Expansion

The expansion operation is performed by taking the wedge product between an object A and the antidual of another object B with higher grade. The result is an object C that contains A and is orthogonal to B, allowing a projection of A onto B through a simple intersection of B 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	Illustration
Dipole containing round point $$\mathbf a$$ and orthogonal to sphere $$\mathbf s$$. $$\begin{split}\mathbf a \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606} =\, &(a_xs_u + a_ws_x)\,\mathbf e_{41} \,&+\, (a_ys_u + a_ws_y)\,\mathbf e_{42} \,&+\, (a_zs_u + a_ws_z)\,\mathbf e_{43} \\ +\, &(a_ys_z - a_zs_y)\,\mathbf e_{23} \,&+\, (a_zs_x - a_xs_z)\,\mathbf e_{31} \,&+\, (a_xs_y - a_ys_x)\,\mathbf e_{12} \\ -\, &(a_xs_w + a_us_x)\,\mathbf e_{15} \,&-\, (a_ys_w + a_us_y)\,\mathbf e_{25} \,&-\, (a_zs_w + a_us_z)\,\mathbf e_{35} + (a_us_u - a_ws_w)\,\mathbf e_{45} \end{split}$$
Dipole containing round point $$\mathbf a$$ and orthogonal to plane $$\mathbf g$$. $$\begin{split}\mathbf a \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &a_wg_x \mathbf e_{41} + a_wg_y \mathbf e_{42} + a_wg_z \mathbf e_{43} \\ +\, &(a_yg_z - a_zg_y)\,\mathbf e_{23} + (a_zg_x - a_xg_z)\,\mathbf e_{31} + (a_xg_y - a_yg_x)\,\mathbf e_{12} \\ -\, &(a_xg_w + a_ug_x)\,\mathbf e_{15} - (a_yg_w + a_ug_y)\,\mathbf e_{25} - (a_zg_w + a_ug_z)\,\mathbf e_{35} - a_wg_w \mathbf e_{45} \end{split}$$
Circle containing dipole $$\mathbf d$$ and orthogonal to sphere $$\mathbf s$$. $$\begin{split}\mathbf d \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606} =\, &(d_{vy}s_z - d_{vz}s_y - d_{mx}s_u)\,\mathbf e_{423} \,&+\, (d_{vz}s_x - d_{vx}s_z - d_{my}s_u)\,\mathbf e_{431} \\ +\, &(d_{vx}s_y - d_{vy}s_x - d_{mz}s_u)\,\mathbf e_{412} \,&-\, (d_{mx}s_x + d_{my}s_y + d_{mz}s_z)\,\mathbf e_{321} \\ -\, &(d_{vx}s_w + d_{pw}s_x + d_{px}s_u)\,\mathbf e_{415} \,&+\, (d_{pz}s_y - d_{py}s_z - d_{mx}s_w)\,\mathbf e_{235} \\ -\, &(d_{vy}s_w + d_{pw}s_y + d_{py}s_u)\,\mathbf e_{425} \,&+\, (d_{px}s_z - d_{pz}s_x - d_{my}s_w)\,\mathbf e_{315} \\ -\, &(d_{vz}s_w + d_{pw}s_z + d_{pz}s_u)\,\mathbf e_{435} \,&+\, (d_{py}s_x - d_{px}s_y - d_{mz}s_w)\,\mathbf e_{125} \end{split}$$
