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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
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| :$$d(\mathbf a, \mathbf b) = \left\Vert\operatorname{att}(\mathbf a \wedge \mathbf b)\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \wedge \operatorname{att}(\mathbf b)\right\Vert_\unicode{x25CB}$$.
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| 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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| The [[points]], [[lines]], and [[planes]] appearing in the distance formulas are defined as follows:
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| :$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
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| :$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
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| :$$\mathbf k = k_{vx} \mathbf e_{41} + k_{vy} \mathbf e_{42} + k_{vz} \mathbf e_{43} + k_{mx} \mathbf e_{23} + k_{my} \mathbf e_{31} + k_{mz} \mathbf e_{12}$$
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| :$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
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| :$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
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| {| class="wikitable"
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| ! Formula !! Interpretation !! Illustration
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| | style="padding: 12px;" | $$d(\mathbf p, \mathbf q) = \sqrt{(q_xp_w - p_xq_w)^2 + (q_yp_w - p_yq_w)^2 + (q_zp_w - p_zq_w)^2} + |p_wq_w|{\large\unicode{x1D7D9}}$$
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| | style="padding: 12px;" | Distance $$d$$ between points $$\mathbf p$$ and $$\mathbf q$$.
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| | style="padding: 12px; text-align: center;" | [[Image:distance_point_point.svg|122px]]
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| | style="padding: 12px;" | $$d(\mathbf p, \boldsymbol l) = \sqrt{(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)^2 + (l_{vz} p_x - l_{vx} p_z + l_{my} p_w)^2 + (l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)^2} + {\large\unicode{x1D7D9}}\sqrt{p_w^2(l_{vx}^2 + l_{vy}^2 + l_{vz}^2)}$$
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| | style="padding: 12px;" | Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.
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| | style="padding: 12px; text-align: center;" | [[Image:distance_point_line.svg|250px]]
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| | style="padding: 12px;" | $$d(\mathbf p, \mathbf g) = |p_xg_x + p_yg_y + p_zg_z + p_wg_w| + {\large\unicode{x1D7D9}}\sqrt{p_w^2(g_x^2 + g_y^2 + g_z^2)}$$
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| | style="padding: 12px;" | Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.
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| | style="padding: 12px; text-align: center;" | [[Image:distance_point_plane.svg|250px]]
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| | style="padding: 12px;" | $$d(\boldsymbol l, \mathbf k) = |l_{vx} k_{mx} + l_{vy} k_{my} + l_{vz} k_{mz} + k_{vx} l_{mx} + k_{vy} l_{my} + k_{vz} l_{mz}| + {\large\unicode{x1D7D9}}\sqrt{(l_{vy} k_{vz} - l_{vz} k_{vy})^2 + (l_{vz} k_{vx} - l_{vx} k_{vz})^2 + (l_{vx} k_{vy} - l_{vy} k_{vx})^2}$$
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| | style="padding: 12px;" | Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.
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| | style="padding: 12px; text-align: center;" | [[Image:distance_line_line.svg|287px]]
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| |}
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| == See Also ==
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| * [[Geometric norm]]
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| * [[Commutators]]
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