Expansion

From Conformal Geometric Algebra
Revision as of 03:19, 6 August 2023 by Eric Lengyel (talk | contribs) (Created page with "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...")
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to navigation Jump to search

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
$$\begin{split}\mathbf s^* \wedge \mathbf a =\, &(s_xa_w + s_ua_x)\,\mathbf e_{41} \,&+\, (s_ya_w + s_ua_y)\,\mathbf e_{42} \,&+\, (s_za_w + s_ua_z)\,\mathbf e_{43} \\ +\, &(s_za_y - s_ya_z)\,\mathbf e_{23} \,&+\, (s_xa_z - s_za_x)\,\mathbf e_{31} \,&+\, (s_ya_x - s_xa_y)\,\mathbf e_{12} \\ -\, &(s_xa_u + s_wa_x)\,\mathbf e_{15} \,&-\, (s_ya_u + s_wa_y)\,\mathbf e_{25} \,&-\, (s_za_u + s_wa_z)\,\mathbf e_{35} + (s_ua_u - s_wa_w)\,\mathbf e_{45} \end{split}$$ Dipole orthogonal to sphere $$\mathbf s$$ and containing round point $$\mathbf a$$.
$$\begin{split}\mathbf g^* \wedge \mathbf a =\, &g_xa_w \mathbf e_{41} + g_ya_w \mathbf e_{42} + g_za_w \mathbf e_{43} \\ +\, &(g_za_y - g_ya_z)\,\mathbf e_{23} + (g_xa_z - g_za_x)\,\mathbf e_{31} + (g_ya_x - g_xa_y)\,\mathbf e_{12} \\ -\, &(g_xa_u + g_wa_x)\,\mathbf e_{15} - (g_ya_u + g_wa_y)\,\mathbf e_{25} - (g_za_u + g_wa_z)\,\mathbf e_{35} - g_wa_w \mathbf e_{45} \end{split}$$ Dipole orthogonal to plane $$\mathbf g$$ and containing round point $$\mathbf a$$.
$$\begin{split}\mathbf s^* \wedge \mathbf d =\, &(s_yd_{vz} - s_zd_{vy} + s_ud_{mx})\,\mathbf e_{423} \,&+\, (s_zd_{vx} - s_xd_{vz} + s_ud_{my})\,\mathbf e_{431} \\ +\, &(s_xd_{vy} - s_yd_{vx} + s_ud_{mz})\,\mathbf e_{412} \,&+\, (s_xd_{mx} + s_yd_{my} + s_zd_{mz})\,\mathbf e_{321} \\ +\, &(s_xd_{pw} + s_wd_{vx} + s_ud_{px})\,\mathbf e_{415} \,&+\, (s_zd_{py} - s_yd_{pz} + s_wd_{mx})\,\mathbf e_{235} \\ +\, &(s_yd_{pw} + s_wd_{vy} + s_ud_{py})\,\mathbf e_{425} \,&+\, (s_xd_{pz} - s_zd_{px} + s_wd_{my})\,\mathbf e_{315} \\ +\, &(s_zd_{pw} + s_wd_{vz} + s_ud_{pz})\,\mathbf e_{435} \,&+\, (s_yd_{px} - s_xd_{py} + s_wd_{mz})\,\mathbf e_{125} \end{split}$$ Circle orthogonal to sphere $$\mathbf s$$ and containing dipole $$\mathbf d$$.
$$\begin{split}\mathbf g^* \wedge \mathbf d =\, &(g_yd_{vz} - g_zd_{vy})\,\mathbf e_{423} \,&+\, (g_zd_{vx} - g_xd_{vz})\,\mathbf e_{431} \\ +\, &(g_xd_{vy} - g_yd_{vx})\,\mathbf e_{412} \,&+\, (g_xd_{mx} + g_yd_{my} + g_zd_{mz})\,\mathbf e_{321} \\ +\, &(g_xd_{pw} + g_wd_{vx})\,\mathbf e_{415} \,&+\, (g_zd_{py} - g_yd_{pz} + g_wd_{mx})\,\mathbf e_{235} \\ +\, &(g_yd_{pw} + g_wd_{vy})\,\mathbf e_{425} \,&+\, (g_xd_{pz} - g_zd_{px} + g_wd_{my})\,\mathbf e_{315} \\ +\, &(g_zd_{pw} + g_wd_{vz})\,\mathbf e_{435} \,&+\, (g_yd_{px} - g_xd_{py} + g_wd_{mz})\,\mathbf e_{125} \end{split}$$ Circle orthogonal to plane $$\mathbf g$$ and containing dipole $$\mathbf d$$.
