Geometric property

From Projective Geometric Algebra
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An element of a projective geometric algebra possesses the geometric property if and only if its bulk norm is a pure scalar and its weight norm is a pure antiscalar. The set of all elements possessing the geometric property is closed under both the geometric product and geometric antiproduct.

Every point $$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$ possesses the geometric property.

A line $$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$ possesses the geometric property when $$v_xm_x + v_ym_y + v_zm_z = 0$$.

Every plane $$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$ possesses the geometric property.

A motor $$\mathbf Q = r_x \mathbf e_{41} + r_y \mathbf e_{42} + r_z \mathbf e_{43} + r_w {\large\unicode{x1d7d9}} + u_x \mathbf e_{23} + u_y \mathbf e_{31} + u_z \mathbf e_{12} + u_w$$ possesses the geometric property when $$r_xy_x + r_yu_y + r_zu_z + r_wu_w = 0$$.

A flector $$\mathbf F = s_x \mathbf e_1 + s_y \mathbf e_2 + s_z \mathbf e_3 + s_w \mathbf e_4 + h_x \mathbf e_{234} + h_y \mathbf e_{314} + h_z \mathbf e_{124} + h_w \mathbf e_{321}$$ possesses the geometric property when $$s_xh_x + s_yh_y + s_zh_z + s_wh_w = 0$$.

See Also