# 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 flat points, lines, planes, round points, dipoles, circles, and spheres appearing in the following tables are defined as follows:

$$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$
$$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$
$$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$
$$\mathbf h = h_x \mathbf e_{4235} + h_y \mathbf e_{4315} + h_z \mathbf e_{4125} + h_w \mathbf e_{3215}$$
$$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
$$\mathbf b = b_x \mathbf e_1 + b_y \mathbf e_2 + b_z \mathbf e_3 + b_w \mathbf e_4 + b_u \mathbf e_5$$
$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
$$\mathbf f = f_{vx} \mathbf e_{41} + f_{vy} \mathbf e_{42} + f_{vz} \mathbf e_{43} + f_{mx} \mathbf e_{23} + f_{my} \mathbf e_{31} + f_{mz} \mathbf e_{12} + f_{px} \mathbf e_{15} + f_{py} \mathbf e_{25} + f_{pz} \mathbf e_{35} + f_{pw} \mathbf e_{45}$$
$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
$$\mathbf o = o_{gx} \mathbf e_{423} + o_{gy} \mathbf e_{431} + o_{gz} \mathbf e_{412} + o_{gw} \mathbf e_{321} + o_{vx} \mathbf e_{415} + o_{vy} \mathbf e_{425} + o_{vz} \mathbf e_{435} + o_{mx} \mathbf e_{235} + o_{my} \mathbf e_{315} + o_{mz} \mathbf e_{125}$$
$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
$$\mathbf t = t_u \mathbf e_{1234} + t_x \mathbf e_{4235} + t_y \mathbf e_{4315} + t_z \mathbf e_{4125} + t_w \mathbf e_{3215}$$

