Euclidean distance - Revision history

Eric Lengyel at 01:30, 8 July 2024

2024-07-08T01:30:31Z

← Older revision		Revision as of 01:30, 8 July 2024
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	The Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two geometric objects '''a''' and '''b''' can be measured by the homogeneous [[magnitude]] given by		The Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two geometric objects '''a''' and '''b''' can be measured by the homogeneous [[magnitude]] given by

	:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.		:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode["segoe ui symbol"]{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode["segoe ui symbol"]{x25CB}$$.

	In the case that the grades of $$\mathbf a$$ and $$\mathbf b$$ sum to $$n$$, the dimension of the algebra, a signed distance can be obtained by using the formula		In the case that the grades of $$\mathbf a$$ and $$\mathbf b$$ sum to $$n$$, the dimension of the algebra, a signed distance can be obtained by using the formula

	:$$d(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.		:$$d(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode["segoe ui symbol"]{x25CB}$$.

	The following table lists formulas for distances between the main types of geometric objects in the 4D rigid geometric algebra over 3D Euclidean space. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.		The following table lists formulas for distances between the main types of geometric objects in the 4D rigid geometric algebra over 3D Euclidean space. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.

Eric Lengyel at 21:15, 21 April 2024

2024-04-21T21:15:59Z

← Older revision		Revision as of 21:15, 21 April 2024
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	== See Also ==		== See Also ==

			* [[Euclidean angle]]
	* [[Geometric norm]]		* [[Geometric norm]]
	* [[Magnitude]]		* [[Magnitude]]
	* [[Attitude]]		* [[Attitude]]

Eric Lengyel at 21:15, 21 April 2024

2024-04-21T21:15:22Z

← Older revision		Revision as of 21:15, 21 April 2024
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	:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.		:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.

	The following table lists formulas for ~~Euclidean~~ distances between the main types of geometric objects in the 4D rigid geometric algebra ~~$$\mathcal G_{3,0,1}$$~~. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.		In the case that the grades of $$\mathbf a$$ and $$\mathbf b$$ sum to $$n$$, the dimension of the algebra, a signed distance can be obtained by using the formula

			:$$d(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.

			The following table lists formulas for distances between the main types of geometric objects in the 4D rigid geometric algebra over 3D Euclidean space. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.

	The [[points]], [[lines]], and [[planes]] appearing in the distance formulas are defined as follows:		The [[points]], [[lines]], and [[planes]] appearing in the distance formulas are defined as follows:
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	\| style="padding: 12px;" \| Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$		\| style="padding: 12px;" \| Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$

	$$d(\mathbf p, \mathbf q) = \~~sqrt~~{~~(q_xp_w~~ - ~~p_xq_w)^2 + (q_yp_w - p_yq_w)^2 + (q_zp_w - p_zq_w)^2~~} + \|p_wq_w\|{\large\unicode{x1D7D9}}$$		$$d(\mathbf p, \mathbf q) = \left\Vert \mathbf q_{xyz}p_w - \mathbf p_{xyz}q_w\right\Vert \mathbf 1 + \|p_wq_w\| {\large\unicode{x1D7D9}}$$
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_point.svg\|122px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_point.svg\|122px]]
	\|-		\|-
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.

	$$d(\mathbf p, \boldsymbol l) = \~~sqrt{(~~l_{vy} ~~p_z - l_~~{vz} ~~p_y~~ + ~~l_{mx}~~ p_w~~)^2 + (~~l_{vz} ~~p_x - l_{vx} p_z~~ + ~~l_{my}~~ p_w~~)^2 + (~~l_{vx} ~~p_y - l_{vy} p_x + l_{mz} p_w)^2} +~~ {\large\unicode{x1D7D9}~~}\sqrt{p_w^2(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)~~}$$		$$d(\mathbf p, \boldsymbol l) = \left\Vert \boldsymbol l_{\mathbf v} \times \mathbf p_{xyz} + p_w \boldsymbol l_{\mathbf m}\right\Vert \mathbf 1 + \left\Vert p_w \boldsymbol l_{\mathbf v}\right\Vert {\large\unicode{x1D7D9}}$$
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_line.svg\|250px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_line.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.

