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 $$\mathbf s$$.
|
|