Circle - Revision history

Eric Lengyel at 04:30, 31 July 2024

2024-07-31T04:30:43Z

← Older revision		Revision as of 04:30, 31 July 2024
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	Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as		Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

	:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .		:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{\mathbf p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

	The various properties of a circle are summarized in the following table.		The various properties of a circle are summarized in the following table.
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	The radius of a circle $$\mathbf c$$ is given by		The radius of a circle $$\mathbf c$$ is given by

	:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .		:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode["segoe ui symbol"]{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

	== Contained Points ==		== Contained Points ==

Eric Lengyel: /* Contained Points */

2024-04-21T20:46:34Z

Contained Points

← Older revision		Revision as of 20:46, 21 April 2024
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	The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as		The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

	:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{~~x2606~~}$$ .		:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{x2605}$$ .

	That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, 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. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.		That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, 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. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

Eric Lengyel: /* Contained Points */

2024-04-21T20:45:31Z

Contained Points

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

	:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \~~underline~~{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .		:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \overline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

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

Eric Lengyel at 23:45, 1 December 2023

2023-12-01T23:45:56Z

Eric Lengyel: /* Contained Points */

2023-11-17T07:24:46Z

Contained Points

← Older revision		Revision as of 07:24, 17 November 2023
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	The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as		The set of round points contained by a circle $$\mathbf c$$ can be expressed parametrically in terms of the center and attitude as

	:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^* \~~vee \operatorname~~{~~att~~}~~(\mathbf c)~~$$ .		:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + \operatorname{att}(\mathbf c) \vee (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^\unicode["segoe ui symbol"]{x2606}$$ .

	That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, 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. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.		That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, 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. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

Eric Lengyel: /* Center and Container */

2023-11-17T07:23:52Z

Center and Container

← Older revision		Revision as of 07:23, 17 November 2023
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	The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by		The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

	:$$\operatorname{cen}(\mathbf c) = -\operatorname{~~car~~}(\mathbf c^*) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .		:$$\operatorname{cen}(\mathbf c) = \operatorname{acr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

	The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]		The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

	:$$\operatorname{con}(\mathbf c) = \operatorname{car}(\mathbf c)^* \~~wedge \mathbf c~~ = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .		:$$\operatorname{con}(\mathbf c) = \mathbf c \wedge \operatorname{car}(\mathbf c)^\unicode["segoe ui symbol"]{x2606} = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

	A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as		A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

Eric Lengyel: /* Carrier and Anticarrier */

2023-11-17T07:20:27Z

Carrier and Anticarrier

← Older revision		Revision as of 07:20, 17 November 2023
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	The [[anticarrier]] of a circle $$\mathbf c$$ is the [[line]]		The [[anticarrier]] of a circle $$\mathbf c$$ is the [[line]]

	:$$\operatorname{acr}(\mathbf c) = \mathbf c^* \wedge \mathbf e_5 = c_{gx} \mathbf e_{415} + c_{gy} \mathbf e_{425} + c_{gz} \mathbf e_{435} + c_{vx} \mathbf e_{235} + c_{vy} \mathbf e_{315} + c_{vz} \mathbf e_{125}$$ .		:$$\operatorname{acr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

	The carrier and anticarrier meet at the flat center of the circle, which is given by the [[flat point]]		The carrier and anticarrier meet at the flat center of the circle, which is given by the [[flat point]]

	:$$\operatorname{car}(\mathbf c) \vee \operatorname{acr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} - (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .		:$$\operatorname{car}(\mathbf c) \vee \operatorname{acr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

	== Center and Container ==		== Center and Container ==

Eric Lengyel: Created page with "NOTOC In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form :$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ . If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ componen..."

2023-08-06T03:15:01Z

Created page with "__NOTOC__ In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form :$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ . If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ componen..."

New page

__NOTOC__
In the 5D conformal geometric algebra $$\mathcal G_{4,1}$$, a ''circle'' $$\mathbf c$$ is a trivector with ten components having the general form

:$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ .

If the $$gx$$, $$gy$$, $$gz$$, and $$gw$$ components are all zero, then the circle contains the [[point at infinity]], and it is thus a straight [[line]].

Given a center $$\mathbf p = (p_x, p_y, p_z)$$, a plane normal $$\mathbf n = (n_x, n_y, n_z)$$, and a radius $$r$$, a circle can be formulated as

:$$\mathbf c = n_x \mathbf e_{423} + n_y \mathbf e_{431} + n_z \mathbf e_{412} + (p_yn_z - p_zn_y) \mathbf e_{415} + (p_zn_x - p_xn_z) \mathbf e_{425} + (p_xn_y - p_yn_x) \mathbf e_{435} + (\mathbf p \cdot \mathbf n)(p_x \mathbf e_{235} + p_y \mathbf e_{315} + p_z \mathbf e_{125} - \mathbf e_{321}) - \dfrac{p^2 - r^2}{2}(n_x \mathbf e_{235} + n_y \mathbf e_{315} + n_z \mathbf e_{125})$$ .

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

[[Image:circle.svg|800px]]

== Constraints ==

A valid circle $$\mathbf c$$ must satisfy the following constraints, where $$\mathbf g = (c_{gx}, c_{gy}, c_{gz})$$, $$\mathbf v = (c_{vx}, c_{vy}, c_{vz})$$, and $$\mathbf m = (c_{mx}, c_{my}, c_{mz})$$ are treated as ordinary 3D vectors.

