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