	$$d(\mathbf p, \mathbf g) = ~~\|p_xg_x + p_yg_y + p_zg_z + p_wg_w\|~~ + {\large\unicode{x1D7D9}~~}\sqrt{p_w^2(g_x^2 + g_y^2 + g_z^2)~~}$$		$$d(\mathbf p, \mathbf g) = (\mathbf p \cdot \mathbf g)\mathbf 1 + \left\Vert p_w \mathbf g_{xyz} \right\Vert {\large\unicode{x1D7D9}}$$
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_plane.svg\|250px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_plane.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.

	$$d(\boldsymbol l, \mathbf k) = \|l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} ~~+ k_{vx} l_{mx} + k_{vy} l_{my} + k_{vz} l_{mz}\| +~~ {\large\unicode{x1D7D9}~~}\sqrt{(l_{vy} k_{vz} - l_{vz} k_{vy})^2 + (l_{vz} k_{vx} - l_{vx} k_{vz})^2 + (l_{vx} k_{vy} - l_{vy} k_{vx})^2~~}$$		$$d(\boldsymbol l, \mathbf k) = -(\boldsymbol l_{\mathbf v} \cdot \mathbf k_{\mathbf m} + \boldsymbol l_{\mathbf m} \cdot \mathbf k_{\mathbf v})\mathbf 1 + \left\Vert \boldsymbol l_{\mathbf v} \times \mathbf k_{\mathbf v}\right\Vert {\large\unicode{x1D7D9}}$$
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_line_line.svg\|287px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_line_line.svg\|287px]]
	\|}		\|}

Eric Lengyel at 23:30, 13 April 2024

2024-04-13T23:30:41Z

← Older revision		Revision as of 23:30, 13 April 2024
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	{\| class="wikitable"		{\| class="wikitable"
	! Formula ~~!! Interpretation~~ !! Illustration		! Distance Formula !! Illustration
	\|-		\|-
	\| style="padding: 12px;" \| $$d(\mathbf p, \mathbf q) = \sqrt{(q_xp_w - p_xq_w)^2 + (q_yp_w - p_yq_w)^2 + (q_zp_w - p_zq_w)^2} + \|p_wq_w\|{\large\unicode{x1D7D9}}$$		\| style="padding: 12px;" \| Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$
	~~\| style="padding: 12px;" \| Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$.~~
			$$d(\mathbf p, \mathbf q) = \sqrt{(q_xp_w - p_xq_w)^2 + (q_yp_w - p_yq_w)^2 + (q_zp_w - p_zq_w)^2} + \|p_wq_w\|{\large\unicode{x1D7D9}}$$
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_point.svg\|122px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_point.svg\|122px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$d(\mathbf p, \boldsymbol l) = \sqrt{(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)^2 + (l_{vz} p_x - l_{vx} p_z + l_{my} p_w)^2 + (l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)^2} + {\large\unicode{x1D7D9}}\sqrt{p_w^2(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)}$$
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.

			$$d(\mathbf p, \boldsymbol l) = \sqrt{(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)^2 + (l_{vz} p_x - l_{vx} p_z + l_{my} p_w)^2 + (l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)^2} + {\large\unicode{x1D7D9}}\sqrt{p_w^2(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)}$$
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_line.svg\|250px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_line.svg\|250px]]
	\|-		\|-
	~~\| style="padding: 12px;" \| $$d(\mathbf p, \mathbf g) = \|p_xg_x + p_yg_y + p_zg_z + p_wg_w\| + {\large\unicode{x1D7D9}}\sqrt{p_w^2(g_x^2 + g_y^2 + g_z^2)}$$~~
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.