:$$\mathbf g \times \mathbf m - c_{gw}\mathbf v = \mathbf 0$$
:$$\mathbf g \cdot \mathbf v = 0$$
:$$\mathbf v \cdot \mathbf m = 0$$

The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

== Carrier and Anticarrier ==

The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]

:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

The [[anticarrier]] of a circle $$\mathbf c$$ is the [[line]]

:$$\operatorname{acr}(\mathbf c) = \mathbf c^* \wedge \mathbf e_5 = c_{gx} \mathbf e_{415} + c_{gy} \mathbf e_{425} + c_{gz} \mathbf e_{435} + c_{vx} \mathbf e_{235} + c_{vy} \mathbf e_{315} + c_{vz} \mathbf e_{125}$$ .

The carrier and anticarrier meet at the flat center of the circle, which is given by the [[flat point]]

:$$\operatorname{car}(\mathbf c) \vee \operatorname{acr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} - (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

== Center and Container ==

The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

:$$\operatorname{cen}(\mathbf c) = -\operatorname{car}(\mathbf c^*) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]

:$$\operatorname{con}(\mathbf c) = \operatorname{car}(\mathbf c)^* \wedge \mathbf c = -(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{1234} + (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{4235} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{4315} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{4125} + (c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz} - c_{gw}^2)\mathbf e_{3215}$$ .

A unitized circle is equal to the meet of its carrier and container, a relationship that can be expressed as

:$$\mathbf c = \operatorname{car}(\mathbf c) \vee \operatorname{con}(\mathbf c)$$ .

== Norms ==

The radius of a circle $$\mathbf c$$ is given by

:$$\operatorname{rad}(\mathbf c) = \dfrac{\left\Vert\mathbf c\right\Vert_R}{\left\Vert\mathbf c\right\Vert_\unicode{x25CB}} = \sqrt{\dfrac{c_{vx}^2 + c_{vy}^2 + c_{vz}^2 - c_{gw}^2 + 2(c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})}{c_{gx}^2 + c_{gy}^2 + c_{gz}^2}}$$ .

== Contained Points ==

The [[attitude]] of a circle $$\mathbf c$$ is given by

:$$\operatorname{att}(\mathbf c) = \mathbf c \vee \underline{\mathbf e_4} = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ .

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

:$$\mathbf p(\boldsymbol \alpha) = \operatorname{cen}(\mathbf c) + (\alpha_x \mathbf e_1 + \alpha_y \mathbf e_2 + \alpha_z \mathbf e_3)^* \vee \operatorname{att}(\mathbf c)$$ .

That is, $$\mathbf c \wedge \mathbf p(\boldsymbol \alpha) = 0$$ for all vectors $$\boldsymbol \alpha = (\alpha_x, \alpha_y, \alpha_z)$$. In particular, points on the surface of a circle are given when $$\boldsymbol \alpha$$ is parallel to the attitude and has magnitude $$\left\Vert\mathbf c\right\Vert_R$$ (the weighted radius). When $$\mathbf c$$ is a real circle, 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. When $$\mathbf c$$ is an imaginary circle, the round point $$\mathbf p(\boldsymbol \alpha)$$ is always imaginary, and it has an absolute radius at least as large as the circle itself.

== See Also ==

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

← Older revision		Revision as of 23:45, 1 December 2023
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	The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.		The last two constraints are not independent since they can be derived from the first constraint by taking dot products with the vectors $$\mathbf g$$ and $$\mathbf m$$.

	== Carrier and ~~Anticarrier~~ ==		== Carrier and Cocarrier ==

	The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]		The [[carrier]] of a circle $$\mathbf c$$ is the [[plane]]
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	:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .		:$$\operatorname{car}(\mathbf c) = \mathbf c \wedge \mathbf e_5 = c_{gx} \mathbf e_{4235} + c_{gy} \mathbf e_{4315} + c_{gz} \mathbf e_{4125} + c_{gw} \mathbf e_{3215}$$ .

	The [[~~anticarrier~~]] of a circle $$\mathbf c$$ is the [[line]]		The [[cocarrier]] of a circle $$\mathbf c$$ is the [[line]]

	:$$\operatorname{~~acr~~}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .		:$$\operatorname{ccr}(\mathbf c) = \mathbf c^\unicode["segoe ui symbol"]{x2606} \wedge \mathbf e_5 = -c_{gx} \mathbf e_{415} - c_{gy} \mathbf e_{425} - c_{gz} \mathbf e_{435} - c_{vx} \mathbf e_{235} - c_{vy} \mathbf e_{315} - c_{vz} \mathbf e_{125}$$ .

	The carrier and ~~anticarrier~~ meet at the flat center of the circle, which is given by the [[flat point]]		The carrier and cocarrier meet at the flat center of the circle, which is given by the [[flat point]]

	:$$\operatorname{car}(\mathbf c) \vee \operatorname{~~acr~~}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .		:$$\operatorname{car}(\mathbf c) \vee \operatorname{ccr}(\mathbf c) = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_{15} + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_{25} + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_{35} + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_{45}$$ .

	== Center and Container ==		== Center and Container ==
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	The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by		The round [[center]] of a circle $$\mathbf c$$ is the [[round point]] having the same center and radius as $$\mathbf c$$, and it is given by

	:$$\operatorname{cen}(\mathbf c) = \operatorname{~~acr~~}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .		:$$\operatorname{cen}(\mathbf c) = \operatorname{ccr}(\mathbf c) \vee \mathbf c = (c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\mathbf e_1 + (c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\mathbf e_2 + (c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\mathbf e_3 + (c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\mathbf e_4 + (c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\mathbf e_5$$ .

	The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]		The [[container]] of a circle $$\mathbf c$$ is the [[sphere]]