Partners - Revision history

Eric Lengyel at 22:52, 3 April 2024

2024-04-03T22:52:16Z

← Older revision		Revision as of 22:52, 3 April 2024
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	The ''partner'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the round object having the same center, same [[carrier]], and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the [[meet]] of the [[~~carrier~~]] of $$\mathbf x$$ ~~with~~ the [[~~container~~]] of $$\mathbf x^*$$:		The ''partner'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the round object having the same center, same [[carrier]], and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf u$$ is denoted by $$\operatorname{par}(\mathbf u)$$, and it is given by the [[meet]] of the [[container]] of $$\mathbf u^\unicode["segoe ui symbol"]{x2606}$$ and the [[carrier]] of $$\mathbf u$$:

	:$$\operatorname{par}(\mathbf x) = -\operatorname{con}(\mathbf x^\unicode["segoe ui symbol"]{~~x2605~~}) \vee \operatorname{car}(\mathbf x)$$ .		:$$\operatorname{par}(\mathbf u) = (-1)^{\operatorname{gr}(\mathbf u) + 1}\operatorname{con}(\mathbf u^\unicode["segoe ui symbol"]{x2606}) \vee \operatorname{car}(\mathbf u)$$ .

	The [[dot product]] between a round object and its partner is always zero. They are orthogonal:		The [[dot product]] between a round object and its partner is always zero. They are orthogonal:

	:$$\mathbf x \mathbin{\unicode{x25CF}} \operatorname{par}(\mathbf x) = 0$$ .		:$$\mathbf u \mathbin{\unicode{x25CF}} \operatorname{par}(\mathbf u) = 0$$ .

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

Eric Lengyel at 01:01, 27 August 2023

2023-08-27T01:01:08Z

← Older revision		Revision as of 01:01, 27 August 2023
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	The ''partner'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the round object having the same center, same [[carrier]], and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the [[meet]] of the [[carrier]] of $$\mathbf x$$ with the [[container]] of $$\mathbf x^*$$:		The ''partner'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the round object having the same center, same [[carrier]], and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the [[meet]] of the [[carrier]] of $$\mathbf x$$ with the [[container]] of $$\mathbf x^*$$:

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

	The [[dot product]] between a round object and its partner is always zero. They are orthogonal:		The [[dot product]] between a round object and its partner is always zero. They are orthogonal:

Eric Lengyel at 21:55, 25 August 2023

2023-08-25T21:55:14Z

Eric Lengyel: Created page with "The ''partner'' of a round object (a round point, dipole, circle, or sphere) is the round object having the same center, same carrier, and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the meet of the carrier of $$\mathbf x$$ with the container of $$\mathbf x^*$$: :$$\operatorname{par}(\mathbf x) = \operatorna..."

2023-08-06T03:16:34Z

Created page with "The ''partner'' of a round object (a round point, dipole, circle, or sphere) is the round object having the same center, same carrier, and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the meet of the carrier of $$\mathbf x$$ with the container of $$\mathbf x^*$$: :$$\operatorname{par}(\mathbf x) = \operatorna..."

New page

The ''partner'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the round object having the same center, same [[carrier]], and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the [[meet]] of the [[carrier]] of $$\mathbf x$$ with the [[container]] of $$\mathbf x^*$$:

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

The [[dot product]] between a round object and its partner is always zero. They are orthogonal:

:$$\mathbf x \mathbin{\unicode{x25CF}} \operatorname{par}(\mathbf x) = 0$$ .

