Join and meet - Revision history

Eric Lengyel at 20:34, 26 October 2023

2023-10-26T20:34:27Z

← Older revision		Revision as of 20:34, 26 October 2023
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	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$
	\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.		\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:plane_connect_point.svg\|~~250px~~]]		\| style="padding: 12px; text-align: center;" \| [[Image:plane_connect_point.svg\|200px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$
	\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.
	\| style="padding: 12px;" \| [[Image:line_connect_point.svg\|~~250px~~]]		\| style="padding: 12px; text-align: center;" \| [[Image:line_connect_point.svg\|200px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$
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	Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.		Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:plane_connect_line.svg\|~~250px~~]]		\| style="padding: 12px; text-align: center;" \| [[Image:plane_connect_line.svg\|200px]]
	\|}		\|}

Eric Lengyel at 07:31, 23 October 2023

2023-10-23T07:31:33Z

← Older revision		Revision as of 07:31, 23 October 2023
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	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$
	\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.		\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:~~plane_weight_join_point~~.svg\|250px]]		\| style="padding: 12px;" \| [[Image:plane_connect_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$
	\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.
	\| style="padding: 12px;" \| [[Image:~~line_weight_join_point~~.svg\|250px]]		\| style="padding: 12px;" \| [[Image:line_connect_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$
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	Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.		Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:~~plane_weight_join_line~~.svg\|250px]]		\| style="padding: 12px;" \| [[Image:plane_connect_line.svg\|250px]]
	\|}		\|}

Eric Lengyel at 07:28, 23 October 2023

2023-10-23T07:28:58Z

← Older revision		Revision as of 07:28, 23 October 2023
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	\| style="padding: 12px;" \| [[Image:line_meet_plane.svg\|250px]]		\| style="padding: 12px;" \| [[Image:line_meet_plane.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf g^\unicode["segoe ui symbol"]{~~x2605~~} ~~\wedge \mathbf p~~ =\, &-~~g_xp_w~~ \mathbf e_{41} - ~~g_yp_w~~ \mathbf e_{42} - ~~g_zp_w~~ \mathbf e_{43} \\ +\, &(~~g_yp_z~~ - ~~g_zp_y~~)\,\mathbf e_{23} + (~~g_zp_x~~ - ~~g_xp_z~~)\,\mathbf e_{31} + (~~g_xp_y~~ - ~~g_yp_x~~)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$
	\| style="padding: 12px;" \| Line ~~perpendicular to plane~~ $$\mathbf g$$ ~~passing through point~~ $$\mathbf p$$.		\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:plane_weight_join_point.svg\|250px]]		\| style="padding: 12px;" \| [[Image:plane_weight_join_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l^\unicode["segoe ui symbol"]{~~x2605~~} ~~\wedge \mathbf p~~ =\, &-l_{vx} ~~p_w~~ \mathbf e_{423} - l_{vy} ~~p_w~~ \mathbf e_{431} - l_{vz} ~~p_w~~ \mathbf e_{412} \\ +\, &(l_{vx} ~~p_x~~ + l_{vy} ~~p_y~~ + l_{vz} ~~p_z~~)\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$
	\| style="padding: 12px;" \| Plane perpendicular to line $$\boldsymbol l~~$$ containing point $$\mathbf p~~$$.		\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.
	\| style="padding: 12px;" \| [[Image:line_weight_join_point.svg\|250px]]		\| style="padding: 12px;" \| [[Image:line_weight_join_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf g^\unicode["segoe ui symbol"]{~~x2605~~} ~~\wedge \boldsymbol l~~ =\, &(~~g_zl_~~{vy} - ~~g_yl_~~{vz})\,\mathbf e_{423} + (~~g_xl_~~{vz} - ~~g_zl_~~{vx})\,\mathbf e_{431} + (~~g_yl_~~{vx} - ~~g_xl_~~{vy})\,\mathbf e_{412} \\ -\, &(~~g_xl_~~{mx} + ~~g_yl_~~{my} + ~~g_zl_~~{mz})\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$
	\| style="padding: 12px;" \| Plane perpendicular to plane $$\mathbf g~~$$ containing line $$\boldsymbol l~~$$.		\| style="padding: 12px;" \| Plane containing line $$\boldsymbol l$$ and perpendicular to plane $$\mathbf g$$.

	Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.		Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.

Eric Lengyel at 20:19, 26 August 2023

2023-08-26T20:19:40Z

← Older revision		Revision as of 20:19, 26 August 2023
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	\| style="padding: 12px;" \| [[Image:line_meet_plane.svg\|250px]]		\| style="padding: 12px;" \| [[Image:line_meet_plane.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf g^* \wedge \mathbf p =\, &-g_xp_w \mathbf e_{41} - g_yp_w \mathbf e_{42} - g_zp_w \mathbf e_{43} \\ +\, &(g_yp_z - g_zp_y)\,\mathbf e_{23} + (g_zp_x - g_xp_z)\,\mathbf e_{31} + (g_xp_y - g_yp_x)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf g^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf p =\, &-g_xp_w \mathbf e_{41} - g_yp_w \mathbf e_{42} - g_zp_w \mathbf e_{43} \\ +\, &(g_yp_z - g_zp_y)\,\mathbf e_{23} + (g_zp_x - g_xp_z)\,\mathbf e_{31} + (g_xp_y - g_yp_x)\,\mathbf e_{12}\end{split}$$
	\| style="padding: 12px;" \| Line perpendicular to plane $$\mathbf g$$ passing through point $$\mathbf p$$.		\| style="padding: 12px;" \| Line perpendicular to plane $$\mathbf g$$ passing through point $$\mathbf p$$.
	\| style="padding: 12px;" \| [[Image:plane_weight_join_point.svg\|250px]]		\| style="padding: 12px;" \| [[Image:plane_weight_join_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l^* \wedge \mathbf p =\, &-l_{vx} p_w \mathbf e_{423} - l_{vy} p_w \mathbf e_{431} - l_{vz} p_w \mathbf e_{412} \\ +\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l^\unicode["segoe ui symbol"]{x2605} \wedge \mathbf p =\, &-l_{vx} p_w \mathbf e_{423} - l_{vy} p_w \mathbf e_{431} - l_{vz} p_w \mathbf e_{412} \\ +\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\,\mathbf e_{321}\end{split}$$
	\| style="padding: 12px;" \| Plane perpendicular to line $$\boldsymbol l$$ containing point $$\mathbf p$$.		\| style="padding: 12px;" \| Plane perpendicular to line $$\boldsymbol l$$ containing point $$\mathbf p$$.
	\| style="padding: 12px;" \| [[Image:line_weight_join_point.svg\|250px]]		\| style="padding: 12px;" \| [[Image:line_weight_join_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf g^* \wedge \boldsymbol l =\, &(g_zl_{vy} - g_yl_{vz})\,\mathbf e_{423} + (g_xl_{vz} - g_zl_{vx})\,\mathbf e_{431} + (g_yl_{vx} - g_xl_{vy})\,\mathbf e_{412} \\ -\, &(g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf g^\unicode["segoe ui symbol"]{x2605} \wedge \boldsymbol l =\, &(g_zl_{vy} - g_yl_{vz})\,\mathbf e_{423} + (g_xl_{vz} - g_zl_{vx})\,\mathbf e_{431} + (g_yl_{vx} - g_xl_{vy})\,\mathbf e_{412} \\ -\, &(g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{321}\end{split}$$
	\| style="padding: 12px;" \| Plane perpendicular to plane $$\mathbf g$$ containing line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Plane perpendicular to plane $$\mathbf g$$ containing line $$\boldsymbol l$$.