## The Join Operation

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

Formula Description Illustration
$$\begin{split}\mathbf a \wedge \mathbf b =\, &(a_wb_x - a_xb_w)\,\mathbf e_{41} \,&+\, (a_wb_y - a_yb_w)\,\mathbf e_{42} \,&+\, (a_wb_z - a_zb_w)\,\mathbf e_{43} \\ +\, &(a_yb_z - a_zb_y)\,\mathbf e_{23} \,&+\, (a_zb_x - a_xb_z)\,\mathbf e_{31} \,&+\, (a_xb_y - a_yb_x)\,\mathbf e_{12} \\ +\, &(a_xb_u - a_ub_x)\,\mathbf e_{15} \,&+\, (a_yb_u - a_ub_y)\,\mathbf e_{25} \,&+\, (a_zb_u - a_ub_z)\,\mathbf e_{35} + (a_wb_u - a_ub_w)\,\mathbf e_{45} \end{split}$$ Dipole containing round points $$\mathbf a$$ and $$\mathbf b$$.
$$\begin{split}\mathbf p \wedge \mathbf a =\, &(p_xa_w - p_wa_x)\,\mathbf e_{415} \,&+\, (p_ya_w - p_wa_y)\,\mathbf e_{425} \,&+\, (p_za_w - p_wa_z)\,\mathbf e_{435} \\ +\, &(p_za_y - p_ya_z)\,\mathbf e_{235} \,&+\, (p_xa_z - p_za_x)\,\mathbf e_{315} \,&+\, (p_ya_x - p_xa_y)\,\mathbf e_{125} \end{split}$$ Line containing flat point $$\mathbf p$$ and round point $$\mathbf a$$.
$$\begin{split}\mathbf d \wedge \mathbf a =\, &(d_{vy}a_z - d_{vz}a_y + d_{mx}a_w)\,\mathbf e_{423} \,&+\, (d_{vz}a_x - d_{vx}a_z + d_{my}a_w)\,\mathbf e_{431} \\ +\, &(d_{vx}a_y - d_{vy}a_x + d_{mz}a_w)\,\mathbf e_{412} \,&-\, (d_{mx}a_x + d_{my}a_y + d_{mz}a_z)\,\mathbf e_{321} \\ +\, &(d_{px}a_w - d_{pw}a_x + d_{vx}a_u)\,\mathbf e_{415} \,&+\, (d_{py}a_w - d_{pw}a_y + d_{vy}a_u)\,\mathbf e_{425} \,&+\, (d_{pz}a_w - d_{pw}a_z + d_{vz}a_u)\,\mathbf e_{435} \\ +\, &(d_{pz}a_y - d_{py}a_z + d_{mx}a_u)\,\mathbf e_{235} \,&+\, (d_{px}a_z - d_{pz}a_x + d_{my}a_u)\,\mathbf e_{315} \,&+\, (d_{py}a_x - d_{px}a_y + d_{mz}a_u)\,\mathbf e_{125} \end{split}$$ Circle containing dipole $$\mathbf d$$ and round point $$\mathbf a$$.
$$\begin{split}\boldsymbol l \wedge \mathbf a =\, &(l_{vz}a_y - l_{vy}a_z - l_{mx}a_w)\,\mathbf e_{4235} \,&+\, (l_{vx}a_z - l_{vz}a_x - l_{my}a_w)\,\mathbf e_{4315} \\ +\, &(l_{vy}a_x - l_{vx}a_y - l_{mz}a_w)\,\mathbf e_{4125} \,&-\, (l_{mx}a_x + l_{my}a_y + l_{mz}a_z)\,\mathbf e_{3215} \end{split}$$ Plane containing line $$\boldsymbol l$$ and round point $$\mathbf a$$.
$$\begin{split}\mathbf d \wedge \mathbf p =\, &(d_{vy}p_z - d_{vz}p_y + d_{mx}p_w)\,\mathbf e_{4235} \,&+\, (d_{vz}p_x - d_{vx}p_z + d_{my}p_w)\,\mathbf e_{4315} \\ +\, &(d_{vx}p_y - d_{vy}p_x + d_{mz}p_w)\,\mathbf e_{4125} \,&-\, (d_{mx}p_x + d_{my}p_y + d_{mz}p_z)\,\mathbf e_{3215} \end{split}$$ Plane containing dipole $$\mathbf d$$ and flat point $$\mathbf p$$.
$$\begin{split}\mathbf c \wedge \mathbf a = -\, &(c_{gx}a_x + c_{gy}a_y + c_{gz}a_z + c_{gw}a_w)\,\mathbf e_{1234} \\ +\, &(c_{vz}a_y - c_{vy}a_z + c_{gx}a_u - c_{mx}a_w)\,\mathbf e_{4235} \\ +\, &(c_{vx}a_z - c_{vz}a_x + c_{gy}a_u - c_{my}a_w)\,\mathbf e_{4315} \\ +\, &(c_{vy}a_x - c_{vx}a_y + c_{gz}a_u - c_{mz}a_w)\,\mathbf e_{4125} \\ +\, &(c_{mx}a_x + c_{my}a_y + c_{mz}a_z + c_{gw}a_u)\,\mathbf e_{3215} \end{split}$$ Sphere containing circle $$\mathbf c$$ and round point $$\mathbf a$$.
$$\begin{split}\mathbf d \wedge \mathbf f = -\, &(d_{vx}f_{mx} + d_{vy}f_{my} + d_{vz}f_{mz} + d_{mx}f_{vx} + d_{my}f_{vy} + d_{mz}f_{vz})\,\mathbf e_{1234} \\ +\, &(d_{vy}f_{pz} - d_{vz}f_{py} + d_{pz}f_{vy} - d_{py}f_{vz} + d_{mx}f_{pw} + d_{pw}f_{mx})\,\mathbf e_{4235} \\ +\, &(d_{vz}f_{px} - d_{vx}f_{pz} + d_{px}f_{vz} - d_{pz}f_{vx} + d_{my}f_{pw} + d_{pw}f_{my})\,\mathbf e_{4315} \\ +\, &(d_{vx}f_{py} - d_{vy}f_{px} + d_{py}f_{vx} - d_{px}f_{vy} + d_{mz}f_{pw} + d_{pw}f_{mz})\,\mathbf e_{4125} \\ -\, &(d_{mx}f_{px} + d_{my}f_{py} + d_{mz}f_{pz} + d_{px}f_{mx} + d_{py}f_{my} + d_{pz}f_{mz})\,\mathbf e_{3215} \end{split}$$ Sphere containing dipoles $$\mathbf d$$ and $$\mathbf f$$.