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

{| class="wikitable"
! Type !! Definition !! Partner
|-
| 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;" | $$\begin{split}\operatorname{par}(\mathbf a) = {\phantom +}\,&a_xa_w^2 \mathbf e_1 + a_ya_w^2 \mathbf e_2 + a_za_w^2 \mathbf e_3 + a_w^3 \mathbf e_4 + (a_x^2 + a_y^2 + a_z^2 - a_wa_u)\,a_w \mathbf e_5\end{split}$$
|-
| 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;" | $$\begin{split}\operatorname{par}(\mathbf d) =
&-(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)(d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{pw} \mathbf e_{45}) \\
&+ (d_{mx}^2 + d_{my}^2 + d_{mz}^2 - d_{pw}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})(d_{vx} \mathbf e_{15} + d_{vy} \mathbf e_{25} + d_{vz} \mathbf e_{35}) \\
&+ (d_{my} d_{vz} - d_{mz} d_{vy})\,d_{pw}\mathbf e_{15} + (d_{mz} d_{vx} - d_{mx} d_{vz})\,d_{pw}\mathbf e_{25} + (d_{mx} d_{vy} - d_{my} d_{vx})\,d_{pw}\mathbf e_{35}\end{split}$$
|-
| 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;" | $$\begin{split}\operatorname{par}(\mathbf c) =
{\phantom +}\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)(c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435}) \\
+\,&(c_{gw}^2 - c_{vx}^2 - c_{vy}^2 - c_{vz}^2 - c_{gx} c_{mx} - c_{gy} c_{my} - c_{gz} c_{mz})(c_{gx} \mathbf e_{235} + c_{gy} \mathbf e_{315} + c_{gz} \mathbf e_{125}) \\
+\,&(c_{vy} c_{gz} - c_{vz} c_{gy})\,c_{gw}\mathbf e_{235} + (c_{vz} c_{gx} - c_{vx} c_{gz})\,c_{gw}\mathbf e_{315} + (c_{vx} c_{gy} - c_{vy} c_{gx})\,c_{gw}\mathbf e_{125}\end{split}$$
|-
| 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;" | $$\begin{split}\operatorname{par}(\mathbf s) = &-s_u^3 \mathbf e_{1234} - s_xs_u^2 \mathbf e_{4235} - s_ys_u^2 \mathbf e_{4315} - s_zs_u^2 \mathbf e_{4125} + (s_ws_u - s_x^2 - s_y^2 - s_z^2)\,s_u \mathbf e_{3215}\end{split}$$
|}

== See Also ==

* [[Carriers]]
* [[Centers]]
* [[Containers]]
* [[Attitude]]

← Older revision		Revision as of 21:55, 25 August 2023
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	The ''partner'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the round object having the same center, same [[carrier]], and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the [[meet]] of the [[carrier]] of $$\mathbf x$$ with the [[container]] of $$\mathbf x^*$$:		The ''partner'' of a round object (a [[round point]], [[dipole]], [[circle]], or [[sphere]]) is the round object having the same center, same [[carrier]], and same absolute size, but having a squared radius of the opposite sign. The partner of an object $$\mathbf x$$ is denoted by $$\operatorname{par}(\mathbf x)$$, and it is given by the [[meet]] of the [[carrier]] of $$\mathbf x$$ with the [[container]] of $$\mathbf x^*$$:

