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| Line 1: |
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| 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 dot products between two unitized objects of the same type are listed in the following table. The vector $$\mathbf v$$ is the difference between their center positions, and the scalars $$r_1$$ and $$r_2$$ are their radii. In the case of dipoles and circles, the vectors $$\mathbf n_1$$ and $$\mathbf n_2$$ correspond to the directions of the carrier lines or the normals of the carrier planes. |
| | |
| The [[flat points]], [[lines]], [[planes]], [[round points]], [[dipoles]], [[circles]], and [[spheres]] appearing in the following tables are defined as follows:
| |
| | |
| :$$\mathbf p = p_x \mathbf e_{15} + p_y \mathbf e_{25} + p_z \mathbf e_{35} + p_w \mathbf e_{45}$$
| |
| :$$\boldsymbol l = l_{vx} \mathbf e_{415} + l_{vy} \mathbf e_{425} + l_{vz} \mathbf e_{435} + l_{mx} \mathbf e_{235} + l_{my} \mathbf e_{315} + l_{mz} \mathbf e_{125}$$
| |
| :$$\mathbf g = g_x \mathbf e_{4235} + g_y \mathbf e_{4315} + g_z \mathbf e_{4125} + g_w \mathbf e_{3215}$$
| |
| | |
| :$$\mathbf a = a_x \mathbf e_1 + a_y \mathbf e_2 + a_z \mathbf e_3 + a_w \mathbf e_4 + a_u \mathbf e_5$$
| |
| :$$\mathbf d = d_{vx} \mathbf e_{41} + d_{vy} \mathbf e_{42} + d_{vz} \mathbf e_{43} + d_{mx} \mathbf e_{23} + d_{my} \mathbf e_{31} + d_{mz} \mathbf e_{12} + d_{px} \mathbf e_{15} + d_{py} \mathbf e_{25} + d_{pz} \mathbf e_{35} + d_{pw} \mathbf e_{45}$$
| |
| :$$\mathbf c = c_{gx} \mathbf e_{423} + c_{gy} \mathbf e_{431} + c_{gz} \mathbf e_{412} + c_{gw} \mathbf e_{321} + c_{vx} \mathbf e_{415} + c_{vy} \mathbf e_{425} + c_{vz} \mathbf e_{435} + c_{mx} \mathbf e_{235} + c_{my} \mathbf e_{315} + c_{mz} \mathbf e_{125}$$
| |
| :$$\mathbf s = s_u \mathbf e_{1234} + s_x \mathbf e_{4235} + s_y \mathbf e_{4315} + s_z \mathbf e_{4125} + s_w \mathbf e_{3215}$$
| |
|
| |
|
| {| class="wikitable" | | {| class="wikitable" |
| ! Formula || Description || Illustration | | ! Type || Dot Product |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf s^* \wedge \mathbf a =
| |
| -\, &(s_xa_w + s_ua_x)\,\mathbf e_{41} \,&-\, (s_ya_w + s_ua_y)\,\mathbf e_{42} \,&-\, (s_za_w + s_ua_z)\,\mathbf e_{43} \\
| |
| +\, &(s_ya_z - s_za_y)\,\mathbf e_{23} \,&+\, (s_za_x - s_xa_z)\,\mathbf e_{31} \,&+\, (s_xa_y - s_ya_x)\,\mathbf e_{12} \\
| |
| +\, &(s_xa_u + s_wa_x)\,\mathbf e_{15} \,&+\, (s_ya_u + s_wa_y)\,\mathbf e_{25} \,&+\, (s_za_u + s_wa_z)\,\mathbf e_{35} + (s_wa_w - s_ua_u)\,\mathbf e_{45}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Dipole orthogonal to sphere $$\mathbf s$$ and containing round point $$\mathbf a$$.
| |
| | style="padding: 12px;" | [[Image:sphere_connect_round.svg|250px]]
| |
| |- | | |- |
| | style="padding: 12px;" | $$\begin{split}\mathbf g^* \wedge \mathbf a = | | | style="padding: 12px;" | Round points $$\mathbf a_1$$ and $$\mathbf a_2$$ |
| -\, &g_xa_w \mathbf e_{41} - g_ya_w \mathbf e_{42} - g_za_w \mathbf e_{43} \\
| | | style="padding: 12px;" | $$\mathbf a_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf a_2 = -\dfrac{1}{2}(\mathbf v^2 + r_1^2 + r_2^2)$$ |
| +\, &(g_ya_z - g_za_y)\,\mathbf e_{23} + (g_za_x - g_xa_z)\,\mathbf e_{31} + (g_xa_y - g_ya_x)\,\mathbf e_{12} \\
| |
| +\, &(g_xa_u + g_wa_x)\,\mathbf e_{15} + (g_ya_u + g_wa_y)\,\mathbf e_{25} + (g_za_u + g_wa_z)\,\mathbf e_{35} + g_wa_w \mathbf e_{45}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Dipole orthogonal to plane $$\mathbf g$$ and containing round point $$\mathbf a$$.
