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 $$\mathcal G_{4,1}$$.

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 $$x \mathbf e_{23} + y \mathbf e_{31} + z \mathbf e_{12}$$.