Expansion - Revision history

Eric Lengyel at 06:50, 24 October 2023

2023-10-24T06:50:19Z

← Older revision		Revision as of 06:50, 24 October 2023
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	$$\begin{split}\mathbf d \wedge \mathbf c^\unicode["segoe ui symbol"]{x2606}		$$\begin{split}\mathbf d \wedge \mathbf c^\unicode["segoe ui symbol"]{x2606}
	=\, &(c_{vx}d_{vx} + c_{vy}d_{vy} + c_{vz}d_{vz} + c_{gx}d_{mx} + c_{gy}d_{my} + c_{gz~~}d_{mz~~})\,\mathbf e_{1234} \\		=\, &(d_{vx}c_{vx} + d_{vy}c_{vy} + d_{vz}c_{vz} + d_{mx}c_{gx} + d_{my}c_{gy} + d_{mz}c_{gz})\,\mathbf e_{1234} \\
	+\, &(c_{my}d_{vz} - c_{mz}d_{vy} - c_{vx}d_{pw} + c_{gz}d_{py} - c_{gy}d_{pz} + c_{gw~~}d_{mx~~})\,\mathbf e_{4235} \\		+\, &(d_{vz}c_{my} - d_{vy}c_{mz} - d_{pw}c_{vx} + d_{py}c_{gz} - d_{pz}c_{gy} + d_{mx}c_{gw})\,\mathbf e_{4235} \\
	+\, &(c_{mz}d_{vx} - c_{mx}d_{vz} - c_{vy}d_{pw} + c_{gx}d_{pz} - c_{gz}d_{px} + c_{gw~~}d_{my~~})\,\mathbf e_{4315} \\		+\, &(d_{vx}c_{mz} - d_{vz}c_{mx} - d_{pw}c_{vy} + d_{pz}c_{gx} - d_{px}c_{gz} + d_{my}c_{gw})\,\mathbf e_{4315} \\
	+\, &(c_{mx}d_{vy} - c_{my}d_{vx} - c_{vz}d_{pw} + c_{gy}d_{px} - c_{gx}d_{py} + c_{gw~~}d_{mz~~})\,\mathbf e_{4125} \\		+\, &(d_{vy}c_{mx} - d_{vx}c_{my} - d_{pw}c_{vz} + d_{px}c_{gy} - d_{py}c_{gx} + d_{mz}c_{gw})\,\mathbf e_{4125} \\
	+\, &(c_{vx}d_{px} + c_{vy}d_{py} + c_{vz}d_{pz} + c_{mx}d_{mx} + c_{my}d_{my} + c_~~{mz}d_~~{mz})\,\mathbf e_{3215}		+\, &(d_{px}c_{vx} + d_{py}c_{vy} + d_{pz}c_{vz} + d_{mx}c_{mx} + d_{my}c_{my} + d_{mz}c_{mz})\,\mathbf e_{3215}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:circle_connect_dipole.svg\|200px]]		\| style="padding: 24px;" \| [[Image:circle_connect_dipole.svg\|200px]]
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	$$\begin{split}\mathbf d \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606}		$$\begin{split}\mathbf d \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606}
	=\, &(l_{vx}d_{vx} + l_{vy}d_{vy} + l_{vz}d_{vz})\,\mathbf e_{1234} \\		=\, &(d_{vx}l_{vx} + d_{vy}l_{vy} + d_{vz}l_{vz})\,\mathbf e_{1234} \\
	+\, &(l_{my}d_{vz} - l_{mz}d_{vy} - l_{vx~~}d_{pw~~})\,\mathbf e_{4235} \\		+\, &(d_{vz}l_{my} - d_{vy}l_{mz} - d_{pw}l_{vx})\,\mathbf e_{4235} \\
	+\, &(l_{mz}d_{vx} - l_{mx}d_{vz} - l_{vy~~}d_{pw~~})\,\mathbf e_{4315} \\		+\, &(d_{vx}l_{mz} - d_{vz}l_{mx} - d_{pw}l_{vy})\,\mathbf e_{4315} \\