## The Meet Operation

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

Formula Description Illustration
$$\begin{split}\mathbf s \vee \mathbf t =\, &(s_ut_x - s_xt_u)\,\mathbf e_{423} \,&+\, (s_ut_y - s_yt_u)\,\mathbf e_{431} \,&+\, (s_ut_z - s_zt_u)\,\mathbf e_{412} + (s_ut_w - s_wt_u)\,\mathbf e_{321} \\ +\, &(s_zt_y - s_yt_z)\,\mathbf e_{415} \,&+\, (s_xt_z - s_zt_x)\,\mathbf e_{425} \,&+\, (s_yt_x - s_xt_y)\,\mathbf e_{435} \\ +\, &(s_xt_w - s_wt_x)\,\mathbf e_{235} \,&+\, (s_yt_w - s_wt_y)\,\mathbf e_{315} \,&+\, (s_zt_w - s_wt_z)\,\mathbf e_{125} \end{split}$$ Circle where spheres $$\mathbf s$$ and $$\mathbf t$$ intersect.

Zero if $$\mathbf s$$ and $$\mathbf t$$ are coincident.

$$\begin{split}\mathbf s \vee \mathbf g =\, &s_ug_x \mathbf e_{423} + s_ug_y \mathbf e_{431} + s_ug_z \mathbf e_{412} + s_ug_w \mathbf e_{321} \\ +\, &(s_zg_y - s_yg_z)\,\mathbf e_{415} + (s_xg_z - s_zg_x)\,\mathbf e_{425} + (s_yg_x - s_xg_y)\,\mathbf e_{435} \\ +\, &(s_xg_w - s_wg_x)\,\mathbf e_{235} + (s_yg_w - s_wg_y)\,\mathbf e_{315} + (s_zg_w - s_wg_z)\,\mathbf e_{125} \end{split}$$ Circle where sphere $$\mathbf s$$ and plane $$\mathbf g$$ intersect.
$$\begin{split}\mathbf g \vee \mathbf h =\, &(g_zh_y - g_yh_z)\,\mathbf e_{415} + (g_xh_z - g_zh_x)\,\mathbf e_{425} + (g_yh_x - g_xh_y)\,\mathbf e_{435} \\ +\, &(g_xh_w - g_wh_x)\,\mathbf e_{235} + (g_yh_w - g_wh_y)\,\mathbf e_{315} + (g_zh_w - g_wh_z)\,\mathbf e_{125} \end{split}$$ Line where planes $$\mathbf g$$ and plane $$\mathbf h$$ intersect.
$$\begin{split}\mathbf s \vee \mathbf c =\, &(s_yc_{gz} - s_zc_{gy} + s_uc_{vx})\,\mathbf e_{41} \,&+\, (s_wc_{gx} - s_xc_{gw} + s_uc_{mx})\,\mathbf e_{23} \\ +\, &(s_zc_{gx} - s_xc_{gz} + s_uc_{vy})\,\mathbf e_{42} \,&+\, (s_wc_{gy} - s_yc_{gw} + s_uc_{my})\,\mathbf e_{31} \\ +\, &(s_xc_{gy} - s_yc_{gx} + s_uc_{vz})\,\mathbf e_{43} \,&+\, (s_wc_{gz} - s_zc_{gw} + s_uc_{mz})\,\mathbf e_{12} \\ +\, &(s_zc_{my} - s_yc_{mz} + s_wc_{vx})\,\mathbf e_{15} \,&+\, (s_xc_{mz} - s_zc_{mx} + s_wc_{vy})\,\mathbf e_{25} \\ +\, &(s_yc_{mx} - s_xc_{my} + s_wc_{vz})\,\mathbf e_{35} \,&-\, (s_xc_{vx} + s_yc_{vy} + s_zc_{vz})\,\mathbf e_{45} \end{split}$$ Dipole where sphere $$\mathbf s$$ and circle $$\mathbf c$$ intersect.

