Partners

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 $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the meet of the carrier of $$\mathbf x$$ with the container of $$\mathbf x^*$$:


 * $$\operatorname{par}(\mathbf x) = \operatorname{car}(\mathbf x) \vee \operatorname{con}(\mathbf x^*)$$.

The dot product between a round object and its partner is always zero. They are orthogonal:


 * $$\mathbf x \mathbin{\unicode{x25CF}} \operatorname{par}(\mathbf x) = 0$$.

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