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

	The [[dot product]] between a round object and its partner is always zero. They are orthogonal:		The [[dot product]] between a round object and its partner is always zero. They are orthogonal:
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	\| style="padding: 12px;" \| [[Round point]]		\| 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;" \| $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
	\| style="padding: 12px;" \| $$\begin{split}\operatorname{par}(\mathbf a) = ~~{\phantom +}\,&~~a_xa_w^2 \mathbf e_1 + a_ya_w^2 \mathbf e_2 + a_za_w^2 \mathbf e_3 + a_w^3 \mathbf e_4 + (a_x^2 + a_y^2 + a_z^2 - a_wa_u)\,a_w \mathbf e_5\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\operatorname{par}(\mathbf a) &= a_xa_w^2 \mathbf e_1 + a_ya_w^2 \mathbf e_2 + a_za_w^2 \mathbf e_3 + a_w^3 \mathbf e_4 + (a_x^2 + a_y^2 + a_z^2 - a_wa_u)\,a_w \mathbf e_5\end{split}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Dipole]]		\| 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;" \| $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
	\| style="padding: 12px;" \| $$\begin{split}\operatorname{par}(\mathbf d) =		\| style="padding: 12px;" \| $$\begin{split}\operatorname{par}(\mathbf d) =\,&(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)(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_{pw} \mathbf e_{45}) \\
	&-(d_{vx}^2 + d_{vy}^2 + d_{vz}^2)(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_{pw} \mathbf e_{45}) \\		+\,&(d_{pw}^2 - d_{mx}^2 - d_{my}^2 - d_{mz}^2 - d_{vx} d_{px} - d_{vy} d_{py} - d_{vz} d_{pz})(d_{vx} \mathbf e_{15} + d_{vy} \mathbf e_{25} + d_{vz} \mathbf e_{35}) \\
	&+ (d_{mx}^2 + d_{my}^2 + d_{mz}^2 - d_{pw}^2 + d_{vx} d_{px} + d_{vy} d_{py} + d_{vz} d_{pz})(d_{vx} \mathbf e_{15} + d_{vy} \mathbf e_{25} + d_{vz} \mathbf e_{35}) \\		+\,&(d_{mz} d_{vy} - d_{my} d_{vz})\,d_{pw}\mathbf e_{15} + (d_{mx} d_{vz} - d_{mz} d_{vx})\,d_{pw}\mathbf e_{25} + (d_{my} d_{vx} - d_{mx} d_{vy})\,d_{pw}\mathbf e_{35}\end{split}$$
	&+ (d_{my} d_{vz} - d_{mz} d_{vy})\,d_{pw}\mathbf e_{15} + (d_{mz} d_{vx} - d_{mx} d_{vz})\,d_{pw}\mathbf e_{25} + (d_{mx} d_{vy} - d_{my} d_{vx})\,d_{pw}\mathbf e_{35}\end{split}$$
	\|-		\|-
	\| style="padding: 12px;" \| [[Circle]]		\| 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;" \| $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
	\| style="padding: 12px;" \| $$\begin{split}\operatorname{par}(\mathbf c) =		\| style="padding: 12px;" \| $$\begin{split}\operatorname{par}(\mathbf c) =\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)(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}) \\
	~~{\phantom +}~~\,&(c_{gx}^2 + c_{gy}^2 + c_{gz}^2)(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_{gw}^2 - c_{vx}^2 - c_{vy}^2 - c_{vz}^2 - c_{gx} c_{mx} - c_{gy} c_{my} - c_{gz} c_{mz})(c_{gx} \mathbf e_{235} + c_{gy} \mathbf e_{315} + c_{gz} \mathbf e_{125}) \\		+\,&(c_{gw}^2 - c_{vx}^2 - c_{vy}^2 - c_{vz}^2 - c_{gx} c_{mx} - c_{gy} c_{my} - c_{gz} c_{mz})(c_{gx} \mathbf e_{235} + c_{gy} \mathbf e_{315} + c_{gz} \mathbf e_{125}) \\
	+\,&(c_{vy} c_{gz} - c_{vz} c_{gy})\,c_{gw}\mathbf e_{235} + (c_{vz} c_{gx} - c_{vx} c_{gz})\,c_{gw}\mathbf e_{315} + (c_{vx} c_{gy} - c_{vy} c_{gx})\,c_{gw}\mathbf e_{125}\end{split}$$		+\,&(c_{vy} c_{gz} - c_{vz} c_{gy})\,c_{gw}\mathbf e_{235} + (c_{vz} c_{gx} - c_{vx} c_{gz})\,c_{gw}\mathbf e_{315} + (c_{vx} c_{gy} - c_{vy} c_{gx})\,c_{gw}\mathbf e_{125}\end{split}$$
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	\| style="padding: 12px;" \| [[Sphere]]		\| 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;" \| $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
	\| style="padding: 12px;" \| $$\begin{split}\operatorname{par}(\mathbf s) = &-s_u^3 \mathbf e_{1234} - s_xs_u^2 \mathbf e_{4235} - s_ys_u^2 \mathbf e_{4315} - s_zs_u^2 \mathbf e_{4125} + (~~s_ws_u -~~ s_x^2 - s_y^2 - s_z^2)\,s_u \mathbf e_{3215}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\operatorname{par}(\mathbf s) &= s_u^3 \mathbf e_{1234} + s_xs_u^2 \mathbf e_{4235} + s_ys_u^2 \mathbf e_{4315} + s_zs_u^2 \mathbf e_{4125} + (s_x^2 + s_y^2 + s_z^2 - s_ws_u)\,s_u \mathbf e_{3215}\end{split}$$
	\|}		\|}