User:Zortwort/sandbox


 * $$ 2s=\theta_1+\theta_2+2\phi $$
 * $$ M=2 sin^{-1}\left(\left[\frac{\sin(s-\theta_1)\sin(s-\theta_2)}{\sin \theta_1 \sin \theta_2} \right]^{\frac{1}{2}} \right)\quad $$
 * $$ \frac{\sin M}{\sin m}=\frac{\sin \frac{\pi}{2}}{\sin \theta_1} $$
 * $$ m=\sin^{-1}{\left[\frac{\sin M \sin \theta_1}{\sin \frac{\pi}{2}} \right]} $$
 * $$ \tan{\left[\frac{1}{2}(m+\theta_1)\right]}=\frac{\cos\left[\frac{1}{2}(M-\frac{\pi}{2})\right]}{\cos\left[\frac{1}{2}(M+\frac{\pi}{2})\right]}tan\left(\frac{n}{2}\right)$$
 * $$ n=2\tan^{-1}{\left[\frac{\cos\left[\frac{1}{2}(M+\frac{\pi}{2})\right]\tan{\left[\frac{1}{2}(m+\theta_1)\right]}}{\cos\left[\frac{1}{2}(M-\frac{\pi}{2})\right]}\right]}$$
 * $$ x=\sin(n-\phi)$$
 * $$ y=\sin M \sin \theta_1 $$
 * $$ \frac{\theta_1+\theta_2}{2}+j+\frac{\theta_1+\theta_2}{2}-j=k$$
 * $$ T\hat a = [\sin \phi, 0, cos\ phi] = \hat a'$$
 * $$ T\hat b = [-\sin \phi, 0, cos\ phi] = \hat b'$$
 * $$ T\hat c=[x,y,z] = \hat c'$$
 * $$ d=\cos^{-1}(n_1\cdot n_2)$$
 * $$ x^2+y^2+z^2=1$$
 * $$ \sin^2(n-\phi)+\sin^2 M \sin^2 \theta_1+z^2=1$$
 * $$ z=\sqrt {1-\sin^2(n-\phi)-\sin^2 M \sin^2 \theta_1}$$
 * $$ T^{-1} \hat c'=\hat c$$
 * $$ T \hat c=\hat c'$$
 * $$ T \hat a=n_1$$
 * $$ T \hat b=n_2$$
 * $$ \theta_2=k-\theta_1$$
 * $$ 2\theta_1+2\phi=k$$
 * $$ \theta_1={k\over 2} - \phi$$
 * $$ \theta_1=k-({k\over 2} - \phi)$$
 * $$ \theta_1={k\over 2} + \phi$$
 * $$ \cos({k \over 2} + j) = -\sin (\phi) x + \cos (\phi) z$$
 * $$ \cos({k \over 2} - j) = \sin (\phi) x + \cos (\phi) z$$
 * $$ \cos \left({k \over 2} \right) \cos (j) - \sin \left({k \over 2}\right ) \sin (j) = -\sin (\phi) x + \cos (\phi) z$$
 * $$ \cos \left({k \over 2} \right) \cos (j) + \sin \left({k \over 2}\right ) \sin (j) = \sin (\phi) x + \cos (\phi) z$$
 * $$ 2 \cos \left({k \over 2} \right ) \cos (j) = 2 z \cos (\phi)$$
 * $$ \cos (j) = \frac{ \cos (\phi)} { \cos \left({k \over 2} \right )} z $$
 * $$ \sin (j) = \frac{ \sin (\phi)} { \sin \left({k \over 2} \right )} x $$
 * $$ 1 = \cos^2 j + \sin^2 j = \frac{ \cos^2 (\phi)} { \cos^2 \left({k \over 2} \right )} z^2 + \frac{ \sin^2 (\phi)} { \sin^2 \left({k \over 2} \right )} x^2 $$
 * $$ {SA \over 2}=\int\limits_{t= -\sin\left(2\phi+{k \over 2}\right)}^{t = \sin \left(2\phi+{k \over 2} \right)} {\cos^{-1}\left[\vec {OC} \cdot \vec {OP} \right ]} \, dt$$
 * $$ {SA } = 2 \int\limits_{t= -\sin\left(2\phi+{k \over 2}\right)}^{t = \sin \left(2\phi+{k \over 2} \right)} \cos^{-1}{\left[{\sqrt {1 - \frac {\sin^2 (\phi)}{\sin^2 \left ({k \over 2} \right)}t^2} \frac {\cos \left ({k \over 2} \right)} {\cos (\phi)}}\right]} \, dt$$
 * $$ P_z = $$
 * $$ D= (0,0,1) $$


 * $$ R_1=R_x(\alpha)R_y(\beta)R_z(\gamma) $$


 * $$R_x(\alpha)= \begin{bmatrix}

0 & 0 & 0\\ 0 & \cos (\alpha) & -\sin (\alpha)\\ 0 & \sin (\alpha) & \cos (\alpha)\\ \end{bmatrix} $$
 * $$R_z(\gamma)= \begin{bmatrix}

