Expansion
The connect operation is performed by taking the wedge product between the dual of an object A and another object B with lower grade. The result is an object C that is orthogonal to A and contains B, allowing a projection of B onto A through a simple intersection of A and C.
The flat points, lines, planes, round points, dipoles, circles, and spheres appearing in the following tables are defined as follows:
- $$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$
- $$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$
- $$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$
- $$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
- $$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
- $$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
- $$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| Formula | Description | Illustration |
|---|---|---|
| $$\begin{split}\mathbf s^* \wedge \mathbf a = -\, &(s_xa_w + s_ua_x)\,\mathbf e_{41} \,&-\, (s_ya_w + s_ua_y)\,\mathbf e_{42} \,&-\, (s_za_w + s_ua_z)\,\mathbf e_{43} \\ +\, &(s_ya_z - s_za_y)\,\mathbf e_{23} \,&+\, (s_za_x - s_xa_z)\,\mathbf e_{31} \,&+\, (s_xa_y - s_ya_x)\,\mathbf e_{12} \\ +\, &(s_xa_u + s_wa_x)\,\mathbf e_{15} \,&+\, (s_ya_u + s_wa_y)\,\mathbf e_{25} \,&+\, (s_za_u + s_wa_z)\,\mathbf e_{35} + (s_wa_w - s_ua_u)\,\mathbf e_{45} \end{split}$$ | Dipole orthogonal to sphere $$\mathbf s$$ and containing round point $$\mathbf a$$. | 
|
| $$\begin{split}\mathbf g^* \wedge \mathbf a = -\, &g_xa_w \mathbf e_{41} - g_ya_w \mathbf e_{42} - g_za_w \mathbf e_{43} \\ +\, &(g_ya_z - g_za_y)\,\mathbf e_{23} + (g_za_x - g_xa_z)\,\mathbf e_{31} + (g_xa_y - g_ya_x)\,\mathbf e_{12} \\ +\, &(g_xa_u + g_wa_x)\,\mathbf e_{15} + (g_ya_u + g_wa_y)\,\mathbf e_{25} + (g_za_u + g_wa_z)\,\mathbf e_{35} + g_wa_w \mathbf e_{45} \end{split}$$ | Dipole orthogonal to plane $$\mathbf g$$ and containing round point $$\mathbf a$$. | 
|
| $$\begin{split}\mathbf s^* \wedge \mathbf d =\, &(s_zd_{vy} - s_yd_{vz} - s_ud_{mx})\,\mathbf e_{423} \,&+\, (s_xd_{vz} - s_zd_{vx} - s_ud_{my})\,\mathbf e_{431} \\ +\, &(s_yd_{vx} - s_xd_{vy} - s_ud_{mz})\,\mathbf e_{412} \,&-\, (s_xd_{mx} + s_yd_{my} + s_zd_{mz})\,\mathbf e_{321} \\ -\, &(s_xd_{pw} + s_wd_{vx} + s_ud_{px})\,\mathbf e_{415} \,&+\, (s_yd_{pz} - s_zd_{py} - s_wd_{mx})\,\mathbf e_{235} \\ -\, &(s_yd_{pw} + s_wd_{vy} + s_ud_{py})\,\mathbf e_{425} \,&+\, (s_zd_{px} - s_xd_{pz} - s_wd_{my})\,\mathbf e_{315} \\ -\, &(s_zd_{pw} + s_wd_{vz} + s_ud_{pz})\,\mathbf e_{435} \,&+\, (s_xd_{py} - s_yd_{px} - s_wd_{mz})\,\mathbf e_{125} \end{split}$$ | Circle orthogonal to sphere $$\mathbf s$$ and containing dipole $$\mathbf d$$. | 
