Round point - Revision history

Eric Lengyel at 04:29, 31 July 2024

2024-07-31T04:29:17Z

← Older revision		Revision as of 04:29, 31 July 2024
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	Given a position $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a round point can be formulated as		Given a position $$\mathbf p = (p_x, p_y, p_z)$$ and a radius $$r$$, a round point can be formulated as

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

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

	:$$\operatorname{rad}(\mathbf a) = \dfrac{\left\Vert\mathbf a\right\Vert_R}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{2a_wa_u - a_x^2 - a_y^2 - a_z^2}{a_w^2}}$$ .		:$$\operatorname{rad}(\mathbf a) = \dfrac{\left\Vert\mathbf a\right\Vert_R}{\left\Vert\mathbf a\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{2a_wa_u - a_x^2 - a_y^2 - a_z^2}{a_w^2}}$$ .

	== See Also ==		== See Also ==

Eric Lengyel: /* Center and Container */

2023-12-01T23:43:19Z

Center and Container

← Older revision		Revision as of 23:43, 1 December 2023
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	The round [[center]] of a round point $$\mathbf a$$ is the round point itself with a different weight:		The round [[center]] of a round point $$\mathbf a$$ is the round point itself with a different weight:

	:$$\operatorname{cen}(\mathbf a) = \operatorname{~~acr~~}(\mathbf a) \vee \mathbf a = a_xa_w \mathbf e_1 + a_ya_w \mathbf e_2 + a_za_w \mathbf e_3 + a_w^2 \mathbf e_4 + a_wa_u \mathbf e_5$$ .		:$$\operatorname{cen}(\mathbf a) = \operatorname{ccr}(\mathbf a) \vee \mathbf a = a_xa_w \mathbf e_1 + a_ya_w \mathbf e_2 + a_za_w \mathbf e_3 + a_w^2 \mathbf e_4 + a_wa_u \mathbf e_5$$ .

	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		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

Eric Lengyel: /* Center and Container */

2023-11-17T07:05:02Z

Center and Container

← Older revision		Revision as of 07:05, 17 November 2023
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	The round [[center]] of a round point $$\mathbf a$$ is the round point itself with a different weight:		The round [[center]] of a round point $$\mathbf a$$ is the round point itself with a different weight:

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

	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		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

	:$$\operatorname{con}(\mathbf a) = \operatorname{car}(\mathbf a)^* \~~wedge \mathbf a~~ = -a_w^2 \mathbf e_{1234} + a_xa_w \mathbf e_{4235} + a_ya_w \mathbf e_{4315} + a_za_w \mathbf e_{4125} + (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}$$ .		:$$\operatorname{con}(\mathbf a) = \mathbf a \wedge \operatorname{car}(\mathbf a)^\unicode["segoe ui symbol"]{x2606} = -a_w^2 \mathbf e_{1234} + a_xa_w \mathbf e_{4235} + a_ya_w \mathbf e_{4315} + a_za_w \mathbf e_{4125} + (a_wa_u - a_x^2 - a_y^2 - a_z^2) \mathbf e_{3215}$$ .

	== Norms ==		== Norms ==

Eric Lengyel: Created page with "NOTOC 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 :$$\mathbf a = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + \mathbf e_4 + \dfrac{p^2 + r^2}{2} \mathbf e_5$$ . Th..."

2023-08-06T03:14:37Z

Created page with "__NOTOC__ 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 :$$\mathbf a = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + \mathbf e_4 + \dfrac{p^2 + r^2}{2} \mathbf e_5$$ . Th..."

New page

__NOTOC__
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.

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

== 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

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

== See Also ==

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