Sphere

(Redirected from Spheres)

In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a sphere $$\mathbf s$$ is a quadrivector having the general form

$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ .

If the $$s_u$$ component is zero, then the sphere contains the point at infinity, and it is thus a flat plane.

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

$$\mathbf s = -\mathbf e_{1234} + p_x \mathbf e_{4235} + p_y \mathbf e_{4315} + p_z \mathbf e_{4125} - \dfrac{\mathbf p^2 - r^2}{2} \mathbf e_{3215}$$ .

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

Center and Container

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

$$\operatorname{cen}(\mathbf s) = \operatorname{ccr}(\mathbf s) \vee \mathbf s = -s_xs_u \mathbf e_1 - s_ys_u \mathbf e_2 - s_zs_u \mathbf e_3 + s_u^2 \mathbf e_4 + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\mathbf e_5$$ .

The container of a sphere $$\mathbf s$$ is the sphere itself with a different weight:

$$\operatorname{con}(\mathbf s) = \mathbf s \wedge \operatorname{car}(\mathbf s)^\unicode["segoe ui symbol"]{x2606} = s_u^2 \mathbf e_{1234} + s_xs_u \mathbf e_{4235} + s_ys_u \mathbf e_{4315} + s_zs_u \mathbf e_{4125} + s_ws_u \mathbf e_{3215}$$ .

Norms

The radius of a sphere $$\mathbf s$$ is given by

$$\operatorname{rad}(\mathbf s) = \dfrac{\left\Vert\mathbf s\right\Vert_R}{\left\Vert\mathbf s\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 - 2s_ws_u}{s_u^2}}$$ .

Contained Points

The attitude of a sphere $$\mathbf s$$ is given by

$$\operatorname{att}(\mathbf s) = \mathbf s \vee \overline{\mathbf e_4} = s_u \mathbf e_{321} + s_x \mathbf e_{235} + s_t \mathbf e_{315} + s_z \mathbf e_{125}$$ .

The set of round points contained by a sphere $$\mathbf s$$ can be expressed parametrically in terms of the center and attitude as

$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + \operatorname{att}(\mathbf s) \vee (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^\unicode["segoe ui symbol"]{x2605}$$ .

That is, $$\mathbf s \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all bivectors $$\boldsymbol \alpha$$. In particular, points on the surface of a sphere are given when $$\boldsymbol \alpha$$ has magnitude $$\left\Vert\mathbf s\right\Vert_R$$ (the weighted radius), and this is precisely where the radius of $$\mathbf p(\boldsymbol \alpha)$$ is zero. For smaller magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is real, and for larger magnitudes of $$\boldsymbol \alpha$$, the round point $$\mathbf p(\boldsymbol \alpha)$$ is imaginary.