|
| $$\begin{split}\mathbf g^* \wedge \mathbf d =\, &(g_zd_{vy} - g_yd_{vz})\,\mathbf e_{423} \,&+\, (g_xd_{vz} - g_zd_{vx})\,\mathbf e_{431} \\ +\, &(g_yd_{vx} - g_xd_{vy})\,\mathbf e_{412} \,&-\, (g_xd_{mx} + g_yd_{my} + g_zd_{mz})\,\mathbf e_{321} \\ -\, &(g_xd_{pw} + g_wd_{vx})\,\mathbf e_{415} \,&+\, (g_yd_{pz} - g_zd_{py} - g_wd_{mx})\,\mathbf e_{235} \\ -\, &(g_yd_{pw} + g_wd_{vy})\,\mathbf e_{425} \,&+\, (g_zd_{px} - g_xd_{pz} - g_wd_{my})\,\mathbf e_{315} \\ -\, &(g_zd_{pw} + g_wd_{vz})\,\mathbf e_{435} \,&+\, (g_xd_{py} - g_yd_{px} - g_wd_{mz})\,\mathbf e_{125} \end{split}$$ | Circle orthogonal to plane $$\mathbf g$$ and containing dipole $$\mathbf d$$. | 
|
| $$\begin{split}\mathbf s^* \wedge \mathbf p = -\, &(s_xp_w + s_up_x)\,\mathbf e_{415} \,&-\, (s_yp_w + s_up_y)\,\mathbf e_{425} \,&-\, (s_zp_w + s_up_z)\,\mathbf e_{435} \\ +\, &(s_yp_z - s_zp_y)\,\mathbf e_{235} \,&+\, (s_zp_x - s_xp_z)\,\mathbf e_{315} \,&+\, (s_xp_y - s_yp_x)\,\mathbf e_{125} \end{split}$$ | Line orthogonal to sphere $$\mathbf s$$ and containing flat point $$\mathbf p$$. | 
|
| $$\begin{split}\mathbf g^* \wedge \mathbf p = -\, &g_xp_w \mathbf e_{415} - g_yp_w \mathbf e_{425} - g_zp_w \mathbf e_{435} \\ +\, &(g_yp_z - g_zp_y)\,\mathbf e_{235} + (g_zp_x - g_xp_z)\,\mathbf e_{315} + (g_xp_y - g_yp_x)\,\mathbf e_{125} \end{split}$$ | Line orthogonal to plane $$\mathbf g$$ and containing flat point $$\mathbf p$$. | 
|
| $$\begin{split}\mathbf s^* \wedge \mathbf c =\, &(s_xc_{gx} + s_yc_{gy} + s_zc_{gz} - s_uc_{gw})\,\mathbf e_{1234} \\ +\, &(s_zc_{vy} - s_yc_{vz} + s_wc_{gx} - s_uc_{mx})\,\mathbf e_{4235} \\ +\, &(s_xc_{vz} - s_zc_{vx} + s_wc_{gy} - s_uc_{my})\,\mathbf e_{4315} \\ +\, &(s_yc_{vx} - s_xc_{vy} + s_wc_{gz} - s_uc_{mz})\,\mathbf e_{4125} \\ +\, &(s_wc_{gw} - s_xc_{mx} - s_yc_{my} - s_zc_{mz})\,\mathbf e_{3215} \end{split}$$ | Sphere orthogonal to sphere $$\mathbf s$$ and containing circle $$\mathbf c$$. | 
|
| $$\begin{split}\mathbf g^* \wedge \mathbf c =\, &(g_xc_{gx} + g_yc_{gy} + g_zc_{gz})\,\mathbf e_{1234} \\ +\, &(g_zc_{vy} - g_yc_{vz} + g_wc_{gx})\,\mathbf e_{4235} \\ +\, &(g_xc_{vz} - g_zc_{vx} + g_wc_{gy})\,\mathbf e_{4315} \\ +\, &(g_yc_{vx} - g_xc_{vy} + g_wc_{gz})\,\mathbf e_{4125} \\ +\, &(g_wc_{gw} - g_xc_{mx} - g_yc_{my} - g_zc_{mz})\,\mathbf e_{3215} \end{split}$$ | Sphere orthogonal to plane $$\mathbf g$$ and containing circle $$\mathbf c$$. | 