			$$d(\mathbf p, \mathbf g) = \|p_xg_x + p_yg_y + p_zg_z + p_wg_w\| + {\large\unicode{x1D7D9}}\sqrt{p_w^2(g_x^2 + g_y^2 + g_z^2)}$$
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_plane.svg\|250px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_plane.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$d(\boldsymbol l, \mathbf k) = \|l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} + k_{vx} l_{mx} + k_{vy} l_{my} + k_{vz} l_{mz}\| + {\large\unicode{x1D7D9}}\sqrt{(l_{vy} k_{vz} - l_{vz} k_{vy})^2 + (l_{vz} k_{vx} - l_{vx} k_{vz})^2 + (l_{vx} k_{vy} - l_{vy} k_{vx})^2}$$
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.

			$$d(\boldsymbol l, \mathbf k) = \|l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} + k_{vx} l_{mx} + k_{vy} l_{my} + k_{vz} l_{mz}\| + {\large\unicode{x1D7D9}}\sqrt{(l_{vy} k_{vz} - l_{vz} k_{vy})^2 + (l_{vz} k_{vx} - l_{vx} k_{vz})^2 + (l_{vx} k_{vy} - l_{vy} k_{vx})^2}$$
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_line_line.svg\|287px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_line_line.svg\|287px]]
	\|}		\|}


			== In the Book ==

			* Euclidean distances are discussed in Section 2.11.

	== See Also ==		== See Also ==

Eric Lengyel at 01:38, 3 August 2023

2023-08-03T01:38:59Z

← Older revision		Revision as of 01:38, 3 August 2023
Line 1:		Line 1:
	The Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two geometric objects '''a''' and '''b''' can be measured by the homogeneous [[magnitude]] given by		The Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two geometric objects '''a''' and '''b''' can be measured by the homogeneous [[magnitude]] given by

	:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \~~vee~~ \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.		:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.

	The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.		The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.

Eric Lengyel at 01:38, 3 August 2023

2023-08-03T01:38:06Z

← Older revision		Revision as of 01:38, 3 August 2023
Line 1:		Line 1:
	The Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two geometric objects '''a''' and '''b''' can be measured by the homogeneous [[magnitude]] given by		The Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two geometric objects '''a''' and '''b''' can be measured by the homogeneous [[magnitude]] given by

	:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a\~~right~~\~~Vert_\unicode~~{~~x25CB~~} ~~\vee \left\Vert~~\mathbf b\right\Vert_\unicode{x25CB}$$.		:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \vee \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.

	The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.		The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.

Eric Lengyel at 05:23, 1 August 2023

2023-08-01T05:23:24Z

← Older revision		Revision as of 05:23, 1 August 2023
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	The ~~Euclidean distance between geometric objects can be measured by homogeneous [[magnitudes]] of [[attitudes]]. In particular, the~~ Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two objects '''a''' and '''b''' is given by		The Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two geometric objects '''a''' and '''b''' can be measured by the homogeneous [[magnitude]] given by

	:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \~~wedge~~ \~~operatorname~~{~~att~~}(\mathbf b)\right\Vert_\unicode{x25CB}$$.		:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a\right\Vert_\unicode{x25CB} \vee \left\Vert\mathbf b\right\Vert_\unicode{x25CB}$$.

	The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.		The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.

Eric Lengyel: /* See Also */

2023-08-01T05:06:01Z

Eric Lengyel at 05:04, 1 August 2023

2023-08-01T05:04:46Z

Eric Lengyel: Created page with "The Euclidean distance between geometric objects can be measured by using commutators to calculate homogeneous magnitudes. The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. The points, lines, and p..."

2023-07-15T06:10:16Z

Created page with "The Euclidean distance between geometric objects can be measured by using commutators to calculate homogeneous magnitudes. The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed. The points, lines, and p..."

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The Euclidean distance between geometric objects can be measured by using [[commutators]] to calculate homogeneous [[magnitudes]].

The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.

