Euclidean distance: Difference between revisions
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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 | ||
:$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$. | :$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode["segoe ui symbol"]{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode["segoe ui symbol"]{x25CB}$$. | ||
The following table lists formulas for | 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 | ||
:$$d(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode["segoe ui symbol"]{x25CB}$$. | |||
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 [[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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{| class="wikitable" | {| class="wikitable" | ||
! Formula | ! Distance Formula !! Illustration | ||
|- | |- | ||
| style="padding: 12px;" | $$d(\mathbf p, \mathbf q) = \ | | style="padding: 12px;" | Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$ | ||
$$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) = \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) = (\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) = -(\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}}$$ | |||
| style="padding: 12px; text-align: center;" | [[Image:distance_line_line.svg|287px]] | | style="padding: 12px; text-align: center;" | [[Image:distance_line_line.svg|287px]] | ||
|} | |} | ||
== In the Book == | |||
* Euclidean distances are discussed in Section 2.11. | |||
== See Also == | == See Also == | ||
* [[Euclidean angle]] | |||
* [[Geometric norm]] | * [[Geometric norm]] | ||
* [[Magnitude]] | * [[Magnitude]] | ||
* [[Attitude]] | * [[Attitude]] |
Latest revision as of 01:30, 8 July 2024
The Euclidean distance $$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["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$$, the dimension of the algebra, a signed distance can be obtained by using the formula
- $$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}$$.
The following table lists formulas for distances between the main types of geometric objects in the 4D rigid geometric algebra over 3D Euclidean space. These formulas are general and do not require the geometric objects to be unitized. Most of them become simpler if unitization can be assumed.
The points, lines, and planes appearing in the distance formulas are defined as follows:
- $$\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}$$
In the Book
- Euclidean distances are discussed in Section 2.11.