Geometric property

From Projective Geometric Algebra
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An element $$\mathbf a$$ of a projective geometric algebra possesses the geometric property if and only if the wedge product between $$\mathbf a$$ and its own reverse is zero. That is,

$$\mathbf a \wedge \mathbf{\tilde a} = 0$$ .

This implies the symmetric property

$$\mathbf a \vee \smash{\mathbf{\underset{\Large\unicode{x7E}}{a}}} = 0$$ .

The set of all elements possessing the geometric property is closed under both the geometric product and geometric antiproduct.

The following table lists the requirements that must be satisfied for the main types in the 4D projective geometric algebra $$\mathcal G_{3,0,1}$$ to possess the geometric property. Points and planes do not have any requirements—they all possess the geometric property.

Type Definition Requirement
Point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
Line $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ $$v_xm_x + v_ym_y + v_zm_z = 0$$
Plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$
Motor $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ $$r_xu_x + r_yu_y + r_zu_z + r_wu_w = 0$$
Flector $$\mathbf G = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ $$s_xh_x + s_yh_y + s_zh_z + s_wh_w = 0$$

See Also