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

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

{| class="wikitable"
! Formula !! Interpretation !! Illustration
|-
| style="padding: 12px;" | $$\left\Vert[\mathbf p, \mathbf q]^{\Large\unicode{x27C7}}_-\right\Vert_\unicode{x25CF} + \left\Vert[\mathbf p, \mathbf q]^{\Large\unicode{x27C7}}_+\right\Vert_\unicode{x25CB} = \sqrt{(q_xp_w - p_xq_w)^2 + (q_yp_w - p_yq_w)^2 + (q_zp_w - p_zq_w)^2} + |p_wq_w|{\large\unicode{x1D7D9}}$$
| style="padding: 12px;" | Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$.
| style="padding: 12px; text-align: center;" | [[Image:distance_point_point.svg|122px]]
|-
| style="padding: 12px;" | $$\left\Vert[\mathbf p, \boldsymbol l]^{\Large\unicode{x27C7}}_-\right\Vert_\unicode{x25CF} + \left\Vert[\mathbf p, \boldsymbol l]^{\Large\unicode{x27C7}}_+\right\Vert_\unicode{x25CB} = \sqrt{(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)^2 + (l_{vz} p_x - l_{vx} p_z + l_{my} p_w)^2 + (l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)^2} + {\large\unicode{x1D7D9}}\sqrt{p_w^2(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)}$$
| style="padding: 12px;" | Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.
| style="padding: 12px; text-align: center;" | [[Image:distance_point_line.svg|250px]]
|-
| style="padding: 12px;" | $$\left\Vert[\mathbf p, \mathbf g]^{\Large\unicode{x27C7}}_-\right\Vert_\unicode{x25CF} + \left\Vert[\mathbf p, \mathbf g]^{\Large\unicode{x27C7}}_+\right\Vert_\unicode{x25CB} = |p_xg_x + p_yg_y + p_zg_z + p_wg_w| + {\large\unicode{x1D7D9}}\sqrt{p_w^2(g_x^2 + g_y^2 + g_z^2)}$$
| style="padding: 12px;" | Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.
| style="padding: 12px; text-align: center;" | [[Image:distance_point_plane.svg|250px]]
|-
| style="padding: 12px;" | $$\left\Vert[\boldsymbol l, \mathbf k]^{\Large\unicode{x27C7}}_+\right\Vert_\unicode{x25CF} + \left\Vert[\boldsymbol l, \mathbf k]^{\Large\unicode{x27C7}}_-\right\Vert_\unicode{x25CB} = |l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} + k_{vx} l_{mx} + k_{vy} l_{my} + k_{vz} l_{mz}| + {\large\unicode{x1D7D9}}\sqrt{(l_{vy} k_{vz} - l_{vz} k_{vy})^2 + (l_{vz} k_{vx} - l_{vx} k_{vz})^2 + (l_{vx} k_{vy} - l_{vy} k_{vx})^2}$$
| style="padding: 12px;" | Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.
| style="padding: 12px; text-align: center;" | [[Image:distance_line_line.svg|287px]]
|}

== See Also ==

* [[Geometric norm]]
* [[Commutators]]

← Older revision		Revision as of 05:06, 1 August 2023
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	* [[Geometric norm]]		* [[Geometric norm]]
	* [[~~Commutators~~]]		* [[Magnitude]]
			* [[Attitude]]

← Older revision		Revision as of 05:04, 1 August 2023
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	The Euclidean distance between geometric objects can be measured by ~~using~~ [[~~commutators~~]] ~~to calculate homogeneous~~ [[~~magnitudes~~]].		The Euclidean distance between geometric objects can be measured by homogeneous [[magnitudes]] of [[attitudes]]. In particular, the Euclidean distance $$d(\mathbf a, \mathbf b)$$ between two objects '''a''' and '''b''' is given by

			:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.