| |
| | style="padding: 12px;" | [[Image:plane_connect_round.svg|250px]]
| |
| |- | | |- |
| | style="padding: 12px;" | $$\begin{split}\mathbf s^* \wedge \mathbf d | | | style="padding: 12px;" | Dipoles $$\mathbf d_1$$ and $$\mathbf d_2$$ |
| =\, &(s_zd_{vy} - s_yd_{vz} - s_ud_{mx})\,\mathbf e_{423} \,&+\, (s_xd_{vz} - s_zd_{vx} - s_ud_{my})\,\mathbf e_{431} \\ | | | style="padding: 12px;" | $$\mathbf d_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf d_2 = -\dfrac{1}{2}(\mathbf n_1 \cdot \mathbf n_2)(\mathbf v^2 + r_1^2 + r_2^2) + (\mathbf n_1 \cdot \mathbf v)(\mathbf n_2 \cdot \mathbf v)$$ |
| +\, &(s_yd_{vx} - s_xd_{vy} - s_ud_{mz})\,\mathbf e_{412} \,&-\, (s_xd_{mx} + s_yd_{my} + s_zd_{mz})\,\mathbf e_{321} \\
| |
| -\, &(s_xd_{pw} + s_wd_{vx} + s_ud_{px})\,\mathbf e_{415} \,&+\, (s_yd_{pz} - s_zd_{py} - s_wd_{mx})\,\mathbf e_{235} \\
| |
| -\, &(s_yd_{pw} + s_wd_{vy} + s_ud_{py})\,\mathbf e_{425} \,&+\, (s_zd_{px} - s_xd_{pz} - s_wd_{my})\,\mathbf e_{315} \\
| |
| -\, &(s_zd_{pw} + s_wd_{vz} + s_ud_{pz})\,\mathbf e_{435} \,&+\, (s_xd_{py} - s_yd_{px} - s_wd_{mz})\,\mathbf e_{125}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Circle orthogonal to sphere $$\mathbf s$$ and containing dipole $$\mathbf d$$.
| |
| | style="padding: 12px;" | [[Image:sphere_connect_dipole.svg|250px]]
| |
| |- | | |- |
| | style="padding: 12px;" | $$\begin{split}\mathbf g^* \wedge \mathbf d | | | style="padding: 12px;" | Circles $$\mathbf c_1$$ and $$\mathbf c_2$$ |
| =\, &(g_zd_{vy} - g_yd_{vz})\,\mathbf e_{423} \,&+\, (g_xd_{vz} - g_zd_{vx})\,\mathbf e_{431} \\ | | | style="padding: 12px;" | $$\mathbf c_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf c_2 = +\dfrac{1}{2}(\mathbf n_1 \cdot \mathbf n_2)(\mathbf v^2 - r_1^2 - r_2^2) - (\mathbf n_1 \cdot \mathbf v)(\mathbf n_2 \cdot \mathbf v)$$ |
| +\, &(g_yd_{vx} - g_xd_{vy})\,\mathbf e_{412} \,&-\, (g_xd_{mx} + g_yd_{my} + g_zd_{mz})\,\mathbf e_{321} \\ | |
| -\, &(g_xd_{pw} + g_wd_{vx})\,\mathbf e_{415} \,&+\, (g_yd_{pz} - g_zd_{py} - g_wd_{mx})\,\mathbf e_{235} \\
| |
| -\, &(g_yd_{pw} + g_wd_{vy})\,\mathbf e_{425} \,&+\, (g_zd_{px} - g_xd_{pz} - g_wd_{my})\,\mathbf e_{315} \\ | |
| -\, &(g_zd_{pw} + g_wd_{vz})\,\mathbf e_{435} \,&+\, (g_xd_{py} - g_yd_{px} - g_wd_{mz})\,\mathbf e_{125}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Circle orthogonal to plane $$\mathbf g$$ and containing dipole $$\mathbf d$$.
