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 | Commutator | Description | Illustration |
|---|---|---|---|
| $$\begin{split}\mathbf p \wedge \mathbf q =\, &(p_wq_x - p_xq_w)\,\mathbf e_{41} + (p_wq_y - p_yq_w)\,\mathbf e_{42} + (p_wq_z - p_zq_w)\,\mathbf e_{43} \\ +\, &(p_yq_z - p_zq_y)\,\mathbf e_{23} + (p_zq_x - p_xq_z)\,\mathbf e_{31} + (p_xq_y - p_yq_x)\,\mathbf e_{12}\end{split}$$ | $$[\mathbf p, \mathbf q]^{\Large\unicode{x27D1}}_-$$ | Line containing points $$\mathbf p$$ and $$\mathbf q$$.
Zero if $$\mathbf p$$ and $$\mathbf q$$ are coincident. |
|
| $$\begin{split}\boldsymbol l \wedge \mathbf p =\, &(l_{vy} p_z - l_{vz} p_y + l_{mx} p_w)\,\mathbf e_{423} \\ +\, &(l_{vz} p_x - l_{vx} p_z + l_{my} p_w)\,\mathbf e_{431} \\ +\, &(l_{vx} p_y - l_{vy} p_x + l_{mz} p_w)\,\mathbf e_{412} \\ -\, &(l_{mx} p_x + l_{my} p_y + l_{mz} p_z)\,\mathbf e_{321}\end{split}$$ | $$[\boldsymbol l, \mathbf p]^{\Large\unicode{x27D1}}_+$$ | Plane containing line $$\boldsymbol l$$ and point $$\mathbf p$$.
Zero if $$\mathbf p$$ lies on the line $$\boldsymbol l$$. |
|
| $$\begin{split}\mathbf g \vee \mathbf h =\, &(g_zh_y - g_yh_z)\,\mathbf e_{41} + (g_xh_z - g_zh_x)\,\mathbf e_{42} + (g_yh_x - g_xh_y)\,\mathbf e_{43} \\ +\, &(g_xh_w - g_wh_x)\,\mathbf e_{23} + (g_yh_w - g_wh_y)\,\mathbf e_{31} + (g_zh_w - g_wh_z)\,\mathbf e_{12}\end{split}$$ | $$[\mathbf g, \mathbf h]^{\Large\unicode{x27C7}}_-$$ | Line where planes $$\mathbf g$$ and $$\mathbf h$$ intersect.
Direction $$\mathbf v$$ is zero if $$\mathbf g$$ and $$\mathbf h$$ are parallel. |
|
| $$\begin{split}\boldsymbol l \vee \mathbf g =\, &(l_{my} g_z - l_{mz} g_y + l_{vx} g_w)\,\mathbf e_1 \\ +\, &(l_{mz} g_x - l_{mx} g_z + l_{vy} g_w)\,\mathbf e_2 \\ +\, &(l_{mx} g_y - l_{my} g_x + l_{vz} g_w)\,\mathbf e_3 \\ -\, &(l_{vx} g_x + l_{vy} g_y + l_{vz} g_z)\,\mathbf e_4\end{split}$$ | $$[\boldsymbol l, \mathbf g]^{\Large\unicode{x27C7}}_+$$ | Point where line $$\boldsymbol l$$ intersects plane $$\mathbf g$$.
Weight $$w$$ is zero if $$\boldsymbol l$$ is parallel to $$\mathbf g$$. |
|
| $$\begin{split}\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-g_xp_w \mathbf e_{41} - g_yp_w \mathbf e_{42} - g_zp_w \mathbf e_{43} \\ +\, &(g_yp_z - g_zp_y)\,\mathbf e_{23} + (g_zp_x - g_xp_z)\,\mathbf e_{31} + (g_xp_y - g_yp_x)\,\mathbf e_{12}\end{split}$$ | $$[\mathbf p, \mathbf g]^{\Large\unicode{x27C7}}_+$$ | Line perpendicular to plane $$\mathbf g$$ passing through point $$\mathbf p$$. | 
|
| $$\begin{split}\underline{\boldsymbol l_\smash{\unicode{x25CB}}} \wedge \mathbf p =\, &-l_{vx} p_w \mathbf e_{423} - l_{vy} p_w \mathbf e_{431} - l_{vz} p_w \mathbf e_{412} \\ +\, &(l_{vx} p_x + l_{vy} p_y + l_{vz} p_z)\,\mathbf e_{321}\end{split}$$ | $$-[\mathbf p, \boldsymbol l]^{\Large\unicode{x27C7}}_+$$ | Plane perpendicular to line $$\boldsymbol l$$ containing point $$\mathbf p$$. | 
|
| $$\begin{split}\underline{\mathbf g_\smash{\unicode{x25CB}}} \wedge \boldsymbol l =\, &(g_zl_{vy} - g_yl_{vz})\,\mathbf e_{423} + (g_xl_{vz} - g_zl_{vx})\,\mathbf e_{431} + (g_yl_{vx} - g_xl_{vy})\,\mathbf e_{412} \\ -\, &(g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{321}\end{split}$$ | $$[\boldsymbol l, \mathbf g]^{\Large\unicode{x27C7}}_-$$ | Plane perpendicular to plane $$\mathbf g$$ containing line $$\boldsymbol l$$.
Normal direction is $$(0,0,0)$$ if $$\boldsymbol l$$ is perpendicular to $$\mathbf g$$. |
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