	The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.		The following table lists formulas for Euclidean distances between the main types of geometric objects in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$. These formulas are general and do not require the geometric objects to be [[unitized]]. Most of them become simpler if unitization can be assumed.
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	! Formula !! Interpretation !! Illustration		! Formula !! Interpretation !! Illustration
	\|-		\|-
	\| style="padding: 12px;" \| $$~~\left\Vert[~~\mathbf p, \mathbf q~~]^{\Large\unicode{x27C7}}_-\right\Vert_\unicode{x25CF} + \left\Vert[\mathbf p, \mathbf q]^{\Large\unicode{x27C7}}_+\right\Vert_\unicode{x25CB}~~ = \sqrt{(q_xp_w - p_xq_w)^2 + (q_yp_w - p_yq_w)^2 + (q_zp_w - p_zq_w)^2} + \|p_wq_w\|{\large\unicode{x1D7D9}}$$		\| style="padding: 12px;" \| $$d(\mathbf p, \mathbf q) = \sqrt{(q_xp_w - p_xq_w)^2 + (q_yp_w - p_yq_w)^2 + (q_zp_w - p_zq_w)^2} + \|p_wq_w\|{\large\unicode{x1D7D9}}$$
	\| style="padding: 12px;" \| Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$.		\| style="padding: 12px;" \| Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$.
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_point.svg\|122px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_point.svg\|122px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$~~\left\Vert[~~\mathbf p, \boldsymbol l~~]^{\Large\unicode{x27C7}}_-\right\Vert_\unicode{x25CF} + \left\Vert[\mathbf p, \boldsymbol l]^{\Large\unicode{x27C7}}_+\right\Vert_\unicode{x25CB}~~ = \sqrt{(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)^2 + (l_{vz} p_x - l_{vx} p_z + l_{my} p_w)^2 + (l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)^2} + {\large\unicode{x1D7D9}}\sqrt{p_w^2(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)}$$		\| style="padding: 12px;" \| $$d(\mathbf p, \boldsymbol l) = \sqrt{(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)^2 + (l_{vz} p_x - l_{vx} p_z + l_{my} p_w)^2 + (l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)^2} + {\large\unicode{x1D7D9}}\sqrt{p_w^2(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)}$$
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_line.svg\|250px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_line.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$~~\left\Vert[~~\mathbf p, \mathbf g~~]^{\Large\unicode{x27C7}}_-\right\Vert_\unicode{x25CF} + \left\Vert[\mathbf p, \mathbf g]^{\Large\unicode{x27C7}}_+\right\Vert_\unicode{x25CB}~~ = \|p_xg_x + p_yg_y + p_zg_z + p_wg_w\| + {\large\unicode{x1D7D9}}\sqrt{p_w^2(g_x^2 + g_y^2 + g_z^2)}$$		\| style="padding: 12px;" \| $$d(\mathbf p, \mathbf g) = \|p_xg_x + p_yg_y + p_zg_z + p_wg_w\| + {\large\unicode{x1D7D9}}\sqrt{p_w^2(g_x^2 + g_y^2 + g_z^2)}$$
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_plane.svg\|250px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_point_plane.svg\|250px]]
	\|-		\|-
	\| style="padding: 12px;" \| $$~~\left\Vert[~~\boldsymbol l, \mathbf k~~]^{\Large\unicode{x27C7}}_+\right\Vert_\unicode{x25CF} + \left\Vert[\boldsymbol l, \mathbf k]^{\Large\unicode{x27C7}}_-\right\Vert_\unicode{x25CB}~~ = \|l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} + k_{vx} l_{mx} + k_{vy} l_{my} + k_{vz} l_{mz}\| + {\large\unicode{x1D7D9}}\sqrt{(l_{vy} k_{vz} - l_{vz} k_{vy})^2 + (l_{vz} k_{vx} - l_{vx} k_{vz})^2 + (l_{vx} k_{vy} - l_{vy} k_{vx})^2}$$		\| style="padding: 12px;" \| $$d(\boldsymbol l, \mathbf k) = \|l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} + k_{vx} l_{mx} + k_{vy} l_{my} + k_{vz} l_{mz}\| + {\large\unicode{x1D7D9}}\sqrt{(l_{vy} k_{vz} - l_{vz} k_{vy})^2 + (l_{vz} k_{vx} - l_{vx} k_{vz})^2 + (l_{vx} k_{vy} - l_{vy} k_{vx})^2}$$
	\| style="padding: 12px;" \| Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.		\| style="padding: 12px;" \| Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.
	\| style="padding: 12px; text-align: center;" \| [[Image:distance_line_line.svg\|287px]]		\| style="padding: 12px; text-align: center;" \| [[Image:distance_line_line.svg\|287px]]