| |
| | style="padding: 12px;" | [[Image:plane_connect_dipole.svg|250px]]
| |
| |- | | |- |
| | style="padding: 12px;" | $$\begin{split}\mathbf s^* \wedge \mathbf p = | | | style="padding: 12px;" | Spheres $$\mathbf s_1$$ and $$\mathbf s_2$$ |
| -\, &(s_xp_w + s_up_x)\,\mathbf e_{415} \,&-\, (s_yp_w + s_up_y)\,\mathbf e_{425} \,&-\, (s_zp_w + s_up_z)\,\mathbf e_{435} \\
| | | style="padding: 12px;" | $$\mathbf s_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf s_2 = +\dfrac{1}{2}(\mathbf v^2 - r_1^2 - r_2^2)$$ |
| +\, &(s_yp_z - s_zp_y)\,\mathbf e_{235} \,&+\, (s_zp_x - s_xp_z)\,\mathbf e_{315} \,&+\, (s_xp_y - s_yp_x)\,\mathbf e_{125}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Line orthogonal to sphere $$\mathbf s$$ and containing flat point $$\mathbf p$$.
| |
| | style="padding: 12px;" | [[Image:sphere_connect_point.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf g^* \wedge \mathbf p = | |
| -\, &g_xp_w \mathbf e_{415} - g_yp_w \mathbf e_{425} - g_zp_w \mathbf e_{435} \\
| |
| +\, &(g_yp_z - g_zp_y)\,\mathbf e_{235} + (g_zp_x - g_xp_z)\,\mathbf e_{315} + (g_xp_y - g_yp_x)\,\mathbf e_{125}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Line orthogonal to plane $$\mathbf g$$ and containing flat point $$\mathbf p$$.
| |
| | style="padding: 12px;" | [[Image:plane_connect_point.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf s^* \wedge \mathbf c
| |
| =\, &(s_xc_{gx} + s_yc_{gy} + s_zc_{gz} - s_uc_{gw})\,\mathbf e_{1234} \\
| |
| +\, &(s_zc_{vy} - s_yc_{vz} + s_wc_{gx} - s_uc_{mx})\,\mathbf e_{4235} \\
| |
| +\, &(s_xc_{vz} - s_zc_{vx} + s_wc_{gy} - s_uc_{my})\,\mathbf e_{4315} \\
| |
| +\, &(s_yc_{vx} - s_xc_{vy} + s_wc_{gz} - s_uc_{mz})\,\mathbf e_{4125} \\
| |
| +\, &(s_wc_{gw} - s_xc_{mx} - s_yc_{my} - s_zc_{mz})\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Sphere orthogonal to sphere $$\mathbf s$$ and containing circle $$\mathbf c$$.
| |
| | style="padding: 12px;" | [[Image:sphere_connect_circle.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf g^* \wedge \mathbf c
| |
| =\, &(g_xc_{gx} + g_yc_{gy} + g_zc_{gz})\,\mathbf e_{1234} \\
| |
| +\, &(g_zc_{vy} - g_yc_{vz} + g_wc_{gx})\,\mathbf e_{4235} \\
| |
| +\, &(g_xc_{vz} - g_zc_{vx} + g_wc_{gy})\,\mathbf e_{4315} \\
| |
| +\, &(g_yc_{vx} - g_xc_{vy} + g_wc_{gz})\,\mathbf e_{4125} \\
| |
| +\, &(g_wc_{gw} - g_xc_{mx} - g_yc_{my} - g_zc_{mz})\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Sphere orthogonal to plane $$\mathbf g$$ and containing circle $$\mathbf c$$.
| |
| | style="padding: 12px;" | [[Image:plane_connect_circle.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf s^* \wedge \boldsymbol l
| |
| =\, &(s_zl_{vy} - s_yl_{vz} - s_ul_{mx})\,\mathbf e_{4235} + (s_xl_{vz} - s_zl_{vx} - s_ul_{my})\,\mathbf e_{4315} \\
| |
| +\, &(s_yl_{vx} - s_xl_{vy} - s_ul_{mz})\,\mathbf e_{4125} - (s_xl_{mx} + s_yl_{my} + s_zl_{mz})\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Plane orthogonal to sphere $$\mathbf s$$ and containing line $$\boldsymbol l$$.
| |
| | style="padding: 12px;" | [[Image:sphere_connect_line.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf g^* \wedge \boldsymbol l
| |
| =\, &(g_zl_{vy} - g_yl_{vz})\,\mathbf e_{4235} + (g_xl_{vz} - g_zl_{vx})\,\mathbf e_{4315} \\
| |
| +\, &(g_yl_{vx} - g_xl_{vy})\,\mathbf e_{4125} - (g_xl_{mx} + g_yl_{my} + g_zl_{mz})\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Plane orthogonal to plane $$\mathbf g$$ and containing line $$\boldsymbol l$$.
