Centers - Revision history

Eric Lengyel at 22:58, 3 April 2024

2024-04-03T22:58:57Z

← Older revision		Revision as of 22:58, 3 April 2024
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	The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[cocarrier]]:		The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf u$$ is denoted by $$\operatorname{cen}(\mathbf u)$$, and it is given by the [[meet]] of $$\mathbf u$$ and its own [[cocarrier]]:

	:$$\operatorname{cen}(\mathbf x) = \operatorname{ccr}(\mathbf x) \vee \mathbf x$$ .		:$$\operatorname{cen}(\mathbf u) = \operatorname{ccr}(\mathbf u) \vee \mathbf u$$ .

	The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.		The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

Eric Lengyel at 23:42, 1 December 2023

2023-12-01T23:42:23Z

← Older revision		Revision as of 23:42, 1 December 2023
Line 1:		Line 1:
	The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[~~anticarrier~~]]:		The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[cocarrier]]:

	:$$\operatorname{cen}(\mathbf x) = \operatorname{~~acr~~}(\mathbf x) \vee \mathbf x$$ .		:$$\operatorname{cen}(\mathbf x) = \operatorname{ccr}(\mathbf x) \vee \mathbf x$$ .

	The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.		The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

Eric Lengyel at 06:56, 17 November 2023

2023-11-17T06:56:17Z

← Older revision		Revision as of 06:56, 17 November 2023
Line 1:		Line 1:
	The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:		The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:

	:$$\operatorname{cen}(\mathbf x) = \operatorname{~~car~~}(\mathbf x~~^\unicode["segoe ui symbol"]{x2605}~~) \vee \mathbf x$$ .		:$$\operatorname{cen}(\mathbf x) = \operatorname{acr}(\mathbf x) \vee \mathbf x$$ .

	The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.		The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

Eric Lengyel at 01:00, 27 August 2023

2023-08-27T01:00:20Z

← Older revision		Revision as of 01:00, 27 August 2023
Line 1:		Line 1:
	The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:		The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:

	:$$\operatorname{cen}(\mathbf x) = \operatorname{car}(\mathbf x^*) \vee \mathbf x$$ .		:$$\operatorname{cen}(\mathbf x) = \operatorname{car}(\mathbf x^\unicode["segoe ui symbol"]{x2605}) \vee \mathbf x$$ .

	The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.		The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

Eric Lengyel at 21:35, 25 August 2023

2023-08-25T21:35:28Z

← Older revision		Revision as of 21:35, 25 August 2023
Line 1:		Line 1:
	The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:		The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:

	:$$\operatorname{cen}(\mathbf x) = -\operatorname{car}(\mathbf x^*) \vee \mathbf x$$ .		:$$\operatorname{cen}(\mathbf x) = \operatorname{car}(\mathbf x^*) \vee \mathbf x$$ .

	~~(The negative sign is not strictly necessary, but is included so the function always produces a result having a positive weight.)~~ The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.		The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

	The following table lists the centers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.		The following table lists the centers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

Eric Lengyel: Created page with "The ''center'' of a round object (a round point, dipole, circle, or sphere) is the round point having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the meet of $$\mathbf x$$ and its own anticarrier: :$$\operatorname{cen}(\mathbf x) = -\operatorname{car}(\mathbf x^*) \vee \mathbf x$$ . (The negative sign is not strictly necessary, but is included so the fu..."

2023-08-06T03:16:06Z

Created page with "The ''center'' of a round object (a round point, dipole, circle, or sphere) is the round point having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the meet of $$\mathbf x$$ and its own anticarrier: :$$\operatorname{cen}(\mathbf x) = -\operatorname{car}(\mathbf x^*) \vee \mathbf x$$ . (The negative sign is not strictly necessary, but is included so the fu..."

New page

The ''center'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the [[round point]] having the same center and radius. The center of an object $$\mathbf x$$ is denoted by $$\operatorname{cen}(\mathbf x)$$, and it is given by the [[meet]] of $$\mathbf x$$ and its own [[anticarrier]]:

:$$\operatorname{cen}(\mathbf x) = -\operatorname{car}(\mathbf x^*) \vee \mathbf x$$ .

(The negative sign is not strictly necessary, but is included so the function always produces a result having a positive weight.) The squared radius of an object's center has the same sign as the squared radius of the object itself. That is, a real object has a real center, and an imaginary object has an imaginary center.

The following table lists the centers for the round objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.

{| class="wikitable"
! Type !! Definition !! Center
|-
| style="padding: 12px;" | [[Round point]]
| style="padding: 12px;" | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf a) =
{\phantom +}\,&a_xa_w \mathbf e_1 \\
+\,&a_ya_w \mathbf e_2 \\
+\,&a_za_w \mathbf e_3 \\
+\,&a_w^2 \mathbf e_4 \\
+\,&a_wa_u \mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Dipole]]
| style="padding: 12px;" | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf d) =
{\phantom +}\,&(d_{vy} d_{mz} - d_{vz} d_{my} + d_{vx} d_{pw})\,\mathbf e_1 \\
+\,&(d_{vz} d_{mx} - d_{vx} d_{mz} + d_{vy} d_{pw})\,\mathbf e_2 \\
+\,&(d_{vx} d_{my} - d_{vy} d_{mx} + d_{vz} d_{pw})\,\mathbf e_3 \\
+\,&(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)\,\mathbf e_4 \\
+\,&(d_{pw}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})\,\mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Circle]]
| style="padding: 12px;" | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf c) =
{\phantom +}\,&(c_{gy} c_{vz} - c_{gz} c_{vy} - c_{gx} c_{gw})\,\mathbf e_1 \\
+\,&(c_{gz} c_{vx} - c_{gx} c_{vz} - c_{gy} c_{gw})\,\mathbf e_2 \\
+\,&(c_{gx} c_{vy} - c_{gy} c_{vx} - c_{gz} c_{gw})\,\mathbf e_3 \\
+\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)\,\mathbf e_4 \\
+\,&(c_{vx}^2 + c_{vy}^2 + c_{vz}^2 + c_{gx} c_{mx} + c_{gy} c_{my} + c_{gz} c_{mz})\,\mathbf e_5\end{split}$$
|-
| style="padding: 12px;" | [[Sphere]]
| style="padding: 12px;" | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| style="padding: 12px;" | $$\begin{split}\operatorname{cen}(\mathbf s) =
-\,&s_xs_u \mathbf e_1 \\
-\,&s_ys_u \mathbf e_2 \\
-\,&s_zs_u \mathbf e_3 \\
+\,&s_u^2 \mathbf e_4 \\
+\,&(s_x^2 + s_y^2 + s_z^2 - s_ws_u)\,\mathbf e_5\end{split}$$
|}

== See Also ==

* [[Containers]]
* [[Carriers]]
* [[Partners]]
* [[Attitude]]