Circle containing dipole $$\mathbf d$$ and orthogonal to plane $$\mathbf g$$. $$\begin{split}\mathbf d \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(d_{vy}g_z - d_{vz}g_y)\,\mathbf e_{423} \,&+\, (d_{vz}g_x - d_{vx}g_z)\,\mathbf e_{431} \\ +\, &(d_{vx}g_y - d_{vy}g_x)\,\mathbf e_{412} \,&-\, (d_{mx}g_x + d_{my}g_y + d_{mz}g_z)\,\mathbf e_{321} \\ -\, &(d_{vx}g_w + d_{pw}g_x)\,\mathbf e_{415} \,&+\, (d_{pz}g_y - d_{py}g_z - d_{mx}g_w)\,\mathbf e_{235} \\ -\, &(d_{vy}g_w + d_{pw}g_y)\,\mathbf e_{425} \,&+\, (d_{px}g_z - d_{pz}g_x - d_{my}g_w)\,\mathbf e_{315} \\ -\, &(d_{vz}g_w + d_{pw}g_z)\,\mathbf e_{435} \,&+\, (d_{py}g_x - d_{px}g_y - d_{mz}g_w)\,\mathbf e_{125} \end{split}$$
Line containing flat point $$\mathbf p$$ and orthogonal to sphere $$\mathbf s$$. $$\begin{split}\mathbf p \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606} = -\, &(p_xs_u + p_ws_x)\,\mathbf e_{415} \,&-\, (p_ys_u + p_ws_y)\,\mathbf e_{425} \,&-\, (p_zs_u + p_ws_z)\,\mathbf e_{435} \\ +\, &(p_zs_y - p_ys_z)\,\mathbf e_{235} \,&+\, (p_xs_z - p_zs_x)\,\mathbf e_{315} \,&+\, (p_ys_x - p_xs_y)\,\mathbf e_{125} \end{split}$$
Line containing flat point $$\mathbf p$$ and orthogonal to plane $$\mathbf g$$. $$\begin{split}\mathbf p \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} = -\, &p_wg_x \mathbf e_{415} - p_wg_y \mathbf e_{425} - p_wg_z \mathbf e_{435} \\ +\, &(p_zg_y - p_yg_z)\,\mathbf e_{235} + (p_xg_z - p_zg_x)\,\mathbf e_{315} + (p_yg_x - p_xg_y)\,\mathbf e_{125} \end{split}$$
Sphere containing circle $$\mathbf c$$ and orthogonal to sphere $$\mathbf s$$. $$\begin{split}\mathbf c \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606} =\, &(s_uc_{gw} - s_xc_{gx} - s_yc_{gy} - s_zc_{gz})\,\mathbf e_{1234} \\ +\, &(s_yc_{vz} - s_zc_{vy} + s_uc_{mx} - s_wc_{gx})\,\mathbf e_{4235} \\ +\, &(s_zc_{vx} - s_xc_{vz} + s_uc_{my} - s_wc_{gy})\,\mathbf e_{4315} \\ +\, &(s_xc_{vy} - s_yc_{vx} + s_uc_{mz} - s_wc_{gz})\,\mathbf e_{4125} \\ +\, &(s_xc_{mx} + s_yc_{my} + s_zc_{mz} - s_wc_{gw})\,\mathbf e_{3215} \end{split}$$
Sphere containing circle $$\mathbf c$$ and orthogonal to plane $$\mathbf g$$. $$\begin{split}\mathbf c \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} = -\, &(g_xc_{gx} + g_yc_{gy} + g_zc_{gz})\,\mathbf e_{1234} \\ +\, &(g_yc_{vz} - g_zc_{vy} - g_wc_{gx})\,\mathbf e_{4235} \\ +\, &(g_zc_{vx} - g_xc_{vz} - g_wc_{gy})\,\mathbf e_{4315} \\ +\, &(g_xc_{vy} - g_yc_{vx} - g_wc_{gz})\,\mathbf e_{4125} \\ +\, &(g_xc_{mx} + g_yc_{my} + g_zc_{mz} - g_wc_{gw})\,\mathbf e_{3215} \end{split}$$
