Attitude
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
- $$\operatorname{att}(\mathbf x) = \mathbf x \vee \overline{\mathbf e_4}$$ .
The following table lists the attitude for the geometric objects in the 5D conformal geometric algebra $$\mathcal G_{4,1}$$.
| Type | Definition | Attitude |
|---|---|---|
| Flat point | $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$ | $$\operatorname{att}(\mathbf p) = p_w \mathbf e_5$$ |
| Line | $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$ | $$\operatorname{att}(\boldsymbol l) = l_{vx} \mathbf e_{15} + l_{vy} \mathbf e_{25} + l_{vz} \mathbf e_{35}$$ |
| Plane | $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$ | $$\operatorname{att}(\mathbf g) = g_x \mathbf e_{235} + g_y \mathbf e_{315} + g_z \mathbf e_{125}$$ |
| Round point | $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$ | $$\operatorname{att}(\mathbf a) = a_w \mathbf 1$$ |
| Dipole | $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$ | $$\operatorname{att}(\mathbf d) = d_{vx} \mathbf e_1 + d_{vy} \mathbf e_2 + d_{vz} \mathbf e_3 + d_{pw} \mathbf e_5$$ |
| Circle | $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$ | $$\operatorname{att}(\mathbf c) = c_{gx} \mathbf e_{23} + c_{gy} \mathbf e_{31} + c_{gz} \mathbf e_{12} + c_{vx} \mathbf e_{15} + c_{vy} \mathbf e_{25} + c_{vz} \mathbf e_{35}$$ |
| Sphere | $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$ | $$\operatorname{att}(\mathbf s) = s_u \mathbf e_{321} + s_x \mathbf e_{235} + s_y \mathbf e_{315} + s_z \mathbf e_{125}$$ |
The round points contained by a round object (dipole, circle, or sphere) differ from the object's center by a multiple of the object's attitude. In general, any point $$\mathbf p$$ contained in a grade $$k$$ object $$\mathbf x$$ is given by
- $$\mathbf p = \operatorname{cen}(\mathbf x) + \boldsymbol \alpha^* \vee \operatorname{att}(\mathbf x)$$ ,
where $$\boldsymbol \alpha$$ is a Euclidean $$(k - 2)$$-vector. If $$\mathbf x$$ is a dipole, then $$\boldsymbol \alpha$$ is a scalar, if $$\mathbf x$$ is a circle, then $$\boldsymbol \alpha$$ is a vector $$x \mathbf e_1 + y \mathbf e_2 + z \mathbf e_3$$, and if $$\mathbf x$$ is a sphere, then $$\boldsymbol \alpha$$ is a bivector $$x \mathbf e_{23} + y \mathbf e_{31} + z \mathbf e_{12}$$.