	+\, &(l_{mx}d_{vy} - l_{my}d_{vx} - l_{vz~~}d_{pw~~})\,\mathbf e_{4125} \\		+\, &(d_{vy}l_{mx} - d_{vx}l_{my} - d_{pw}l_{vz})\,\mathbf e_{4125} \\
	+\, &(l_{vx}d_{px} + l_{vy}d_{~~py} + l_~~{vz}~~d_{pz~~} + l_{mx}d_{mx} + l_{my}d_{my} + l_{mz}d_{mz})\,\mathbf e_{3215}		+\, &(d_{px}l_{vx} + d_{py}l_{vy} + d_{pzl_{vz}} + d_{mx}l_{mx} + d_{my}l_{my} + d_{mz}l_{mz})\,\mathbf e_{3215}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:line_connect_dipole.svg\|200px]]		\| style="padding: 24px;" \| [[Image:line_connect_dipole.svg\|200px]]
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	$$\begin{split}\mathbf a \wedge \mathbf d^\unicode["segoe ui symbol"]{x2606}		$$\begin{split}\mathbf a \wedge \mathbf d^\unicode["segoe ui symbol"]{x2606}
	=\, &(d_{vx}~~a_x~~ + d_{vy}~~a_y~~ + d_{vz}~~a_z~~ - d_{pw}~~a_w~~)\,\mathbf e_{1234} \\		=\, &(a_xd_{vx} + a_yd_{vy} + a_zd_{vz} - a_wd_{pw})\,\mathbf e_{1234} \\
	+\, &(d_{my}~~a_z~~ - d_{mz}~~a_y~~ + d_{px}~~a_w~~ - d_{vx}~~a_u~~)\,\mathbf e_{4235} \\		+\, &(a_zd_{my} - a_yd_{mz} + a_wd_{px} - a_ud_{vx})\,\mathbf e_{4235} \\
	+\, &(d_{mz}~~a_x~~ - d_{mx}~~a_z~~ + d_{py}~~a_w~~ - d_{vy}~~a_u~~)\,\mathbf e_{4315} \\		+\, &(a_xd_{mz} - a_zd_{mx} + a_wd_{py} - a_ud_{vy})\,\mathbf e_{4315} \\
	+\, &(d_{mx}~~a_y~~ - d_{my}~~a_x~~ + d_{pz}~~a_w~~ - d_{vz}~~a_u~~)\,\mathbf e_{4125} \\		+\, &(a_yd_{mx} - a_xd_{my} + a_wd_{pz} - a_ud_{vz})\,\mathbf e_{4125} \\
	+\, &(d_{pw}~~a_u~~ - d_{px}~~a_x~~ - d_{py}~~a_y~~ - d_{pz}~~a_z~~)\,\mathbf e_{3215}		+\, &(a_ud_{pw} - a_xd_{px} - a_yd_{py} - a_zd_{pz})\,\mathbf e_{3215}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:dipole_connect_round.svg\|200px]]		\| style="padding: 24px;" \| [[Image:dipole_connect_round.svg\|200px]]
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	$$\begin{split}\mathbf a \wedge \mathbf p^\unicode["segoe ui symbol"]{x2606} =		$$\begin{split}\mathbf a \wedge \mathbf p^\unicode["segoe ui symbol"]{x2606} =
	-\, &~~p_wa_w~~ \mathbf e_{1234} + ~~p_xa_w~~ \mathbf e_{4235} + ~~p_ya_w~~ \mathbf e_{4315} + ~~p_za_w~~ \mathbf e_{4125} \\		-\, &a_wp_w \mathbf e_{1234} + a_wp_x \mathbf e_{4235} + a_wp_y \mathbf e_{4315} + a_wp_z \mathbf e_{4125} \\
	+\, &(~~p_wa_u~~ - ~~p_xa_x~~ - ~~p_ya_y~~ - ~~p_za_z~~)\,\mathbf e_{3215}		+\, &(a_up_w - a_xp_x - a_yp_y - a_zp_z)\,\mathbf e_{3215}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:point_connect_round.svg\|200px]]		\| style="padding: 24px;" \| [[Image:point_connect_round.svg\|200px]]