Zero if $$\mathbf c$$ lies in $$\mathbf s$$.

$$\begin{split}\mathbf s \vee \boldsymbol l =\, &s_ul_{vx} \mathbf e_{41} + s_ul_{vy} \mathbf e_{42} + s_ul_{vz} \mathbf e_{43} + s_ul_{mx}\,\mathbf e_{23} + s_ul_{my}\,\mathbf e_{31} + s_ul_{mz}\,\mathbf e_{12} \\ +\, &(s_zl_{my} - s_yl_{mz} + s_wl_{vx})\,\mathbf e_{15} + (s_xl_{mz} - s_zl_{mx} + s_wl_{vy})\,\mathbf e_{25} \\ +\, &(s_yl_{mx} - s_xl_{my} + s_wl_{vz})\,\mathbf e_{35} - (s_xl_{vx} + s_yl_{vy} + s_zl_{vz})\,\mathbf e_{45} \end{split}$$ Dipole where sphere $$\mathbf s$$ and line $$\boldsymbol l$$ intersect.
$$\begin{split}\mathbf g \vee \boldsymbol l =\, &(g_zl_{my} - g_yl_{mz} + g_wl_{vx})\,\mathbf e_{15} + (g_xl_{mz} - g_zl_{mx} + g_wl_{vy})\,\mathbf e_{25} \\ +\, &(g_yl_{mx} - g_xl_{my} + g_wl_{vz})\,\mathbf e_{35} - (g_xl_{vx} + g_yl_{vy} + g_zl_{vz})\,\mathbf e_{45} \end{split}$$ Flat point where plane $$\mathbf g$$ and line $$\boldsymbol l$$ intersect.
$$\begin{split}\mathbf g \vee \mathbf c =\, &(g_yc_{gz} - g_zc_{gy})\,\mathbf e_{41} + (g_zc_{gx} - g_xc_{gz})\,\mathbf e_{42} + (g_xc_{gy} - g_yc_{gx})\,\mathbf e_{43} \\ +\, &(g_wc_{gx} - g_xc_{gw})\,\mathbf e_{23} + (g_wc_{gy} - g_yc_{gw})\,\mathbf e_{31} + (g_wc_{gz} - g_zc_{gw})\,\mathbf e_{12} \\ +\, &(g_zc_{my} - g_yc_{mz} + g_wc_{vx})\,\mathbf e_{15} + (g_xc_{mz} - g_zc_{mx} + g_wc_{vy})\,\mathbf e_{25} \\ +\, &(g_yc_{mx} - g_xc_{my} + g_wc_{vz})\,\mathbf e_{35} - (g_xc_{vx} + g_yc_{vy} + g_zc_{vz})\,\mathbf e_{45} \end{split}$$ Dipole where plane $$\mathbf g$$ and circle $$\mathbf c$$ intersect.

Zero if $$\mathbf c$$ lies in $$\mathbf g$$.

