Eric Lengyel: Redirected page to Centers

2023-08-06T03:16:03Z

Redirected page to Centers

← Older revision		Revision as of 03:16, 6 August 2023
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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 [[anticarrier]]:		#REDIRECT [[Centers]]

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

	(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]]

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..."

2023-08-06T03:15:28Z

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]]

Center - Revision history

Eric Lengyel: Redirected page to Centers