Eric Lengyel at 06:46, 24 October 2023

2023-10-24T06:46:27Z

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Eric Lengyel at 06:40, 24 October 2023

2023-10-24T06:40:00Z

← Older revision		Revision as of 06:40, 24 October 2023
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	$$\begin{split}\mathbf d \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606}		$$\begin{split}\mathbf d \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606}
	=\, &(~~s_zd_~~{vy} - ~~s_yd_~~{vz} - ~~s_ud_~~{mx})\,\mathbf e_{423} \,&+\, (~~s_xd_~~{vz} - ~~s_zd_~~{vx} - ~~s_ud_~~{my})\,\mathbf e_{431} \\		=\, &(d_{vy}s_z - d_{vz}s_y - d_{mx}s_u)\,\mathbf e_{423} \,&+\, (d_{vz}s_x - d_{vx}s_z - d_{my}s_u)\,\mathbf e_{431} \\
	+\, &(~~s_yd_~~{vx} - ~~s_xd_~~{vy} - ~~s_ud_~~{mz})\,\mathbf e_{412} \,&-\, (~~s_xd_~~{mx} + ~~s_yd_~~{my} + ~~s_zd_~~{mz})\,\mathbf e_{321} \\		+\, &(d_{vx}s_y - d_{vy}s_x - d_{mz}s_u)\,\mathbf e_{412} \,&-\, (d_{mx}s_x + d_{my}s_y + d_{mz}s_z)\,\mathbf e_{321} \\
	-\, &(~~s_xd_~~{pw} + ~~s_wd_~~{vx} + ~~s_ud_~~{px})\,\mathbf e_{415} \,&+\, (~~s_yd_~~{pz} - ~~s_zd_~~{py} - ~~s_wd_~~{mx})\,\mathbf e_{235} \\		-\, &(d_{vx}s_w + d_{pw}s_x + d_{px}s_u)\,\mathbf e_{415} \,&+\, (d_{pz}s_y - d_{py}s_z - d_{mx}s_w)\,\mathbf e_{235} \\
	-\, &(~~s_yd_~~{pw} + ~~s_wd_~~{vy} + ~~s_ud_~~{py})\,\mathbf e_{425} \,&+\, (~~s_zd_~~{px} - ~~s_xd_~~{pz} - ~~s_wd_~~{my})\,\mathbf e_{315} \\		-\, &(d_{vy}s_w + d_{pw}s_y + d_{py}s_u)\,\mathbf e_{425} \,&+\, (d_{px}s_z - d_{pz}s_x - d_{my}s_w)\,\mathbf e_{315} \\
	-\, &(~~s_zd_~~{pw} + ~~s_wd_~~{vz} + ~~s_ud_~~{pz})\,\mathbf e_{435} \,&+\, (~~s_xd_~~{py} - ~~s_yd_~~{px} - ~~s_wd_~~{mz})\,\mathbf e_{125}		-\, &(d_{vz}s_w + d_{pw}s_z + d_{pz}s_u)\,\mathbf e_{435} \,&+\, (d_{py}s_x - d_{px}s_y - d_{mz}s_w)\,\mathbf e_{125}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:sphere_connect_dipole.svg\|200px]]		\| style="padding: 24px;" \| [[Image:sphere_connect_dipole.svg\|200px]]
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	$$\begin{split}\mathbf d \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}		$$\begin{split}\mathbf d \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}
	=\, &(~~g_zd_~~{vy} - ~~g_yd_~~{vz})\,\mathbf e_{423} \,&+\, (~~g_xd_~~{vz} - ~~g_zd_~~{vx})\,\mathbf e_{431} \\		=\, &(d_{vy}g_z - d_{vz}g_y)\,\mathbf e_{423} \,&+\, (d_{vz}g_x - d_{vx}g_z)\,\mathbf e_{431} \\
	+\, &(~~g_yd_~~{vx} - ~~g_xd_~~{vy})\,\mathbf e_{412} \,&-\, (~~g_xd_~~{mx} + ~~g_yd_~~{my} + ~~g_zd_~~{mz})\,\mathbf e_{321} \\		+\, &(d_{vx}g_y - d_{vy}g_x)\,\mathbf e_{412} \,&-\, (d_{mx}g_x + d_{my}g_y + d_{mz}g_z)\,\mathbf e_{321} \\