$$\begin{split}\mathbf c \vee \mathbf o =\, &(c_{gz}o_{my} - c_{gy}o_{mz} + c_{my}o_{gz} - c_{mz}o_{gy} + c_{vx}o_{gw} + c_{gw}o_{vx})\,\mathbf e_1 \\ +\, &(c_{gx}o_{mz} - c_{gz}o_{mx} + c_{mz}o_{gx} - c_{mx}o_{gz} + c_{vy}o_{gw} + c_{gw}o_{vy})\,\mathbf e_2 \\ +\, &(c_{gy}o_{mx} - c_{gx}o_{my} + c_{mx}o_{gy} - c_{my}o_{gx} + c_{vz}o_{gw} + c_{gw}o_{vz})\,\mathbf e_3 \\ -\, &(c_{gx}o_{vx} + c_{gy}o_{vy} + c_{gz}o_{vz} + c_{vx}o_{gx} + c_{vy}o_{gy} + c_{vz}o_{gz})\,\mathbf e_4 \\ -\, &(c_{mx}o_{vx} + c_{my}o_{vy} + c_{mz}o_{vz} + c_{vx}o_{mx} + c_{vy}o_{my} + c_{vz}o_{mz})\,\mathbf e_5 \end{split}$$ Round point contained by circles $$\mathbf c$$ and $$\mathbf o$$.

Result is real if circles are linked and imaginary otherwise.

$$\begin{split}\mathbf c \vee \boldsymbol l =\, &(c_{gz}l_{my} - c_{gy}l_{mz} + c_{gw}l_{vx})\,\mathbf e_1 \\ +\, &(c_{gx}l_{mz} - c_{gz}l_{mx} + c_{gw}l_{vy})\,\mathbf e_2 \\ +\, &(c_{gy}l_{mx} - c_{gx}l_{my} + c_{gw}l_{vz})\,\mathbf e_3 \\ -\, &(c_{gx}l_{vx} + c_{gy}l_{vy} + c_{gz}l_{vz})\,\mathbf e_4 \\ -\, &(c_{mx}l_{vx} + c_{my}l_{vy} + c_{mz}l_{vz} + c_{vx}l_{mx} + c_{vy}l_{my} + c_{vz}l_{mz})\,\mathbf e_5 \end{split}$$ Round point centered on line $$\boldsymbol l$$ and contained by circle $$\mathbf c$$.

Result is real if line passes through interior of circle and imaginary otherwise.

$$\begin{split}\mathbf s \vee \mathbf d =\, &(s_yd_{mz} - s_zd_{my} - s_wd_{vx} + s_ud_{px})\,\mathbf e_1 \\ +\, &(s_zd_{mx} - s_xd_{mz} - s_wd_{vy} + s_ud_{py})\,\mathbf e_2 \\ +\, &(s_xd_{my} - s_yd_{mx} - s_wd_{vz} + s_ud_{pz})\,\mathbf e_3 \\ +\, &(s_xd_{vx} + s_yd_{vy} + s_zd_{vz} + s_ud_{pw})\,\mathbf e_4 \\ -\, &(s_xd_{px} + s_yd_{py} + s_zd_{pz} + s_wd_{pw})\,\mathbf e_5 \end{split}$$ Round point contained by sphere $$\mathbf s$$ and dipole $$\mathbf d$$.
$$\begin{split}\mathbf g \vee \mathbf d =\, &(g_yd_{mz} - g_zd_{my} - g_wd_{vx})\,\mathbf e_1 \\ +\, &(g_zd_{mx} - g_xd_{mz} - g_wd_{vy})\,\mathbf e_2 \\ +\, &(g_xd_{my} - g_yd_{mx} - g_wd_{vz})\,\mathbf e_3 \\ +\, &(g_xd_{vx} + g_yd_{vy} + g_zd_{vz})\,\mathbf e_4 \\ -\, &(g_xd_{px} + g_yd_{py} + g_zd_{pz} + g_wd_{pw})\,\mathbf e_5 \end{split}$$ Round point centered in plane $$\mathbf g$$ and contained by dipole $$\mathbf d$$.
$$\begin{split}\mathbf s \vee \mathbf p =\, &s_up_x\mathbf e_1 + s_up_y\mathbf e_2 + s_up_z\mathbf e_3 + s_up_w\mathbf e_4 \\ -\, &(s_xp_x + s_yp_y + s_zp_z + s_wp_w)\,\mathbf e_5 \end{split}$$ Round point centered at flat point $$\mathbf p$$ and contained by sphere $$\mathbf s$$.