx} - c_{vx} c_{gz})\,c_{gw}\mathbf e_{315} + (c_{vx} c_{gy} - c_{vy} c_{gx})\,c_{gw}\mathbf e_{125}\end{split}$$ |
| Sphere | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ | $$\begin{split}\operatorname{par}(\mathbf s) &= s_u^3 \mathbf e_{1234} + s_xs_u^2 \mathbf e_{4235} + s_ys_u^2 \mathbf e_{4315} + s_zs_u^2 \mathbf e_{4125} + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\,s_u \mathbf e_{3215}\end{split}$$ |