File:Sphere connect point.svg and Round point: Difference between pages
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Eric Lengyel (talk | contribs) (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...") |
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__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]] |
Revision as of 03:14, 6 August 2023
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
- $$\operatorname{rad}(\mathbf a) = \dfrac{\left\Vert\mathbf a\right\Vert_R}{\left\Vert\mathbf a\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{2a_wa_u - a_x^2 - a_y^2 - a_z^2}{a_w^2}}$$ .
See Also
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