Euclidean angle

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The cosine of the Euclidean angle $$\cos \phi(\mathbf a, \mathbf b)$$ between two geometric objects a and b can be measured by the homogeneous magnitude given by

$$\cos \phi(\mathbf a, \mathbf b) = \left\Vert \mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606}\right\Vert_\unicode{x25CF} + \left\Vert\mathbf a \right\Vert_\unicode{x25CB}\left\Vert\mathbf b \right\Vert_\unicode{x25CB}$$.

In the case that the grades of $$\mathbf a$$ and $$\mathbf b$$ are equal, a signed angle can be obtained by using the formula

$$\cos \phi(\mathbf a, \mathbf b) = \mathbf a \vee \mathbf b^\unicode["segoe ui symbol"]{x2606} + \left\Vert\mathbf a \right\Vert_\unicode{x25CB}\left\Vert\mathbf b \right\Vert_\unicode{x25CB}$$.

The following table lists formulas for angles 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 lines and planes appearing in the distance formulas are defined as follows:

$$\mathbf k = k_{vx} \mathbf e_{41} + k_{vy} \mathbf e_{42} + k_{vz} \mathbf e_{43} + k_{mx} \mathbf e_{23} + k_{my} \mathbf e_{31} + k_{mz} \mathbf e_{12}$$
$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$
Angle Formula Illustration
Cosine of angle $$\phi$$ between planes $$\mathbf g$$ and $$\mathbf h$$.

$$\cos \phi(\mathbf g, \mathbf h) = (\mathbf g_{xyz} \cdot \mathbf h_{xyz}) \mathbf 1 + \left\Vert\mathbf g \right\Vert_\unicode{x25CB}\left\Vert\mathbf h \right\Vert_\unicode{x25CB}$$

Cosine of angle $$\phi$$ between plane $$\mathbf g$$ and line $$\boldsymbol l$$.

$$\cos \phi(\mathbf g, \boldsymbol l) = \left\Vert \mathbf g_{xyz} \times \boldsymbol l_{\mathbf v}\right\Vert \mathbf 1 + \left\Vert\mathbf g \right\Vert_\unicode{x25CB}\left\Vert\boldsymbol l \right\Vert_\unicode{x25CB}$$

Cosine of angle $$\phi$$ between lines $$\boldsymbol l$$ and line $$\mathbf k$$.

$$\cos \phi(\boldsymbol l, \mathbf k) = (\boldsymbol l_{\mathbf v} \cdot \mathbf k_{\mathbf v})\mathbf 1 + \left\Vert\boldsymbol l \right\Vert_\unicode{x25CB}\left\Vert\mathbf k \right\Vert_\unicode{x25CB}$$

In the Book

  • Euclidean angles are discussed in Section 2.13.3.

See Also