Plane containing line $$\boldsymbol l$$ and orthogonal to sphere $$\mathbf s$$. $$\begin{split}\boldsymbol l \wedge \mathbf s^\unicode["segoe ui symbol"]{x2606} =\, &(s_yl_{vz} - s_zl_{vy} + s_ul_{mx})\,\mathbf e_{4235} + (s_zl_{vx} - s_xl_{vz} + s_ul_{my})\,\mathbf e_{4315} \\ +\, &(s_xl_{vy} - s_yl_{vx} + s_ul_{mz})\,\mathbf e_{4125} + (s_xl_{mx} + s_yl_{my} + s_zl_{mz})\,\mathbf e_{3215} \end{split}$$
Plane containing line $$\boldsymbol l$$ and orthogonal to plane $$\mathbf g$$. $$\begin{split}\boldsymbol l \wedge \mathbf g^\unicode["segoe ui symbol"]{x2606} =\, &(g_yl_{vz} - g_zl_{vy})\,\mathbf e_{4235} + (g_zl_{vx} - g_xl_{vz})\,\mathbf e_{4315} \\ +\, &(g_xl_{vy} - g_yl_{vx})\,\mathbf e_{4125} + (g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{3215} \end{split}$$
Circle containing round point $$\mathbf a$$ and orthogonal to circle $$\mathbf c$$. $$\begin{split}\mathbf a \wedge \mathbf c^\unicode["segoe ui symbol"]{x2606} =\, &(c_{gz}a_y - c_{gy}a_z - c_{vx}a_w)\,\mathbf e_{423} \,&+\, (c_{gx}a_z - c_{gz}a_x - c_{vy}a_w)\,\mathbf e_{431} \\ +\, &(c_{gy}a_x - c_{gx}a_y - c_{vz}a_w)\,\mathbf e_{412} \,&+\, (c_{vx}a_x + c_{vy}a_y + c_{vz}a_z)\,\mathbf e_{321} \\ -\, &(c_{mx}a_w + c_{gw}a_x + c_{gx}a_u)\,\mathbf e_{415} \,&+\, (c_{my}a_z - c_{mz}a_y - c_{vx}a_u)\,\mathbf e_{235} \\ -\, &(c_{my}a_w + c_{gw}a_y + c_{gy}a_u)\,\mathbf e_{425} \,&+\, (c_{mz}a_x - c_{mx}a_z - c_{vy}a_u)\,\mathbf e_{315} \\ -\, &(c_{mz}a_w + c_{gw}a_z + c_{gz}a_u)\,\mathbf e_{435} \,&+\, (c_{mx}a_y - c_{my}a_x - c_{vz}a_u)\,\mathbf e_{125} \end{split}$$
Circle containing round point $$\mathbf a$$ and orthogonal to line $$\boldsymbol l$$. $$\begin{split}\mathbf a \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} = -\, &l_{vx}a_w \mathbf e_{423} - l_{vy}a_w \mathbf e_{431} - l_{vz}a_w \mathbf e_{412}\\ +\, &(l_{vx}a_x + l_{vy}a_y + l_{vz}a_z)\,\mathbf e_{321} \\ -\, &l_{mx}a_w \mathbf e_{415} + (l_{my}a_z - l_{mz}a_y - l_{vx}a_u)\,\mathbf e_{235} \\ -\, &l_{my}a_w \mathbf e_{425} + (l_{mz}a_x - l_{mx}a_z - l_{vy}a_u)\,\mathbf e_{315} \\ -\, &l_{mz}a_w \mathbf e_{435} + (l_{mx}a_y - l_{my}a_x - l_{vz}a_u)\,\mathbf e_{125} \end{split}$$
Plane containing flat point $$\mathbf p$$ and orthogonal to circle $$\mathbf c$$. $$\begin{split}\mathbf p \wedge \mathbf c^\unicode["segoe ui symbol"]{x2606} =\, &(c_{gz}p_y - c_{gy}p_z - c_{vx}p_w)\,\mathbf e_{4235} \,&+\, (c_{gx}p_z - c_{gz}p_x - c_{vy}p_w)\,\mathbf e_{4315} \\ +\, &(c_{gy}p_x - c_{gx}p_y - c_{vz}p_w)\,\mathbf e_{4125} \,&+\, (c_{vx}p_x + c_{vy}p_y + c_{vz}p_z)\,\mathbf e_{3215} \end{split}$$
