# Join and meet

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:

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

## Join

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

Formula Commutator Description
$$\begin{split}\mathbf p \wedge \mathbf q =\, &(q_xp_w - p_xq_w)\,\mathbf e_{41} + (q_yp_w - p_yq_w)\,\mathbf e_{42} + (q_zp_w - p_zq_w)\,\mathbf e_{43} \\ +\, &(p_yq_z - p_zq_y)\,\mathbf e_{23} + (p_zq_x - p_xq_z)\,\mathbf e_{31} + (p_xq_y - p_yq_x)\,\mathbf e_{12}\end{split}$$ $$\mathbf p \mathbin{\unicode{x27D1}\kern{-0.2em}\unicode{x2212}\kern{-0.2em}\unicode{x27D1}} \mathbf q$$ Line containing points $$\mathbf p$$ and $$\mathbf q$$.

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

$$\begin{split}\mathbf p \wedge \mathbf q \wedge \mathbf s =\, &(p_yq_zs_w \,&-\, p_zq_ys_w \,&+\, q_ys_zp_w \,&-\, q_zs_yp_w \,&+\, s_yp_zq_w \,&-\, s_zp_yq_w)\,\mathbf e_{234}\hspace{5em} \\ +\, &(p_zq_xs_w \,&-\, p_xq_zs_w \,&+\, q_zs_xp_w \,&-\, q_xs_zp_w \,&+\, s_zp_xq_w \,&-\, s_xp_zq_w)\,\mathbf e_{314} \\ +\, &(p_xq_ys_w \,&-\, p_yq_xs_w \,&+\, q_xs_yp_w \,&-\, q_ys_xp_w \,&+\, s_xp_yq_w \,&-\, s_yp_xq_w)\,\mathbf e_{124} \\ +\, &\rlap{(p_xq_zs_y - p_xq_ys_z + p_yq_xs_z - p_yq_zs_x + p_zq_ys_x - p_zq_xs_y)\,\mathbf e_{321}}\end{split}$$ Plane containing points $$\mathbf p$$, $$\mathbf q$$, and $$\mathbf s$$.

Zero if all three points are collinear.

$$\begin{split}\mathbf L \wedge \mathbf p =\, &(v_yp_z - v_zp_y + m_xp_w)\,\mathbf e_{234} \\ +\, &(v_zp_x - v_xp_z + m_yp_w)\,\mathbf e_{314} \\ +\, &(v_xp_y - v_yp_x + m_zp_w)\,\mathbf e_{124} \\ -\, &(m_xp_x + m_yp_y + m_zp_z)\,\mathbf e_{321}\end{split}$$ $$\mathbf L \mathbin{\unicode{x27D1}\kern{-0.2em}\unicode{x2212}\kern{-0.2em}\unicode{x27D1}} \mathbf p$$ Plane containing line $$\mathbf L$$ and point $$\mathbf p$$.

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

$$\begin{split}\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-f_xp_w \mathbf e_{41} - f_yp_w \mathbf e_{42} - f_zp_w \mathbf e_{43} \\ +\, &(f_yp_z - f_zp_y)\,\mathbf e_{23} + (f_zp_x - f_xp_z)\,\mathbf e_{31} + (f_xp_y - f_yp_x)\,\mathbf e_{12}\end{split}$$ $$\mathbf p \mathbin{\unicode{x27C7}\kern{-0.2em}\unicode{x2212}\kern{-0.2em}\unicode{x27C7}} \mathbf f$$ Line perpendicular to plane $$\mathbf f$$ passing through point $$\mathbf p$$.
$$\begin{split}\underline{\mathbf L_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-v_xp_w \mathbf e_{234} - v_yp_w \mathbf e_{314} - v_zp_w \mathbf e_{124} \\ +\, &(v_xp_x + v_yp_y + v_zp_z)\,\mathbf e_{321}\end{split}$$ $$\mathbf p \mathbin{\unicode{x27C7}\kern{-0.2em}\unicode{x2B}\kern{-0.2em}\unicode{x27C7}} \mathbf L$$ Plane perpendicular to line $$\mathbf L$$ containing point $$\mathbf p$$.
$$\begin{split}\underline{\mathbf f_\smash{\unicode{x25CB}}} \wedge \mathbf L =\, &(v_yf_z - v_zf_y)\,\mathbf e_{234} + (v_zf_x - v_xf_z)\,\mathbf e_{314} + (v_xf_y - v_yf_x)\,\mathbf e_{124} \\ -\, &(m_xf_x + m_yf_y + m_zf_z)\,\mathbf e_{321}\end{split}$$ $$\mathbf L \mathbin{\unicode{x27C7}\kern{-0.2em}\unicode{x2B}\kern{-0.2em}\unicode{x27C7}} \mathbf f$$ Plane perpendicular to plane $$\mathbf f$$ containing line $$\mathbf L$$.