|
| $$\begin{split}\mathbf s^* \wedge \boldsymbol l =\, &(s_zl_{vy} - s_yl_{vz} - s_ul_{mx})\,\mathbf e_{4235} + (s_xl_{vz} - s_zl_{vx} - s_ul_{my})\,\mathbf e_{4315} \\ +\, &(s_yl_{vx} - s_xl_{vy} - s_ul_{mz})\,\mathbf e_{4125} - (s_xl_{mx} + s_yl_{my} + s_zl_{mz})\,\mathbf e_{3215} \end{split}$$ | Plane orthogonal to sphere $$\mathbf s$$ and containing line $$\boldsymbol l$$. | 
|
| $$\begin{split}\mathbf g^* \wedge \boldsymbol l =\, &(g_zl_{vy} - g_yl_{vz})\,\mathbf e_{4235} + (g_xl_{vz} - g_zl_{vx})\,\mathbf e_{4315} \\ +\, &(g_yl_{vx} - g_xl_{vy})\,\mathbf e_{4125} - (g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{3215} \end{split}$$ | Plane orthogonal to plane $$\mathbf g$$ and containing line $$\boldsymbol l$$. | 
|
| $$\begin{split}\mathbf c^* \wedge \mathbf a =\, &(c_{gy}a_z - c_{gz}a_y + c_{vx}a_w)\,\mathbf e_{423} \,&+\, (c_{gz}a_x - c_{gx}a_z + c_{vy}a_w)\,\mathbf e_{431} \\ +\, &(c_{gx}a_y - c_{gy}a_x + c_{vz}a_w)\,\mathbf e_{412} \,&-\, (c_{vx}a_x + c_{vy}a_y + c_{vz}a_z)\,\mathbf e_{321} \\ +\, &(c_{mx}a_w + c_{gw}a_x + c_{gx}a_u)\,\mathbf e_{415} \,&+\, (c_{mz}a_y - c_{my}a_z + c_{vx}a_u)\,\mathbf e_{235} \\ +\, &(c_{my}a_w + c_{gw}a_y + c_{gy}a_u)\,\mathbf e_{425} \,&+\, (c_{mx}a_z - c_{mz}a_x + c_{vy}a_u)\,\mathbf e_{315} \\ +\, &(c_{mz}a_w + c_{gw}a_z + c_{gz}a_u)\,\mathbf e_{435} \,&+\, (c_{my}a_x - c_{mx}a_y + c_{vz}a_u)\,\mathbf e_{125} \end{split}$$ | Circle orthogonal to circle $$\mathbf c$$ and containing round point $$\mathbf a$$. | 
|
| $$\begin{split}\boldsymbol l^* \wedge \mathbf a =\, &l_{vx}a_w \mathbf e_{423} + l_{vy}a_w \mathbf e_{431} + l_{vz}a_w \mathbf e_{412}\\ -\, &(l_{vx}a_x + l_{vy}a_y + l_{vz}a_z)\,\mathbf e_{321} \\ +\, &l_{mx}a_w \mathbf e_{415} + (l_{mz}a_y - l_{my}a_z + l_{vx}a_u)\,\mathbf e_{235} \\ +\, &l_{my}a_w \mathbf e_{425} + (l_{mx}a_z - l_{mz}a_x + l_{vy}a_u)\,\mathbf e_{315} \\ +\, &l_{mz}a_w \mathbf e_{435} + (l_{my}a_x - l_{mx}a_y + l_{vz}a_u)\,\mathbf e_{125} \end{split}$$ | Circle orthogonal to line $$\boldsymbol l$$ and containing round point $$\mathbf a$$. | 
|
| $$\begin{split}\mathbf c^* \wedge \mathbf p =\, &(c_{gy}p_z - c_{gz}p_y + c_{vx}p_w)\,\mathbf e_{4235} \,&+\, (c_{gz}p_x - c_{gx}p_z + c_{vy}p_w)\,\mathbf e_{4315} \\ +\, &(c_{gx}p_y - c_{gy}p_x + c_{vz}p_w)\,\mathbf e_{4125} \,&-\, (c_{vx}p_x + c_{vy}p_y + c_{vz}p_z)\,\mathbf e_{3215} \end{split}$$ | Plane orthogonal to circle $$\mathbf c$$ and containing flat point $$\mathbf p$$. | 
