# Join and meet

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The join is a binary operation that calculates the higher-dimensional geometry containing its two operands, similar to a union. The meet is another binary operation that calculates the lower-dimensional geometry shared by its two operands, similar to an intersection.

The points, lines, and planes appearing in the following tables are defined as follows:

$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
$$\mathbf K = w_x \mathbf e_{41} + w_y \mathbf e_{42} + w_z \mathbf e_{43} + n_x \mathbf e_{23} + n_y \mathbf e_{31} + n_z \mathbf e_{12}$$
$$\mathbf L = v_x \mathbf e_{41} + v_y \mathbf e_{42} + v_z \mathbf e_{43} + m_x \mathbf e_{23} + m_y \mathbf e_{31} + m_z \mathbf e_{12}$$
$$\mathbf f = f_x \mathbf e_{234} + f_y \mathbf e_{314} + f_z \mathbf e_{124} + f_w \mathbf e_{321}$$
$$\mathbf g = g_x \mathbf e_{234} + g_y \mathbf e_{314} + g_z \mathbf e_{124} + g_w \mathbf e_{321}$$

The join operation is performed by taking the wedge product between two geometric objects. The meet operation is performed by taking the antiwedge product between two geometric objects.

Formula Commutator Description Illustration
$$\begin{split}\mathbf p \wedge \mathbf q =\, &(q_xp_w - p_xq_w)\,\mathbf e_{41} + (q_yp_w - p_yq_w)\,\mathbf e_{42} + (q_zp_w - p_zq_w)\,\mathbf e_{43} \\ +\, &(p_yq_z - p_zq_y)\,\mathbf e_{23} + (p_zq_x - p_xq_z)\,\mathbf e_{31} + (p_xq_y - p_yq_x)\,\mathbf e_{12}\end{split}$$ $$[\mathbf p, \mathbf q]^{\Large\unicode{x27D1}}_-$$ Line containing points $$\mathbf p$$ and $$\mathbf q$$.

Zero if $$\mathbf p$$ and $$\mathbf q$$ are coincident.

$$\begin{split}\mathbf L \wedge \mathbf p =\, &(v_yp_z - v_zp_y + m_xp_w)\,\mathbf e_{234} \\ +\, &(v_zp_x - v_xp_z + m_yp_w)\,\mathbf e_{314} \\ +\, &(v_xp_y - v_yp_x + m_zp_w)\,\mathbf e_{124} \\ -\, &(m_xp_x + m_yp_y + m_zp_z)\,\mathbf e_{321}\end{split}$$ $$[\mathbf L, \mathbf p]^{\Large\unicode{x27D1}}_+$$ Plane containing line $$\mathbf L$$ and point $$\mathbf p$$.

Zero if $$\mathbf p$$ lies on the line $$\mathbf L$$.

$$\begin{split}\mathbf f \vee \mathbf g =\, &(f_zg_y - f_yg_z)\,\mathbf e_{41} + (f_xg_z - f_zg_x)\,\mathbf e_{42} + (f_yg_x - f_xg_y)\,\mathbf e_{43} \\ +\, &(f_xg_w - g_xf_w)\,\mathbf e_{23} + (f_yg_w - g_yf_w)\,\mathbf e_{31} + (f_zg_w - g_zf_w)\,\mathbf e_{12}\end{split}$$ $$[\mathbf f, \mathbf g]^{\Large\unicode{x27C7}}_-$$ Line where planes $$\mathbf f$$ and $$\mathbf g$$ intersect.

Direction $$\mathbf v$$ is zero if $$\mathbf f$$ and $$\mathbf g$$ are parallel.

$$\begin{split}\mathbf L \vee \mathbf f =\, &(m_yf_z - m_zf_y + v_xf_w)\,\mathbf e_1 \\ +\, &(m_zf_x - m_xf_z + v_yf_w)\,\mathbf e_2 \\ +\, &(m_xf_y - m_yf_x + v_zf_w)\,\mathbf e_3 \\ -\, &(v_xf_x + v_yf_y + v_zf_z)\,\mathbf e_4\end{split}$$ $$[\mathbf L, \mathbf f]^{\Large\unicode{x27C7}}_+$$ Point where line $$\mathbf L$$ intersects plane $$\mathbf f$$.

Weight $$w$$ is zero if $$\mathbf L$$ is parallel to $$\mathbf f$$.

$$\begin{split}\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-f_xp_w \mathbf e_{41} - f_yp_w \mathbf e_{42} - f_zp_w \mathbf e_{43} \\ +\, &(f_yp_z - f_zp_y)\,\mathbf e_{23} + (f_zp_x - f_xp_z)\,\mathbf e_{31} + (f_xp_y - f_yp_x)\,\mathbf e_{12}\end{split}$$ $$[\mathbf p, \mathbf f]^{\Large\unicode{x27C7}}_+$$ Line perpendicular to plane $$\mathbf f$$ passing through point $$\mathbf p$$.
$$\begin{split}\underline{\mathbf L_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-v_xp_w \mathbf e_{234} - v_yp_w \mathbf e_{314} - v_zp_w \mathbf e_{124} \\ +\, &(v_xp_x + v_yp_y + v_zp_z)\,\mathbf e_{321}\end{split}$$ $$-[\mathbf p, \mathbf L]^{\Large\unicode{x27C7}}_+$$ Plane perpendicular to line $$\mathbf L$$ containing point $$\mathbf p$$.
$$\begin{split}\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf L =\, &(v_yf_z - v_zf_y)\,\mathbf e_{234} + (v_zf_x - v_xf_z)\,\mathbf e_{314} + (v_xf_y - v_yf_x)\,\mathbf e_{124} \\ -\, &(m_xf_x + m_yf_y + m_zf_z)\,\mathbf e_{321}\end{split}$$ $$[\mathbf L, \mathbf f]^{\Large\unicode{x27C7}}_-$$ Plane perpendicular to plane $$\mathbf f$$ containing line $$\mathbf L$$.

Normal direction is $$(0,0,0)$$ if $$\mathbf L$$ is perpendicular to $$\mathbf f$$.