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Eric Lengyel (talk | contribs) (Created page with "__NOTOC__ 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...") |
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Given a center $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a sphere can be formulated as | Given a center $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a sphere can be formulated as | ||
:$$\mathbf s = -\mathbf e_{1234} + p_x \mathbf e_{4235} + p_y \mathbf e_{4315} + p_z \mathbf e_{4125} - \dfrac{p^2 - r^2}{2} \mathbf e_{3215}$$ . | :$$\mathbf s = -\mathbf e_{1234} + p_x \mathbf e_{4235} + p_y \mathbf e_{4315} + p_z \mathbf e_{4125} - \dfrac{\mathbf p^2 - r^2}{2} \mathbf e_{3215}$$ . | ||
The various properties of a sphere are summarized in the following table. | The various properties of a sphere are summarized in the following table. | ||
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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 | 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{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$$ is the sphere itself with a different weight: | The [[container]] of a sphere $$\mathbf s$$ is the sphere itself with a different weight: | ||
:$$\operatorname{con}(\mathbf s) = \operatorname{car}(\mathbf s)^ | :$$\operatorname{con}(\mathbf s) = \mathbf s \wedge \operatorname{car}(\mathbf s)^\unicode["segoe ui symbol"]{x2606} = s_u^2 \mathbf e_{1234} + s_xs_u \mathbf e_{4235} + s_ys_u \mathbf e_{4315} + s_zs_u \mathbf e_{4125} + s_ws_u \mathbf e_{3215}$$ . | ||
== Norms == | == Norms == | ||
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The radius of a sphere $$\mathbf s$$ is given by | The radius of a sphere $$\mathbf s$$ is given by | ||
:$$\operatorname{rad}(\mathbf s) = \dfrac{\left\Vert\mathbf s\right\Vert_R}{\left\Vert\mathbf s\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 - 2s_ws_u}{s_u^2}}$$ . | :$$\operatorname{rad}(\mathbf s) = \dfrac{\left\Vert\mathbf s\right\Vert_R}{\left\Vert\mathbf s\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{s_x^2 + s_y^2 + s_z^2 - 2s_ws_u}{s_u^2}}$$ . | ||
== Contained Points == | == Contained Points == | ||
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The [[attitude]] of a sphere $$\mathbf s$$ is given by | The [[attitude]] of a sphere $$\mathbf s$$ is given by | ||
:$$\operatorname{att}(\mathbf s) = \mathbf s \vee \ | :$$\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$$ can be expressed parametrically in terms of the center and attitude as | The set of round points contained by a sphere $$\mathbf s$$ can be expressed parametrically in terms of the center and attitude as | ||
:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^ | :$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + \operatorname{att}(\mathbf s) \vee (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^\unicode["segoe ui symbol"]{x2605}$$ . | ||
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. | 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. | ||
Latest revision as of 04:31, 31 July 2024
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.