|
| $$\begin{split}\boldsymbol l^* \wedge \mathbf p =\, &l_{vx}p_w \mathbf e_{4235} + l_{vy}p_w \mathbf e_{4315} + l_{vz}p_w \mathbf e_{4125} \\ -\, &(l_{vx}p_x + l_{vy}p_y + l_{vz}p_z)\,\mathbf e_{3215} \end{split}$$ | Plane orthogonal to line $$\boldsymbol l$$ and containing flat point $$\mathbf p$$. | 
|
| $$\begin{split}\mathbf c^* \wedge \mathbf d = -\, &(c_{vx}d_{vx} + c_{vy}d_{vy} + c_{vz}d_{vz} + c_{gx}d_{mx} + c_{gy}d_{my} + c_{gz}d_{mz})\,\mathbf e_{1234} \\ +\, &(c_{mz}d_{vy} - c_{my}d_{vz} + c_{vx}d_{pw} + c_{gy}d_{pz} - c_{gz}d_{py} - c_{gw}d_{mx})\,\mathbf e_{4235} \\ +\, &(c_{mx}d_{vz} - c_{mz}d_{vx} + c_{vy}d_{pw} + c_{gz}d_{px} - c_{gx}d_{pz} - c_{gw}d_{my})\,\mathbf e_{4315} \\ +\, &(c_{my}d_{vx} - c_{mx}d_{vy} + c_{vz}d_{pw} + c_{gx}d_{py} - c_{gy}d_{px} - c_{gw}d_{mz})\,\mathbf e_{4125} \\ -\, &(c_{vx}d_{px} + c_{vy}d_{py} + c_{vz}d_{pz} + c_{mx}d_{mx} + c_{my}d_{my} + c_{mz}d_{mz})\,\mathbf e_{3215} \end{split}$$ | Sphere orthogonal to circle $$\mathbf c$$ and containing dipole $$\mathbf d$$. | 
|
| $$\begin{split}\boldsymbol l^* \wedge \mathbf d = -\, &(l_{vx}d_{vx} + l_{vy}d_{vy} + l_{vz}d_{vz})\,\mathbf e_{1234} \\ +\, &(l_{mz}d_{vy} - l_{my}d_{vz} + l_{vx}d_{pw})\,\mathbf e_{4235} \\ +\, &(l_{mx}d_{vz} - l_{mz}d_{vx} + l_{vy}d_{pw})\,\mathbf e_{4315} \\ +\, &(l_{my}d_{vx} - l_{mx}d_{vy} + l_{vz}d_{pw})\,\mathbf e_{4125} \\ -\, &(l_{vx}d_{px} + l_{vy}d_{py} + l_{vz}d_{pz} + l_{mx}d_{mx} + l_{my}d_{my} + l_{mz}d_{mz})\,\mathbf e_{3215} \end{split}$$ | Sphere orthogonal to line $$\boldsymbol l$$ and containing dipole $$\mathbf d$$. | 
|
| $$\begin{split}\mathbf d^* \wedge \mathbf a =\, &(d_{vx}a_x + d_{vy}a_y + d_{vz}a_z - d_{pw}a_w)\,\mathbf e_{1234} \\ +\, &(d_{my}a_z - d_{mz}a_y + d_{px}a_w - d_{vx}a_u)\,\mathbf e_{4235} \\ +\, &(d_{mz}a_x - d_{mx}a_z + d_{py}a_w - d_{vy}a_u)\,\mathbf e_{4315} \\ +\, &(d_{mx}a_y - d_{my}a_x + d_{pz}a_w - d_{vz}a_u)\,\mathbf e_{4125} \\ +\, &(d_{pw}a_u - d_{px}a_x - d_{py}a_y - d_{pz}a_z)\,\mathbf e_{3215} \end{split}$$ | Sphere orthogonal to dipole $$\mathbf d$$ and containing round point $$\mathbf a$$. | 
|
| $$\begin{split}\mathbf p^* \wedge \mathbf a = -\, &p_wa_w \mathbf e_{1234} + p_xa_w \mathbf e_{4235} + p_ya_w \mathbf e_{4315} + p_za_w \mathbf e_{4125} \\ +\, &(p_wa_u - p_xa_x - p_ya_y - p_za_z)\,\mathbf e_{3215} \end{split}$$ | Sphere centered at flat point $$\mathbf p$$ and containing round point $$\mathbf a$$. | 
|