Eric Lengyel: /* See Also */

2023-08-26T00:20:26Z

Eric Lengyel at 00:20, 26 August 2023

2023-08-26T00:20:15Z

← Older revision		Revision as of 00:20, 26 August 2023
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	{\| class="wikitable"		{\| class="wikitable"
	! Formula ~~\|\| Commutator~~ \|\| Description \|\| Illustration		! Formula \|\| Description \|\| Illustration
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf q =\, &(p_wq_x - p_xq_w)\,\mathbf e_{41} + (p_wq_y - p_yq_w)\,\mathbf e_{42} + (p_wq_z - p_zq_w)\,\mathbf e_{43} \\ +\, &(p_yq_z - p_zq_y)\,\mathbf e_{23} + (p_zq_x - p_xq_z)\,\mathbf e_{31} + (p_xq_y - p_yq_x)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf q =\, &(p_wq_x - p_xq_w)\,\mathbf e_{41} + (p_wq_y - p_yq_w)\,\mathbf e_{42} + (p_wq_z - p_zq_w)\,\mathbf e_{43} \\ +\, &(p_yq_z - p_zq_y)\,\mathbf e_{23} + (p_zq_x - p_xq_z)\,\mathbf e_{31} + (p_xq_y - p_yq_x)\,\mathbf e_{12}\end{split}$$
	~~\| style="padding: 12px;" \| $$[\mathbf p, \mathbf q]^{\Large\unicode{x27D1}}_-$$~~
	\| style="padding: 12px;" \| Line containing points $$\mathbf p$$ and $$\mathbf q$$.		\| style="padding: 12px;" \| Line containing points $$\mathbf p$$ and $$\mathbf q$$.

Line 22:		Line 21:
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf p =\, &(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)\,\mathbf e_{423} \\ +\, &(l_{vz} p_x - l_{vx} p_z + l_{my} p_w)\,\mathbf e_{431} \\ +\, &(l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)\,\mathbf e_{412} \\ -\, &(l_{mx} p_x + l_{my} p_y + l_{mz} p_z)\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf p =\, &(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)\,\mathbf e_{423} \\ +\, &(l_{vz} p_x - l_{vx} p_z + l_{my} p_w)\,\mathbf e_{431} \\ +\, &(l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)\,\mathbf e_{412} \\ -\, &(l_{mx} p_x + l_{my} p_y + l_{mz} p_z)\,\mathbf e_{321}\end{split}$$
	~~\| style="padding: 12px;" \| $$[\boldsymbol l, \mathbf p]^{\Large\unicode{x27D1}}_+$$~~
	\| style="padding: 12px;" \| Plane containing line $$\boldsymbol l$$ and point $$\mathbf p$$.		\| style="padding: 12px;" \| Plane containing line $$\boldsymbol l$$ and point $$\mathbf p$$.

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	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf g \vee \mathbf h =\, &(g_zh_y - g_yh_z)\,\mathbf e_{41} + (g_xh_z - g_zh_x)\,\mathbf e_{42} + (g_yh_x - g_xh_y)\,\mathbf e_{43} \\ +\, &(g_xh_w - g_wh_x)\,\mathbf e_{23} + (g_yh_w - g_wh_y)\,\mathbf e_{31} + (g_zh_w - g_wh_z)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf g \vee \mathbf h =\, &(g_zh_y - g_yh_z)\,\mathbf e_{41} + (g_xh_z - g_zh_x)\,\mathbf e_{42} + (g_yh_x - g_xh_y)\,\mathbf e_{43} \\ +\, &(g_xh_w - g_wh_x)\,\mathbf e_{23} + (g_yh_w - g_wh_y)\,\mathbf e_{31} + (g_zh_w - g_wh_z)\,\mathbf e_{12}\end{split}$$
	~~\| style="padding: 12px;" \| $$[\mathbf g, \mathbf h]^{\Large\unicode{x27C7}}_-$$~~
	\| style="padding: 12px;" \| Line where planes $$\mathbf g$$ and $$\mathbf h$$ intersect.		\| style="padding: 12px;" \| Line where planes $$\mathbf g$$ and $$\mathbf h$$ intersect.

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	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \vee \mathbf g =\, &(l_{my} g_z - l_{mz} g_y + l_{vx} g_w)\,\mathbf e_1 \\ +\, &(l_{mz} g_x - l_{mx} g_z + l_{vy} g_w)\,\mathbf e_2 \\ +\, &(l_{mx} g_y - l_{my} g_x + l_{vz} g_w)\,\mathbf e_3 \\ -\, &(l_{vx} g_x + l_{vy} g_y + l_{vz} g_z)\,\mathbf e_4\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \vee \mathbf g =\, &(l_{my} g_z - l_{mz} g_y + l_{vx} g_w)\,\mathbf e_1 \\ +\, &(l_{mz} g_x - l_{mx} g_z + l_{vy} g_w)\,\mathbf e_2 \\ +\, &(l_{mx} g_y - l_{my} g_x + l_{vz} g_w)\,\mathbf e_3 \\ -\, &(l_{vx} g_x + l_{vy} g_y + l_{vz} g_z)\,\mathbf e_4\end{split}$$
	~~\| style="padding: 12px;" \| $$[\boldsymbol l, \mathbf g]^{\Large\unicode{x27C7}}_+$$~~
	\| style="padding: 12px;" \| Point where line $$\boldsymbol l$$ intersects plane $$\mathbf g$$.		\| style="padding: 12px;" \| Point where line $$\boldsymbol l$$ intersects plane $$\mathbf g$$.

