Attitude

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The attitude function, denoted by $$\operatorname{att}$$, extracts the attitude of a geometry and returns a purely directional object. The attitude function is defined as

$$\operatorname{att}(\mathbf x) = \mathbf x \vee \underline{\mathbf e_4}$$ .

The attitude of a line is the line's direction as a vector, and the attitude of a plane is the plane's normal as a bivector.

The following table lists the attitude for the main types in the 4D rigid geometric algebra $$\mathcal G_{3,0,1}$$.

Type Definition Attitude
Magnitude $$\mathbf z = x \mathbf 1 + y {\large\unicode{x1d7d9}}$$ $$\operatorname{att}(\mathbf z) = y \mathbf e_{321}$$
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ $$\operatorname{att}(\mathbf p) = p_w \mathbf 1$$
Line $$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$ $$\operatorname{att}(\boldsymbol l) = l_{vx} \mathbf e_1 + l_{vy} \mathbf e_2 + l_{vz} \mathbf e_3$$
Plane $$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$ $$\operatorname{att}(\mathbf g) = g_x \mathbf e_{23} + g_y \mathbf e_{31} + g_z \mathbf e_{12}$$