| |
| | style="padding: 12px;" | [[Image:plane_connect_line.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf c^* \wedge \mathbf a
| |
| =\, &(c_{gz}a_y - c_{gy}a_z - c_{vx}a_w)\,\mathbf e_{423} \,&+\, (c_{gx}a_z - c_{gz}a_x - c_{vy}a_w)\,\mathbf e_{431} \\
| |
| +\, &(c_{gy}a_x - c_{gx}a_y - c_{vz}a_w)\,\mathbf e_{412} \,&+\, (c_{vx}a_x + c_{vy}a_y + c_{vz}a_z)\,\mathbf e_{321} \\
| |
| -\, &(c_{mx}a_w + c_{gw}a_x + c_{gx}a_u)\,\mathbf e_{415} \,&+\, (c_{my}a_z - c_{mz}a_y - c_{vx}a_u)\,\mathbf e_{235} \\
| |
| -\, &(c_{my}a_w + c_{gw}a_y + c_{gy}a_u)\,\mathbf e_{425} \,&+\, (c_{mz}a_x - c_{mx}a_z - c_{vy}a_u)\,\mathbf e_{315} \\
| |
| -\, &(c_{mz}a_w + c_{gw}a_z + c_{gz}a_u)\,\mathbf e_{435} \,&+\, (c_{mx}a_y - c_{my}a_x - c_{vz}a_u)\,\mathbf e_{125}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Circle orthogonal to circle $$\mathbf c$$ and containing round point $$\mathbf a$$.
| |
| | style="padding: 12px;" | [[Image:circle_connect_round.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\boldsymbol l^* \wedge \mathbf a =
| |
| -\, &l_{vx}a_w \mathbf e_{423} - l_{vy}a_w \mathbf e_{431} - l_{vz}a_w \mathbf e_{412}\\
| |
| +\, &(l_{vx}a_x + l_{vy}a_y + l_{vz}a_z)\,\mathbf e_{321} \\
| |
| -\, &l_{mx}a_w \mathbf e_{415} + (l_{my}a_z - l_{mz}a_y - l_{vx}a_u)\,\mathbf e_{235} \\
| |
| -\, &l_{my}a_w \mathbf e_{425} + (l_{mz}a_x - l_{mx}a_z - l_{vy}a_u)\,\mathbf e_{315} \\
| |
| -\, &l_{mz}a_w \mathbf e_{435} + (l_{mx}a_y - l_{my}a_x - l_{vz}a_u)\,\mathbf e_{125}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Circle orthogonal to line $$\boldsymbol l$$ and containing round point $$\mathbf a$$.
| |
| | style="padding: 12px;" | [[Image:line_connect_round.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf c^* \wedge \mathbf p
| |
| =\, &(c_{gz}p_y - c_{gy}p_z - c_{vx}p_w)\,\mathbf e_{4235} \,&+\, (c_{gx}p_z - c_{gz}p_x - c_{vy}p_w)\,\mathbf e_{4315} \\
| |
| +\, &(c_{gy}p_x - c_{gx}p_y - c_{vz}p_w)\,\mathbf e_{4125} \,&+\, (c_{vx}p_x + c_{vy}p_y + c_{vz}p_z)\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Plane orthogonal to circle $$\mathbf c$$ and containing flat point $$\mathbf p$$.
| |
| | style="padding: 12px;" | [[Image:circle_connect_point.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\boldsymbol l^* \wedge \mathbf p =
| |
| -\, &l_{vx}p_w \mathbf e_{4235} - l_{vy}p_w \mathbf e_{4315} - l_{vz}p_w \mathbf e_{4125} \\
| |
| +\, &(l_{vx}p_x + l_{vy}p_y + l_{vz}p_z)\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Plane orthogonal to line $$\boldsymbol l$$ and containing flat point $$\mathbf p$$.
| |
| | style="padding: 12px;" | [[Image:line_connect_point.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf c^* \wedge \mathbf d
| |
| =\, &(c_{vx}d_{vx} + c_{vy}d_{vy} + c_{vz}d_{vz} + c_{gx}d_{mx} + c_{gy}d_{my} + c_{gz}d_{mz})\,\mathbf e_{1234} \\
| |
| +\, &(c_{my}d_{vz} - c_{mz}d_{vy} - c_{vx}d_{pw} + c_{gz}d_{py} - c_{gy}d_{pz} + c_{gw}d_{mx})\,\mathbf e_{4235} \\
| |
| +\, &(c_{mz}d_{vx} - c_{mx}d_{vz} - c_{vy}d_{pw} + c_{gx}d_{pz} - c_{gz}d_{px} + c_{gw}d_{my})\,\mathbf e_{4315} \\
| |
| +\, &(c_{mx}d_{vy} - c_{my}d_{vx} - c_{vz}d_{pw} + c_{gy}d_{px} - c_{gx}d_{py} + c_{gw}d_{mz})\,\mathbf e_{4125} \\
| |
| +\, &(c_{vx}d_{px} + c_{vy}d_{py} + c_{vz}d_{pz} + c_{mx}d_{mx} + c_{my}d_{my} + c_{mz}d_{mz})\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Sphere orthogonal to circle $$\mathbf c$$ and containing dipole $$\mathbf d$$.