Line 42:		Line 38:
	\| style="padding: 12px;" \| [[Image:line_meet_plane.svg\|250px]]		\| style="padding: 12px;" \| [[Image:line_meet_plane.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}~~\underline{~~\mathbf ~~g_\smash{\unicode{x25CB}}}~~ \wedge \mathbf p =\, &-g_xp_w \mathbf e_{41} - g_yp_w \mathbf e_{42} - g_zp_w \mathbf e_{43} \\ +\, &(g_yp_z - g_zp_y)\,\mathbf e_{23} + (g_zp_x - g_xp_z)\,\mathbf e_{31} + (g_xp_y - g_yp_x)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf g^* \wedge \mathbf p =\, &-g_xp_w \mathbf e_{41} - g_yp_w \mathbf e_{42} - g_zp_w \mathbf e_{43} \\ +\, &(g_yp_z - g_zp_y)\,\mathbf e_{23} + (g_zp_x - g_xp_z)\,\mathbf e_{31} + (g_xp_y - g_yp_x)\,\mathbf e_{12}\end{split}$$
	~~\| style="padding: 12px;" \| $$[\mathbf p, \mathbf g]^{\Large\unicode{x27C7}}_+~~$$
	\| style="padding: 12px;" \| Line perpendicular to plane $$\mathbf g$$ passing through point $$\mathbf p$$.		\| style="padding: 12px;" \| Line perpendicular to plane $$\mathbf g$$ passing through point $$\mathbf p$$.
	\| style="padding: 12px;" \| [[Image:plane_weight_join_point.svg\|250px]]		\| style="padding: 12px;" \| [[Image:plane_weight_join_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}~~\underline{~~\boldsymbol ~~l_\smash{\unicode{x25CB}}}~~ \wedge \mathbf p =\, &-l_{vx} p_w \mathbf e_{423} - l_{vy} p_w \mathbf e_{431} - l_{vz} p_w \mathbf e_{412} \\ +\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l^* \wedge \mathbf p =\, &-l_{vx} p_w \mathbf e_{423} - l_{vy} p_w \mathbf e_{431} - l_{vz} p_w \mathbf e_{412} \\ +\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\,\mathbf e_{321}\end{split}$$
	~~\| style="padding: 12px;" \| $$-[\mathbf p, \boldsymbol l]^{\Large\unicode{x27C7}}_+~~$$
	\| style="padding: 12px;" \| Plane perpendicular to line $$\boldsymbol l$$ containing point $$\mathbf p$$.		\| style="padding: 12px;" \| Plane perpendicular to line $$\boldsymbol l$$ containing point $$\mathbf p$$.
	\| style="padding: 12px;" \| [[Image:line_weight_join_point.svg\|250px]]		\| style="padding: 12px;" \| [[Image:line_weight_join_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}~~\underline{~~\mathbf ~~g_\smash{\unicode{x25CB}}}~~ \wedge \boldsymbol l =\, &(g_zl_{vy} - g_yl_{vz})\,\mathbf e_{423} + (g_xl_{vz} - g_zl_{vx})\,\mathbf e_{431} + (g_yl_{vx} - g_xl_{vy})\,\mathbf e_{412} \\ -\, &(g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf g^* \wedge \boldsymbol l =\, &(g_zl_{vy} - g_yl_{vz})\,\mathbf e_{423} + (g_xl_{vz} - g_zl_{vx})\,\mathbf e_{431} + (g_yl_{vx} - g_xl_{vy})\,\mathbf e_{412} \\ -\, &(g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{321}\end{split}$$
	~~\| style="padding: 12px;" \| $$[\boldsymbol l, \mathbf g]^{\Large\unicode{x27C7}}_-~~$$
	\| style="padding: 12px;" \| Plane perpendicular to plane $$\mathbf g$$ containing line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Plane perpendicular to plane $$\mathbf g$$ containing line $$\boldsymbol l$$.

