Round point

In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a round point $$\mathbf a$$ is a vector having the general form


 * $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$.

Given a position $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a round point can be formulated as


 * $$\mathbf a = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + \mathbf e_4 + \dfrac{p^2 + r^2}{2} \mathbf e_5$$.

The various properties of a sphere are summarized in the following table.



Center and Container
The round center of a round point $$\mathbf a$$ is the round point itself with a different weight:


 * $$\operatorname{cen}(\mathbf a) = -\operatorname{car}(\mathbf a^*) \vee \mathbf a = a_xa_w \mathbf e_1 + a_ya_w \mathbf e_2 + a_za_w \mathbf e_3 + a_w^2 \mathbf e_4 + a_wa_u \mathbf e_5$$.

The container of a round point $$\mathbf a$$ is the sphere having the same center and radius as $$\mathbf a$$, and it is given by


 * $$\operatorname{con}(\mathbf a) = \operatorname{car}(\mathbf a)^* \wedge \mathbf a = -a_w^2 \mathbf e_{1234} + a_xa_w \mathbf e_{4235} + a_ya_w \mathbf e_{4315} + a_za_w \mathbf e_{4125} + (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}$$.

Norms
The radius of a round point $$\mathbf a$$ is given by


 * $$\operatorname{rad}(\mathbf a) = \dfrac{\left\Vert\mathbf a\right\Vert_R}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{2a_wa_u - a_x^2 - a_y^2 - a_z^2}{a_w^2}}$$.