Euclidean distance

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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

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

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:

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

$$d(\mathbf p, \mathbf q) = \left\Vert \mathbf q_{xyz}p_w - \mathbf p_{xyz}q_w\right\Vert \mathbf 1 + |p_wq_w| {\large\unicode{x1D7D9}}$$

Perpendicular distance $$d$$ between point $$\mathbf p$$ and line $$\boldsymbol l$$.

$$d(\mathbf p, \boldsymbol l) = \left\Vert \boldsymbol l_{\mathbf v} \times \mathbf p_{xyz} + p_w \boldsymbol l_{\mathbf m}\right\Vert \mathbf 1 + \left\Vert p_w \boldsymbol l_{\mathbf v}\right\Vert {\large\unicode{x1D7D9}}$$

Perpendicular distance $$d$$ between point $$\mathbf p$$ and plane $$\mathbf g$$.

$$d(\mathbf p, \mathbf g) = (\mathbf p \cdot \mathbf g)\mathbf 1 + \left\Vert p_w \mathbf g_{xyz} \right\Vert {\large\unicode{x1D7D9}}$$

Perpendicular distance $$d$$ between lines $$\mathbf k$$ and $$\boldsymbol l$$.

$$d(\boldsymbol l, \mathbf k) = -(\boldsymbol l_{\mathbf v} \cdot \mathbf k_{\mathbf m} + \boldsymbol l_{\mathbf m} \cdot \mathbf k_{\mathbf v})\mathbf 1 + \left\Vert \boldsymbol l_{\mathbf v} \times \mathbf k_{\mathbf v}\right\Vert {\large\unicode{x1D7D9}}$$

In the Book

  • Euclidean distances are discussed in Section 2.11.

See Also