Plane containing flat point $$\mathbf p$$ and orthogonal to line $$\boldsymbol l$$. $$\begin{split}\mathbf p \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} = -\, &l_{vx}p_w \mathbf e_{4235} - l_{vy}p_w \mathbf e_{4315} - l_{vz}p_w \mathbf e_{4125} \\ +\, &(l_{vx}p_x + l_{vy}p_y + l_{vz}p_z)\,\mathbf e_{3215} \end{split}$$
Sphere containing dipole $$\mathbf d$$ orthogonal to circle $$\mathbf c$$. $$\begin{split}\mathbf d \wedge \mathbf c^\unicode["segoe ui symbol"]{x2606} =\, &(c_{vx}d_{vx} + c_{vy}d_{vy} + c_{vz}d_{vz} + c_{gx}d_{mx} + c_{gy}d_{my} + c_{gz}d_{mz})\,\mathbf e_{1234} \\ +\, &(c_{my}d_{vz} - c_{mz}d_{vy} - c_{vx}d_{pw} + c_{gz}d_{py} - c_{gy}d_{pz} + c_{gw}d_{mx})\,\mathbf e_{4235} \\ +\, &(c_{mz}d_{vx} - c_{mx}d_{vz} - c_{vy}d_{pw} + c_{gx}d_{pz} - c_{gz}d_{px} + c_{gw}d_{my})\,\mathbf e_{4315} \\ +\, &(c_{mx}d_{vy} - c_{my}d_{vx} - c_{vz}d_{pw} + c_{gy}d_{px} - c_{gx}d_{py} + c_{gw}d_{mz})\,\mathbf e_{4125} \\ +\, &(c_{vx}d_{px} + c_{vy}d_{py} + c_{vz}d_{pz} + c_{mx}d_{mx} + c_{my}d_{my} + c_{mz}d_{mz})\,\mathbf e_{3215} \end{split}$$
Sphere containing dipole $$\mathbf d$$ and orthogonal to line $$\boldsymbol l$$. $$\begin{split}\mathbf d \wedge \boldsymbol l^\unicode["segoe ui symbol"]{x2606} =\, &(l_{vx}d_{vx} + l_{vy}d_{vy} + l_{vz}d_{vz})\,\mathbf e_{1234} \\ +\, &(l_{my}d_{vz} - l_{mz}d_{vy} - l_{vx}d_{pw})\,\mathbf e_{4235} \\ +\, &(l_{mz}d_{vx} - l_{mx}d_{vz} - l_{vy}d_{pw})\,\mathbf e_{4315} \\ +\, &(l_{mx}d_{vy} - l_{my}d_{vx} - l_{vz}d_{pw})\,\mathbf e_{4125} \\ +\, &(l_{vx}d_{px} + l_{vy}d_{py} + l_{vz}d_{pz} + l_{mx}d_{mx} + l_{my}d_{my} + l_{mz}d_{mz})\,\mathbf e_{3215} \end{split}$$
Sphere containing round point $$\mathbf a$$ and orthogonal to dipole $$\mathbf d$$. $$\begin{split}\mathbf a \wedge \mathbf d^\unicode["segoe ui symbol"]{x2606} =\, &(d_{vx}a_x + d_{vy}a_y + d_{vz}a_z - d_{pw}a_w)\,\mathbf e_{1234} \\ +\, &(d_{my}a_z - d_{mz}a_y + d_{px}a_w - d_{vx}a_u)\,\mathbf e_{4235} \\ +\, &(d_{mz}a_x - d_{mx}a_z + d_{py}a_w - d_{vy}a_u)\,\mathbf e_{4315} \\ +\, &(d_{mx}a_y - d_{my}a_x + d_{pz}a_w - d_{vz}a_u)\,\mathbf e_{4125} \\ +\, &(d_{pw}a_u - d_{px}a_x - d_{py}a_y - d_{pz}a_z)\,\mathbf e_{3215} \end{split}$$
Sphere containing round point $$\mathbf a$$ and centered at flat point $$\mathbf p$$. $$\begin{split}\mathbf a \wedge \mathbf p^\unicode["segoe ui symbol"]{x2606} = -\, &p_wa_w \mathbf e_{1234} + p_xa_w \mathbf e_{4235} + p_ya_w \mathbf e_{4315} + p_za_w \mathbf e_{4125} \\ +\, &(p_wa_u - p_xa_x - p_ya_y - p_za_z)\,\mathbf e_{3215} \end{split}$$

See Also

• Join and meet
• Exterior products