Eric Lengyel: Created page with "The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection. The points, lines, and planes appearing in the following tables are defined as follows: :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ :$$\mathbf..."

2023-07-15T05:38:56Z

Created page with "The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection. The points, lines, and planes appearing in the following tables are defined as follows: :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ :$$\mathbf..."

New page

The ''join'' is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The ''meet'' is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection.

The [[points]], [[lines]], and [[planes]] appearing in the following tables are defined as follows:

:$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
:$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
:$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
:$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
:$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$

The join operation is performed by taking the [[wedge product]] between two geometric objects. The meet operation is performed by taking the [[antiwedge product]] between two geometric objects.

{| class="wikitable"
! Formula || Commutator || Description || Illustration
|-
| style="padding: 12px;" | $$\begin{split}\mathbf p \wedge \mathbf q =\, &(p_wq_x - p_xq_w)\,\mathbf e_{41} + (p_wq_y - p_yq_w)\,\mathbf e_{42} + (p_wq_z - p_zq_w)\,\mathbf e_{43} \\ +\, &(p_yq_z - p_zq_y)\,\mathbf e_{23} + (p_zq_x - p_xq_z)\,\mathbf e_{31} + (p_xq_y - p_yq_x)\,\mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$[\mathbf p, \mathbf q]^{\Large\unicode{x27D1}}_-$$
| style="padding: 12px;" | Line containing points $$\mathbf p$$ and $$\mathbf q$$.

Zero if $$\mathbf p$$ and $$\mathbf q$$ are coincident.
| style="padding: 12px;" | [[Image:point_join_point.svg|250px]]
|-
| style="padding: 12px;" | $$\begin{split}\boldsymbol l \wedge \mathbf p =\, &(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)\,\mathbf e_{423} \\ +\, &(l_{vz} p_x - l_{vx} p_z + l_{my} p_w)\,\mathbf e_{431} \\ +\, &(l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)\,\mathbf e_{412} \\ -\, &(l_{mx} p_x + l_{my} p_y + l_{mz} p_z)\,\mathbf e_{321}\end{split}$$
| style="padding: 12px;" | $$[\boldsymbol l, \mathbf p]^{\Large\unicode{x27D1}}_+$$
| style="padding: 12px;" | Plane containing line $$\boldsymbol l$$ and point $$\mathbf p$$.

Zero if $$\mathbf p$$ lies on the line $$\boldsymbol l$$.
| style="padding: 12px;" | [[Image:line_join_point.svg|250px]]
|-
| style="padding: 12px;" | $$\begin{split}\mathbf g \vee \mathbf h =\, &(g_zh_y - g_yh_z)\,\mathbf e_{41} + (g_xh_z - g_zh_x)\,\mathbf e_{42} + (g_yh_x - g_xh_y)\,\mathbf e_{43} \\ +\, &(g_xh_w - g_wh_x)\,\mathbf e_{23} + (g_yh_w - g_wh_y)\,\mathbf e_{31} + (g_zh_w - g_wh_z)\,\mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$[\mathbf g, \mathbf h]^{\Large\unicode{x27C7}}_-$$
| style="padding: 12px;" | Line where planes $$\mathbf g$$ and $$\mathbf h$$ intersect.

Direction $$\mathbf v$$ is zero if $$\mathbf g$$ and $$\mathbf h$$ are parallel.
| style="padding: 12px;" | [[Image:plane_meet_plane.svg|250px]]
|-
| style="padding: 12px;" | $$\begin{split}\boldsymbol l \vee \mathbf g =\, &(l_{my} g_z - l_{mz} g_y + l_{vx} g_w)\,\mathbf e_1 \\ +\, &(l_{mz} g_x - l_{mx} g_z + l_{vy} g_w)\,\mathbf e_2 \\ +\, &(l_{mx} g_y - l_{my} g_x + l_{vz} g_w)\,\mathbf e_3 \\ -\, &(l_{vx} g_x + l_{vy} g_y + l_{vz} g_z)\,\mathbf e_4\end{split}$$
| style="padding: 12px;" | $$[\boldsymbol l, \mathbf g]^{\Large\unicode{x27C7}}_+$$
| style="padding: 12px;" | Point where line $$\boldsymbol l$$ intersects plane $$\mathbf g$$.