$$\begin{split}\mathbf s^* \wedge \mathbf p =\, &(s_xp_w + s_up_x)\,\mathbf e_{415} \,&+\, (s_yp_w + s_up_y)\,\mathbf e_{425} \,&+\, (s_zp_w + s_up_z)\,\mathbf e_{435} \\ +\, &(s_zp_y - s_yp_z)\,\mathbf e_{235} \,&+\, (s_xp_z - s_zp_x)\,\mathbf e_{315} \,&+\, (s_yp_x - s_xp_y)\,\mathbf e_{125} \end{split}$$ Line orthogonal to sphere $$\mathbf s$$ and containing flat point $$\mathbf p$$.
$$\begin{split}\mathbf g^* \wedge \mathbf p =\, &g_xp_w \mathbf e_{415} + g_yp_w \mathbf e_{425} + g_zp_w \mathbf e_{435} \\ +\, &(g_zp_y - g_yp_z)\,\mathbf e_{235} + (g_xp_z - g_zp_x)\,\mathbf e_{315} + (g_yp_x - g_xp_y)\,\mathbf e_{125} \end{split}$$ Line orthogonal to plane $$\mathbf g$$ and containing flat point $$\mathbf p$$.
$$\begin{split}\mathbf s^* \wedge \mathbf c =\, &(s_uc_{gw} - s_xc_{gx} - s_yc_{gy} - s_zc_{gz})\,\mathbf e_{1234} \\ +\, &(s_yc_{vz} - s_zc_{vy} - s_wc_{gx} + s_uc_{mx})\,\mathbf e_{4235} \\ +\, &(s_zc_{vx} - s_xc_{vz} - s_wc_{gy} + s_uc_{my})\,\mathbf e_{4315} \\ +\, &(s_xc_{vy} - s_yc_{vx} - s_wc_{gz} + s_uc_{mz})\,\mathbf e_{4125} \\ +\, &(s_xc_{mx} + s_yc_{my} + s_zc_{mz} - s_wc_{gw})\,\mathbf e_{3215} \end{split}$$ Sphere orthogonal to sphere $$\mathbf s$$ and containing circle $$\mathbf c$$.
$$\begin{split}\mathbf g^* \wedge \mathbf c = -\, &(g_xc_{gx} + g_yc_{gy} + g_zc_{gz})\,\mathbf e_{1234} \\ +\, &(g_yc_{vz} - g_zc_{vy} - g_wc_{gx})\,\mathbf e_{4235} \\ +\, &(g_zc_{vx} - g_xc_{vz} - g_wc_{gy})\,\mathbf e_{4315} \\ +\, &(g_xc_{vy} - g_yc_{vx} - g_wc_{gz})\,\mathbf e_{4125} \\ +\, &(g_xc_{mx} + g_yc_{my} + g_zc_{mz} - g_wc_{gw})\,\mathbf e_{3215} \end{split}$$ Sphere orthogonal to plane $$\mathbf g$$ and containing circle $$\mathbf c$$.
$$\begin{split}\mathbf s^* \wedge \boldsymbol l =\, &(s_yl_{vz} - s_zl_{vy} - s_ul_{mx})\,\mathbf e_{4235} + (s_zl_{vx} - s_xl_{vz} - s_ul_{my})\,\mathbf e_{4315} \\ +\, &(s_xl_{vy} - s_yl_{vx} - s_ul_{mz})\,\mathbf e_{4125} + (s_xl_{mx} + s_yl_{my} + s_zl_{mz})\,\mathbf e_{3215} \end{split}$$ Plane orthogonal to sphere $$\mathbf s$$ and containing line $$\boldsymbol l$$.
$$\begin{split}\mathbf g^* \wedge \boldsymbol l =\, &(g_yl_{vz} - g_zl_{vy})\,\mathbf e_{4235} + (g_zl_{vx} - g_xl_{vz})\,\mathbf e_{4315} \\ +\, &(g_xl_{vy} - g_yl_{vx})\,\mathbf e_{4125} + (g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{3215} \end{split}$$ 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 $$\mathbf a$$.

See Also