	-\, &(~~g_xd_~~{pw} + ~~g_wd_~~{vx})\,\mathbf e_{415} \,&+\, (~~g_yd_~~{pz} - ~~g_zd_~~{py} - ~~g_wd_~~{mx})\,\mathbf e_{235} \\		-\, &(d_{vx}g_w + d_{pw}g_x)\,\mathbf e_{415} \,&+\, (d_{pz}g_y - d_{py}g_z - d_{mx}g_w)\,\mathbf e_{235} \\
	-\, &(~~g_yd_~~{pw} + ~~g_wd_~~{vy})\,\mathbf e_{425} \,&+\, (~~g_zd_~~{px} - ~~g_xd_~~{pz} - ~~g_wd_~~{my})\,\mathbf e_{315} \\		-\, &(d_{vy}g_w + d_{pw}g_y)\,\mathbf e_{425} \,&+\, (d_{px}g_z - d_{pz}g_x - d_{my}g_w)\,\mathbf e_{315} \\
	-\, &(~~g_zd_~~{pw} + ~~g_wd_~~{vz})\,\mathbf e_{435} \,&+\, (~~g_xd_~~{py} - ~~g_yd_~~{px} - ~~g_wd_~~{mz})\,\mathbf e_{125}		-\, &(d_{vz}g_w + d_{pw}g_z)\,\mathbf e_{435} \,&+\, (d_{py}g_x - d_{px}g_y - d_{mz}g_w)\,\mathbf e_{125}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:plane_connect_dipole.svg\|200px]]		\| style="padding: 24px;" \| [[Image:plane_connect_dipole.svg\|200px]]
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	$$\begin{split}\mathbf p \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606} =		$$\begin{split}\mathbf p \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606} =
	-\, &(~~s_xp_w~~ + ~~s_up_x~~)\,\mathbf e_{415} \,&-\, (~~s_yp_w~~ + ~~s_up_y~~)\,\mathbf e_{425} \,&-\, (~~s_zp_w~~ + ~~s_up_z~~)\,\mathbf e_{435} \\		-\, &(p_xs_u + p_ws_x)\,\mathbf e_{415} \,&-\, (p_ys_u + p_ws_y)\,\mathbf e_{425} \,&-\, (p_zs_u + p_ws_z)\,\mathbf e_{435} \\
	+\, &(~~s_yp_z~~ - ~~s_zp_y~~)\,\mathbf e_{235} \,&+\, (~~s_zp_x~~ - ~~s_xp_z~~)\,\mathbf e_{315} \,&+\, (~~s_xp_y~~ - ~~s_yp_x~~)\,\mathbf e_{125}		+\, &(p_zs_y - p_ys_z)\,\mathbf e_{235} \,&+\, (p_xs_z - p_zs_x)\,\mathbf e_{315} \,&+\, (p_ys_x - p_xs_y)\,\mathbf e_{125}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:sphere_connect_point.svg\|200px]]		\| style="padding: 24px;" \| [[Image:sphere_connect_point.svg\|200px]]
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	$$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =		$$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =
	-\, &~~g_xp_w~~ \mathbf e_{415} - ~~g_yp_w~~ \mathbf e_{425} - ~~g_zp_w~~ \mathbf e_{435} \\		-\, &p_wg_x \mathbf e_{415} - p_wg_y \mathbf e_{425} - p_wg_z \mathbf e_{435} \\
	+\, &(~~g_yp_z~~ - ~~g_zp_y~~)\,\mathbf e_{235} + (~~g_zp_x~~ - ~~g_xp_z~~)\,\mathbf e_{315} + (~~g_xp_y~~ - ~~g_yp_x~~)\,\mathbf e_{125}		+\, &(p_zg_y - p_yg_z)\,\mathbf e_{235} + (p_xg_z - p_zg_x)\,\mathbf e_{315} + (p_yg_x - p_xg_y)\,\mathbf e_{125}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:plane_connect_point.svg\|200px]]		\| style="padding: 24px;" \| [[Image:plane_connect_point.svg\|200px]]

