# 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:

$$\mathbf p = p_x \mathbf e_1 + p_y \mathbf e_2 + p_z \mathbf e_3 + p_w \mathbf e_4$$
$$\mathbf q = q_x \mathbf e_1 + q_y \mathbf e_2 + q_z \mathbf e_3 + q_w \mathbf e_4$$
$$\boldsymbol l = l_{vx} \mathbf e_{41} + l_{vy} \mathbf e_{42} + l_{vz} \mathbf e_{43} + l_{mx} \mathbf e_{23} + l_{my} \mathbf e_{31} + l_{mz} \mathbf e_{12}$$
$$\mathbf g = g_x \mathbf e_{423} + g_y \mathbf e_{431} + g_z \mathbf e_{412} + g_w \mathbf e_{321}$$
$$\mathbf h = h_x \mathbf e_{423} + h_y \mathbf e_{431} + h_z \mathbf e_{412} + h_w \mathbf e_{321}$$

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 Description Illustration
$$\begin{split}\mathbf p \wedge \mathbf q =\, &(p_wq_x - p_xq_w)\,\mathbf e_{41} + (p_wq_y - p_yq_w)\,\mathbf e_{42} + (p_wq_z - p_zq_w)\,\mathbf e_{43} \\ +\, &(p_yq_z - p_zq_y)\,\mathbf e_{23} + (p_zq_x - p_xq_z)\,\mathbf e_{31} + (p_xq_y - p_yq_x)\,\mathbf e_{12}\end{split}$$ Line containing points $$\mathbf p$$ and $$\mathbf q$$.

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

$$\begin{split}\boldsymbol l \wedge \mathbf p =\, &(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)\,\mathbf e_{423} \\ +\, &(l_{vz} p_x - l_{vx} p_z + l_{my} p_w)\,\mathbf e_{431} \\ +\, &(l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)\,\mathbf e_{412} \\ -\, &(l_{mx} p_x + l_{my} p_y + l_{mz} p_z)\,\mathbf e_{321}\end{split}$$ Plane containing line $$\boldsymbol l$$ and point $$\mathbf p$$.

Zero if $$\mathbf p$$ lies on the line $$\boldsymbol l$$.

$$\begin{split}\mathbf g \vee \mathbf h =\, &(g_zh_y - g_yh_z)\,\mathbf e_{41} + (g_xh_z - g_zh_x)\,\mathbf e_{42} + (g_yh_x - g_xh_y)\,\mathbf e_{43} \\ +\, &(g_xh_w - g_wh_x)\,\mathbf e_{23} + (g_yh_w - g_wh_y)\,\mathbf e_{31} + (g_zh_w - g_wh_z)\,\mathbf e_{12}\end{split}$$ Line where planes $$\mathbf g$$ and $$\mathbf h$$ intersect.

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

$$\begin{split}\boldsymbol l \vee \mathbf g =\, &(l_{my} g_z - l_{mz} g_y + l_{vx} g_w)\,\mathbf e_1 \\ +\, &(l_{mz} g_x - l_{mx} g_z + l_{vy} g_w)\,\mathbf e_2 \\ +\, &(l_{mx} g_y - l_{my} g_x + l_{vz} g_w)\,\mathbf e_3 \\ -\, &(l_{vx} g_x + l_{vy} g_y + l_{vz} g_z)\,\mathbf e_4\end{split}$$ Point where line $$\boldsymbol l$$ intersects plane $$\mathbf g$$.

Weight $$w$$ is zero if $$\boldsymbol l$$ is parallel to $$\mathbf g$$.

$$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &-p_wg_x \mathbf e_{41} - p_wg_y \mathbf e_{42} - p_wg_z \mathbf e_{43} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{23} + (p_xg_z - p_zg_x)\,\mathbf e_{31} + (p_yg_x - p_xg_y)\,\mathbf e_{12}\end{split}$$ Line containing point $$\mathbf p$$ and perpendicular to plane $$\mathbf g$$.
$$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &-p_w l_{vx} \mathbf e_{423} - p_w l_{vy} \mathbf e_{431} - p_w l_{vz} \mathbf e_{412} \\ +\, &(p_x l_{vx} + p_y l_{vy} + p_z l_{vz})\,\mathbf e_{321}\end{split}$$ Plane containing point $$\mathbf p$$ and perpendicular to line $$\boldsymbol l$$.
$$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vy} g_z - l_{vz} g_y)\,\mathbf e_{423} + (l_{vz} g_x - l_{vx} g_z)\,\mathbf e_{431} + (l_{vx} g_y - l_{vy} g_x)\,\mathbf e_{412} \\ -\, &(l_{mx} g_x + l_{my} g_y + l_{mz} g_z)\,\mathbf e_{321}\end{split}$$ Plane containing line $$\boldsymbol l$$ and perpendicular to plane $$\mathbf g$$.

Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$.