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

## Meet

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

Formula Commutator Description
$$\begin{split}\mathbf f \vee \mathbf g =\, &(f_zg_y - f_yg_z)\,\mathbf e_{41} + (f_xg_z - f_zg_x)\,\mathbf e_{42} + (f_yg_x - f_xg_y)\,\mathbf e_{43} \\ +\, &(f_xg_w - g_xf_w)\,\mathbf e_{23} + (f_yg_w - g_yf_w)\,\mathbf e_{31} + (f_zg_w - g_zf_w)\,\mathbf e_{12}\end{split}$$ $$\mathbf f \mathbin{\unicode{x27C7}\kern{-0.2em}\unicode{x2212}\kern{-0.2em}\unicode{x27C7}} \mathbf g$$ Line where planes $$\mathbf f$$ and $$\mathbf g$$ intersect.

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

$$\begin{split}\mathbf f \vee \mathbf g \vee \mathbf h =\, &(f_zg_yh_w \,&-\, f_yg_zh_w \,&+\, g_zh_yf_w \,&-\, g_yh_zf_w \,&+\, h_zf_yg_w \,&-\, h_yf_zg_w&)\,\mathbf e_1 \\ +\, &(f_xg_zh_w \,&-\, f_zg_xh_w \,&+\, g_xh_zf_w \,&-\, g_zh_xf_w \,&+\, h_xf_zg_w \,&-\, h_zf_xg_w&)\,\mathbf e_2 \\ +\, &(f_yg_xh_w \,&-\, f_xg_yh_w \,&+\, g_yh_xf_w \,&-\, g_xh_yf_w \,&+\, h_yf_xg_w \,&-\, h_xf_yg_w&)\,\mathbf e_3 \\ +\, &(f_xg_yh_z \,&-\, f_xg_zh_y \,&+\, f_yg_zh_x \,&-\, f_yg_xh_z \,&+\, f_zg_xh_y \,&-\, f_zg_yh_x&)\,\mathbf e_4\end{split}$$ Point where planes $$\mathbf f$$, $$\mathbf g$$, and $$\mathbf h$$ intersect.

Weight $$w$$ is zero if all three planes intersect at parallel lines.

$$\begin{split}\mathbf L \vee \mathbf f =\, &(m_yf_z - m_zf_y + v_xf_w)\,\mathbf e_1 \\ +\, &(m_zf_x - m_xf_z + v_yf_w)\,\mathbf e_2 \\ +\, &(m_xf_y - m_yf_x + v_zf_w)\,\mathbf e_3 \\ -\, &(v_xf_x + v_yf_y + v_zf_z)\,\mathbf e_4\end{split}$$ $$\mathbf L \mathbin{\unicode{x27C7}\kern{-0.2em}\unicode{x2212}\kern{-0.2em}\unicode{x27C7}} \mathbf f$$ Point where line $$\mathbf L$$ intersects plane $$\mathbf f$$.

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

## Skew Lines

For two skew lines $$\mathbf K$$ and $$\mathbf L$$, the wedge product and antiwedge product both calculate a volume proportional to the Euclidean distance between the two lines. To calculate a third line $$\mathbf J$$ that contains a point on each of the lines $$K$$ and $$L$$, we use the commutator

$$\mathbf J = \mathbf K \mathbin{\unicode{x27C7}\kern{-0.2em}\unicode{x2212}\kern{-0.2em}\unicode{x27C7}} \mathbf L = (v_yw_z - v_zw_y)\mathbf e_{41} + (v_zw_x - v_xw_z)\mathbf e_{42} + (v_xw_y - v_yw_x)\mathbf e_{43} + (v_yn_z - v_zn_y + m_yw_z - m_zw_y)\mathbf e_{23} + (v_zn_x - v_xn_z + m_zw_x - m_xw_z)\mathbf e_{31} + (v_xn_y - v_yn_x + m_xw_y - m_yw_x)\mathbf e_{12}$$ .

The direction of $$\mathbf J$$ is perpendicular to the directions of $$\mathbf K$$ and $$\mathbf L$$, and it contains the closest points of approach between $$\mathbf K$$ and $$\mathbf L$$.