Weight $$w$$ is zero if $$\boldsymbol l$$ is parallel to $$\mathbf g$$.
| style="padding: 12px;" | [[Image:line_meet_plane.svg|250px]]
|-
| style="padding: 12px;" | $$\begin{split}\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-g_xp_w \mathbf e_{41} - g_yp_w \mathbf e_{42} - g_zp_w \mathbf e_{43} \\ +\, &(g_yp_z - g_zp_y)\,\mathbf e_{23} + (g_zp_x - g_xp_z)\,\mathbf e_{31} + (g_xp_y - g_yp_x)\,\mathbf e_{12}\end{split}$$
| style="padding: 12px;" | $$[\mathbf p, \mathbf g]^{\Large\unicode{x27C7}}_+$$
| style="padding: 12px;" | Line perpendicular to plane $$\mathbf g$$ passing through point $$\mathbf p$$.
| style="padding: 12px;" | [[Image:plane_weight_join_point.svg|250px]]
|-
| style="padding: 12px;" | $$\begin{split}\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-l_{vx} p_w \mathbf e_{423} - l_{vy} p_w \mathbf e_{431} - l_{vz} p_w \mathbf e_{412} \\ +\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\,\mathbf e_{321}\end{split}$$
| style="padding: 12px;" | $$-[\mathbf p, \boldsymbol l]^{\Large\unicode{x27C7}}_+$$
| style="padding: 12px;" | Plane perpendicular to line $$\boldsymbol l$$ containing point $$\mathbf p$$.
| style="padding: 12px;" | [[Image:line_weight_join_point.svg|250px]]
|-
| style="padding: 12px;" | $$\begin{split}\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \boldsymbol l =\, &(g_zl_{vy} - g_yl_{vz})\,\mathbf e_{423} + (g_xl_{vz} - g_zl_{vx})\,\mathbf e_{431} + (g_yl_{vx} - g_xl_{vy})\,\mathbf e_{412} \\ -\, &(g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{321}\end{split}$$
| style="padding: 12px;" | $$[\boldsymbol l, \mathbf g]^{\Large\unicode{x27C7}}_-$$
| style="padding: 12px;" | Plane perpendicular to plane $$\mathbf g$$ containing line $$\boldsymbol l$$.

Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.
| style="padding: 12px;" | [[Image:plane_weight_join_line.svg|250px]]
|}

== See Also ==

* [[Exterior products]]
* [[Commutators]]

← Older revision		Revision as of 20:34, 26 October 2023
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	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$
	\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.		\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:plane_connect_point.svg\|~~250px~~]]		\| style="padding: 12px; text-align: center;" \| [[Image:plane_connect_point.svg\|200px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$
	\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.
	\| style="padding: 12px;" \| [[Image:line_connect_point.svg\|~~250px~~]]		\| style="padding: 12px; text-align: center;" \| [[Image:line_connect_point.svg\|200px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$
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	Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.		Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:plane_connect_line.svg\|~~250px~~]]		\| style="padding: 12px; text-align: center;" \| [[Image:plane_connect_line.svg\|200px]]
	\|}		\|}

← Older revision		Revision as of 07:31, 23 October 2023
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	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$
	\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.		\| style="padding: 12px;" \| Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:~~plane_weight_join_point~~.svg\|250px]]		\| style="padding: 12px;" \| [[Image:plane_connect_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$
	\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.
	\| style="padding: 12px;" \| [[Image:~~line_weight_join_point~~.svg\|250px]]		\| style="padding: 12px;" \| [[Image:line_connect_point.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$		\| style="padding: 12px;" \| $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$
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	Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.		Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.
	\| style="padding: 12px;" \| [[Image:~~plane_weight_join_line~~.svg\|250px]]		\| style="padding: 12px;" \| [[Image:plane_connect_line.svg\|250px]]
	\|}		\|}

← Older revision		Revision as of 00:20, 26 August 2023
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	* [[Exterior products]]		* [[Exterior products]]
	* [[Commutators]]