Sphere - Revision history

Eric Lengyel at 04:31, 31 July 2024

2024-07-31T04:31:32Z

← Older revision		Revision as of 04:31, 31 July 2024
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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 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 ==

Eric Lengyel: /* Contained Points */

2024-04-21T20:45:52Z

Contained Points

← Older revision		Revision as of 20:45, 21 April 2024
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	The [[attitude]] of a sphere $$\mathbf s$$ is given by		The [[attitude]] of a sphere $$\mathbf s$$ is given by

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

	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

Eric Lengyel: /* Center and Container */

2023-12-01T23:43:38Z

Center and Container

← Older revision		Revision as of 23:43, 1 December 2023
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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

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

	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:

Eric Lengyel: /* Contained Points */

2023-11-17T07:10:00Z

Contained Points

← Older revision		Revision as of 07:10, 17 November 2023
Line 34:		Line 34:
	The [[attitude]] of a sphere $$\mathbf s$$ is given by		The [[attitude]] of a sphere $$\mathbf s$$ is given by

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

	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

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

	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.

Eric Lengyel: /* Center and Container */

2023-11-17T07:06:54Z

Center and Container

← Older revision		Revision as of 07:06, 17 November 2023
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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

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

	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:

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

	== Norms ==		== Norms ==

Eric Lengyel: 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..."

2023-08-06T03:15:16Z

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..."

New page

__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 formulated as

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

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

[[Image:sphere.svg|512px]]

== 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

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

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

:$$\operatorname{con}(\mathbf s) = \operatorname{car}(\mathbf s)^* \wedge \mathbf s = -s_u^2 \mathbf e_{1234} - s_xs_u \mathbf e_{4235} - s_ys_u \mathbf e_{4315} - s_zs_u \mathbf e_{4125} - s_ws_u \mathbf e_{3215}$$ .

== Norms ==

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

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

== Contained Points ==

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

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

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

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf s) + (\alpha_x \mathbf e_{23} + \alpha_y \mathbf e_{31} + \alpha_z \mathbf e_{12})^* \vee \operatorname{att}(\mathbf s)$$ .

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.

== See Also ==

* [[Flat point]]
* [[Line]]
* [[Plane]]
* [[Round point]]
* [[Dipole]]
* [[Circle]]