\cos (\gamma) & -\sin (\gamma) & 0\\ \sin (\gamma) & \cos (\gamma) & 0\\ 0 & 0 & 1\\ \end{bmatrix} $$
 * $$R_y(\beta)= \begin{bmatrix}

\cos (\beta) & 0 & \sin (\beta)\\ 0 & 1 & 0\\ -\sin (\beta) & 0 & \cos (\beta)\\ \end{bmatrix} $$ \cos (\beta) -\sin (\beta)


 * $$ \frac {\vec a + \vec b}{2}$$
 * $$ \alpha=\cos^{-1}\left[proj_{yz}\left(\frac {\hat a + \hat b}{2}\right)\cdot \hat z \right]$$
 * $$ \beta=\cos^{-1}\left[proj_{xz}\left(\frac {\hat a + \hat b}{2}\right)\cdot \hat x \right]$$
 * $$ \gamma=\cos^{-1}\left[proj_{xy}\left(\frac {\hat a + \hat b}{2}\right)\cdot \hat y \right]$$
 * $$ R_2=$$
 * $$ T=R_1 R_2$$
 * $$P_z = \sqrt {1 - \frac {\sin^2 (\phi)}{\sin^2 \left ({k \over 2} \right)}t^2} \frac {\cos \left ({k \over 2} \right)} {\cos (\phi)}$$
 * $$ \frac {\cos^2 (k \over 2)}{\cos^2 (\phi)}$$
 * $$ \sqrt {1 - \frac {\sin^2 (\phi) t^2}{\sin^2 \left ({k \over 2} \right)}}$$
 * $$ d=\cos^{-1}\left(\hat a' \cdot \hat b' \right )$$
 * $ T$
 * $ \vec a$
 * $ \vec b$
 * $ \vec c$
 * $ \theta_1$
 * $ \theta_2$
 * $ k$
 * $ \phi$
 * $ n$
 * $ m$
 * $ \hat a$
 * $ \hat b$
 * $ \hat c$
 * $ R_1$
 * $ R_2$
 * $ xz$
 * $ xy$
 * $ yz$
 * $ \frac {(\vec a + \vec b)}{2}$
 * $ D=(0,0,1)$
 * $ M$
 * $ m$
 * $ x$
 * $ y$
 * $ z$
 * $ x$
 * $ \vec {OC}$
 * $ \vec {OP}$
 * $ \phi=0$
 * $ k=0$
 * $ z$
 * $ z$
 * $ R_1 \hat a$
 * $ R_1 \hat b$
 * $ R_1 \hat c$
 * $ z$
 * $ \hat a '$
 * $ \hat b '$
 * $ \hat c '$
 * $$ T^{-1} \hat c ' = \hat c$$
 * $ P$
 * $ \theta_1+\theta_2=k$
 * $ k=2$
 * $ \phi = 0.5$
 * $ \mathbb {R}^3$
 * $ j$
 * $ \vec {OP} = \hat c '$
 * $$ cos^{-1} \left( \hat a' \cdot \vec {OP} \right)= \theta_1$$
 * $$ cos^{-1} \left( \hat b' \cdot \vec {OP} \right)= \theta_2$$
 * $ t$
 * $$ P= (x,y,z)$$
 * $ D$
 * $ \phi=0$
 * $ k=\pi$
 * $\pi$
 * $ 2\pi$
 * $ 4\pi$


 * $$ R_2=\begin{cases}

R_z\left(\cos^{-1}\left[proj_{xz} \left( R_1 \hat a \right)\cdot \hat x\right] \right), & \text{if } proj_{xz} \left( R_1 \hat a \right)_y \le 0 \\ R_z \left( 360-\cos^{-1}\left[proj_{xz} \left( R_1 \hat a \right)\cdot \hat x\right] \right), & \text{if } proj_{xz} \left( R_1 \hat a \right)_y > 0 \end{cases} $$


 * $$T=R_z \left(\cos^{-1}\left[proj_{xz} \left( R_1 \frac {\hat a + \hat b}{2} \right)\cdot \hat x)\right]\right) R_1 $$
 * $$ R_2=R_z \left({\begin{cases}

\cos^{-1}\left[proj_{xz} \left( R_1 \hat a \right)\cdot \hat x\right], & \text{if } proj_{xz} \left( R_1 \hat a \right)_y \le 0 \\ 360-\cos^{-1}\left[proj_{xz} \left( R_1 \hat a \right)\cdot \hat x\right], & \text{if } proj_{xz} \left( R_1 \hat a \right)_y > 0 \end{cases}} $$ \right)
 * $$ P = \left(\sin (n-\phi), \sin M \sin \theta_1, \sqrt{\left[1-\sin^2(n-\phi)-\sin^2 M \sin ^2 \theta_1 \right]}\right), {k \over 2} - \phi \le \theta_1 \le {k \over 2} + \phi $$