Eric Lengyel at 06:19, 24 October 2023

2023-10-24T06:19:11Z

← Older revision		Revision as of 06:19, 24 October 2023
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	$$\begin{split}\mathbf a \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606}		$$\begin{split}\mathbf a \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606}
	=\, &(~~s_xa_w~~ + ~~s_ua_x~~)\,\mathbf e_{41} \,&+\, (~~s_ya_w~~ + ~~s_ua_y~~)\,\mathbf e_{42} \,&+\, (~~s_za_w~~ + ~~s_ua_z~~)\,\mathbf e_{43} \\		=\, &(a_xs_u + a_ws_x)\,\mathbf e_{41} \,&+\, (a_ys_u + a_ws_y)\,\mathbf e_{42} \,&+\, (a_zs_u + a_ws_z)\,\mathbf e_{43} \\
	+\, &(~~s_za_y~~ - ~~s_ya_z~~)\,\mathbf e_{23} \,&+\, (~~s_xa_z~~ - ~~s_za_x~~)\,\mathbf e_{31} \,&+\, (~~s_ya_x~~ - ~~s_xa_y~~)\,\mathbf e_{12} \\		+\, &(a_ys_z - a_zs_y)\,\mathbf e_{23} \,&+\, (a_zs_x - a_xs_z)\,\mathbf e_{31} \,&+\, (a_xs_y - a_ys_x)\,\mathbf e_{12} \\
	-\, &(~~s_xa_u~~ + ~~s_wa_x~~)\,\mathbf e_{15} \,&-\, (~~s_ya_u~~ + ~~s_wa_y~~)\,\mathbf e_{25} \,&-\, (~~s_za_u~~ + ~~s_wa_z~~)\,\mathbf e_{35} + (~~s_ua_u~~ - ~~s_wa_w~~)\,\mathbf e_{45}		-\, &(a_xs_w + a_us_x)\,\mathbf e_{15} \,&-\, (a_ys_w + a_us_y)\,\mathbf e_{25} \,&-\, (a_zs_w + a_us_z)\,\mathbf e_{35} + (a_us_u - a_ws_w)\,\mathbf e_{45}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:sphere_connect_round.svg\|200px]]		\| style="padding: 24px;" \| [[Image:sphere_connect_round.svg\|200px]]
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	$$\begin{split}\mathbf a \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}		$$\begin{split}\mathbf a \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606}
	=\, &~~g_xa_w~~ \mathbf e_{41} + ~~g_ya_w~~ \mathbf e_{42} + ~~g_za_w~~ \mathbf e_{43} \\		=\, &a_wg_x \mathbf e_{41} + a_wg_y \mathbf e_{42} + a_wg_z \mathbf e_{43} \\
	+\, &(~~g_za_y~~ - ~~g_ya_z~~)\,\mathbf e_{23} + (~~g_xa_z~~ - ~~g_za_x~~)\,\mathbf e_{31} + (~~g_ya_x~~ - ~~g_xa_y~~)\,\mathbf e_{12} \\		+\, &(a_yg_z - a_zg_y)\,\mathbf e_{23} + (a_zg_x - a_xg_z)\,\mathbf e_{31} + (a_xg_y - a_yg_x)\,\mathbf e_{12} \\
	-\, &(~~g_xa_u~~ + ~~g_wa_x~~)\,\mathbf e_{15} - (~~g_ya_u~~ + ~~g_wa_y~~)\,\mathbf e_{25} - (~~g_za_u~~ + ~~g_wa_z~~)\,\mathbf e_{35} - ~~g_wa_w~~ \mathbf e_{45}		-\, &(a_xg_w + a_ug_x)\,\mathbf e_{15} - (a_yg_w + a_ug_y)\,\mathbf e_{25} - (a_zg_w + a_ug_z)\,\mathbf e_{35} - a_wg_w \mathbf e_{45}
	\end{split}$$		\end{split}$$
	\| style="padding: 24px;" \| [[Image:plane_connect_round.svg\|200px]]		\| style="padding: 24px;" \| [[Image:plane_connect_round.svg\|200px]]

Eric Lengyel: Eric Lengyel moved page Connect to Expansion without leaving a redirect

2023-10-23T03:18:21Z

Eric Lengyel moved page Connect to Expansion without leaving a redirect

← Older revision	Revision as of 03:18, 23 October 2023
(No difference)

Eric Lengyel at 03:14, 23 October 2023

2023-10-23T03:14:59Z

← Older revision		Revision as of 03:14, 23 October 2023
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	The ''~~connect~~'' operation is performed by taking the [[wedge product]] between ~~the dual of~~ an object ''A'' and another object ''B'' with ~~lower~~ grade. The result is an object ''C'' that ~~is orthogonal to~~ ''A'' and ~~contains~~ ''B'', allowing a [[projection]] of ''B'' onto ''A'' through a simple intersection of ''A'' and ''C''.		The ''expansion'' operation is performed by taking the [[wedge product]] between an object ''A'' and the [[antidual]] of another object ''B'' with higher grade. The result is an object ''C'' that contains ''A'' and is orthogonal to ''B'', allowing a [[projection]] of ''A'' onto ''B'' through a simple intersection of ''B'' and ''C''.

	The [[flat points]], [[lines]], [[planes]], [[round points]], [[dipoles]], [[circles]], and [[spheres]] appearing in the following tables are defined as follows:		The [[flat points]], [[lines]], [[planes]], [[round points]], [[dipoles]], [[circles]], and [[spheres]] appearing in the following tables are defined as follows:

Eric Lengyel at 03:12, 23 October 2023

2023-10-23T03:12:16Z

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Eric Lengyel at 03:08, 23 October 2023

2023-10-23T03:08:04Z

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Eric Lengyel at 02:46, 23 October 2023

2023-10-23T02:46:59Z

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Eric Lengyel at 02:41, 23 October 2023

2023-10-23T02:41:29Z

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