| |
| | style="padding: 12px;" | [[Image:circle_connect_dipole.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\boldsymbol l^* \wedge \mathbf d
| |
| =\, &(l_{vx}d_{vx} + l_{vy}d_{vy} + l_{vz}d_{vz})\,\mathbf e_{1234} \\
| |
| +\, &(l_{my}d_{vz} - l_{mz}d_{vy} - l_{vx}d_{pw})\,\mathbf e_{4235} \\
| |
| +\, &(l_{mz}d_{vx} - l_{mx}d_{vz} - l_{vy}d_{pw})\,\mathbf e_{4315} \\
| |
| +\, &(l_{mx}d_{vy} - l_{my}d_{vx} - l_{vz}d_{pw})\,\mathbf e_{4125} \\
| |
| +\, &(l_{vx}d_{px} + l_{vy}d_{py} + l_{vz}d_{pz} + l_{mx}d_{mx} + l_{my}d_{my} + l_{mz}d_{mz})\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Sphere orthogonal to line $$\boldsymbol l$$ and containing dipole $$\mathbf d$$.
| |
| | style="padding: 12px;" | [[Image:line_connect_dipole.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf d^* \wedge \mathbf a
| |
| =\, &(d_{vx}a_x + d_{vy}a_y + d_{vz}a_z - d_{pw}a_w)\,\mathbf e_{1234} \\
| |
| +\, &(d_{my}a_z - d_{mz}a_y + d_{px}a_w - d_{vx}a_u)\,\mathbf e_{4235} \\
| |
| +\, &(d_{mz}a_x - d_{mx}a_z + d_{py}a_w - d_{vy}a_u)\,\mathbf e_{4315} \\
| |
| +\, &(d_{mx}a_y - d_{my}a_x + d_{pz}a_w - d_{vz}a_u)\,\mathbf e_{4125} \\
| |
| +\, &(d_{pw}a_u - d_{px}a_x - d_{py}a_y - d_{pz}a_z)\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Sphere orthogonal to dipole $$\mathbf d$$ and containing round point $$\mathbf a$$.
| |
| | style="padding: 12px;" | [[Image:dipole_connect_round.svg|250px]]
| |
| |-
| |
| | style="padding: 12px;" | $$\begin{split}\mathbf p^* \wedge \mathbf a =
| |
| -\, &p_wa_w \mathbf e_{1234} + p_xa_w \mathbf e_{4235} + p_ya_w \mathbf e_{4315} + p_za_w \mathbf e_{4125} \\
| |
| +\, &(p_wa_u - p_xa_x - p_ya_y - p_za_z)\,\mathbf e_{3215}
| |
| \end{split}$$
| |
| | style="padding: 12px;" | Sphere centered at flat point $$\mathbf p$$ and containing round point $$\mathbf a$$.
| |
| | style="padding: 12px;" | [[Image:point_connect_round.svg|250px]]
| |
| |} | | |} |
| | |
| | In the case of two spheres $$\mathbf s_1$$ and $$\mathbf s_2$$, the dot product gives the product of the radii $$r_1$$ and $$r_2$$ times the cosine of the angle $$\phi$$ between the tangent planes where they intersect, as shown in the following figure. This can be demonstrating by considering the law of cosines for the angle $$\gamma$$, which states |
| | |
| | :$$\mathbf v^2 = r_1^2 + r_2^2 - 2r_1r_2 \cos \gamma$$ . |
| | |
| | Plugging this into the formula for the dot product yields $$\mathbf s_1 \mathbin{\unicode["segoe ui symbol"]{x2022}} \mathbf s_2 = r_1r_2 \cos \phi$$. |
| | |
| | [[Image:sphere-dot-sphere.svg|512px]] |
|
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|
| == See Also == | | == See Also == |
|
| |
|
| * [[Join and meet]] | | * [[Geometric products]] |
| * [[Exterior products]] | | * [[Exterior products]] |
