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\form#0:$ (x ,y ,z ) $
\form#1:$ (x',y',z') $
\form#2:$ \vec{\theta}=(\theta_x,\theta_y,\theta_z) $
\form#3:\[ \left[ \begin{array}{c} x' \\ y' \\ z' \\ \end{array} \right] = \left[ \begin{array}{ccc} 2(n_x^2 - 1) \sin^2\phi + 1 & 2n_x n_y \sin^2\phi - 2n_z\cos \phi\sin \phi & 2n_x n_z \sin^2\phi + 2n_y\cos \phi\sin \phi \\ 2n_y n_x \sin^2\phi + 2n_z\cos \phi\sin \phi & 2(n_y^2 - 1) \sin^2\phi + 1 & 2n_y n_z \sin^2\phi - 2n_x\cos \phi\sin \phi \\ 2n_z n_x \sin^2\phi - 2n_y\cos \phi\sin \phi & 2n_z n_y \sin^2\phi + 2n_x\cos \phi\sin \phi & 2(n_z^2 - 1) \sin^2\phi + 1 \\ \end{array} \right] \left[ \begin{array}{c} x \\ y \\ z \\ \end{array} \right] \]
\form#4:$ \phi $
\form#5:$ \vec{\theta} $
\form#6:$ \phi = \frac{\left|\vec{\theta}\right|}{2} = \frac{1}{2}\sqrt{\theta_x^2 + \theta_y^2 + \theta_z^2} $
\form#7:$ \vec{n} $
\form#8:$ \vec{n} = (n_x,n_y,n_z) = \vec{\theta} / 2\phi $
\form#9:\[ \left[ \begin{array}{ccc} 2(n_x^2 - 1) \sin^2\phi + 1 & 2n_x n_y \sin^2\phi - 2n_z\cos \phi\sin \phi & 2n_x n_z \sin^2\phi + 2n_y\cos \phi\sin \phi \\ 2n_y n_x \sin^2\phi + 2n_z\cos \phi\sin \phi & 2(n_y^2 - 1) \sin^2\phi + 1 & 2n_y n_z \sin^2\phi - 2n_x\cos \phi\sin \phi \\ 2n_z n_x \sin^2\phi - 2n_y\cos \phi\sin \phi & 2n_z n_y \sin^2\phi + 2n_x\cos \phi\sin \phi & 2(n_z^2 - 1) \sin^2\phi + 1 \\ \end{array} \right] \]
\form#10:\[ \left[ \begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & -1 \\ 0 & 1 & 0 \\ \end{array} \right] \left[ \begin{array}{ccc} 2(n_x^2 - 1) \sin^2\phi + 1 & 2n_x n_y \sin^2\phi - 2n_z\cos \phi\sin \phi & 2n_x n_z \sin^2\phi + 2n_y\cos \phi\sin \phi \\ 2n_y n_x \sin^2\phi + 2n_z\cos \phi\sin \phi & 2(n_y^2 - 1) \sin^2\phi + 1 & 2n_y n_z \sin^2\phi - 2n_x\cos \phi\sin \phi \\ 2n_z n_x \sin^2\phi - 2n_y\cos \phi\sin \phi & 2n_z n_y \sin^2\phi + 2n_x\cos \phi\sin \phi & 2(n_z^2 - 1) \sin^2\phi + 1 \\ \end{array} \right] \left[ \begin{array}{c} x \\ y \\ z \\ \end{array} \right] \]
\form#11:\[ \left[ \begin{array}{ccc} 0 & 0 & 1 \\ 0 & 0 & 0 \\ -1 & 0 & 0 \\ \end{array} \right] \left[ \begin{array}{ccc} 2(n_x^2 - 1) \sin^2\phi + 1 & 2n_x n_y \sin^2\phi - 2n_z\cos \phi\sin \phi & 2n_x n_z \sin^2\phi + 2n_y\cos \phi\sin \phi \\ 2n_y n_x \sin^2\phi + 2n_z\cos \phi\sin \phi & 2(n_y^2 - 1) \sin^2\phi + 1 & 2n_y n_z \sin^2\phi - 2n_x\cos \phi\sin \phi \\ 2n_z n_x \sin^2\phi - 2n_y\cos \phi\sin \phi & 2n_z n_y \sin^2\phi + 2n_x\cos \phi\sin \phi & 2(n_z^2 - 1) \sin^2\phi + 1 \\ \end{array} \right] \left[ \begin{array}{c} x \\ y \\ z \\ \end{array} \right] \]
\form#12:\[ \left[ \begin{array}{ccc} 0 & -1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \\ \end{array} \right] \left[ \begin{array}{ccc} 2(n_x^2 - 1) \sin^2\phi + 1 & 2n_x n_y \sin^2\phi - 2n_z\cos \phi\sin \phi & 2n_x n_z \sin^2\phi + 2n_y\cos \phi\sin \phi \\ 2n_y n_x \sin^2\phi + 2n_z\cos \phi\sin \phi & 2(n_y^2 - 1) \sin^2\phi + 1 & 2n_y n_z \sin^2\phi - 2n_x\cos \phi\sin \phi \\ 2n_z n_x \sin^2\phi - 2n_y\cos \phi\sin \phi & 2n_z n_y \sin^2\phi + 2n_x\cos \phi\sin \phi & 2(n_z^2 - 1) \sin^2\phi + 1 \\ \end{array} \right] \left[ \begin{array}{c} x \\ y \\ z \\ \end{array} \right] \]
\form#13:$ (x,y,z) $
\form#14:$ \vec{n}, \phi $
\form#15:$ N $
\form#16:$ p_0 $
\form#17:$ P $
\form#18:$ M $
\form#19:\[ \begin{aligned} & (1 - p_0^N)^M \leq(1 - P) \\ \Rightarrow & M \log(1 - p_0^N) \leq \log(1 - P) \\ \Rightarrow & M \geq \frac{\log(1 - p)}{\log(1 - p_0^N)},~~ \because (1-p_0^N<1 \Rightarrow \log(1-p_0^N)<0) \end{aligned} \]
\form#20:$ M = \lceil \frac{\log(1 - P)}{\log(1 - p_0^N)} \rceil $
\form#21:$ F: \mathbb{R} ^N \mapsto \mathbb{R}^M $
\form#22:$ v $
\form#23:$ F(v)^T F(v) = 0$
\form#24:$ \epsilon $
\form#25:$ F(v)^T F(v) < \epsilon $
\form#26:$ v_0 $
\form#27:$ v_1, v_2, v_3, v_4... $
\form#28:$ v_k $
\form#29:$ F(v_k)^TF(v_k)<\epsilon $
\form#30:\[ v_{i+1} = v_i + (J(v_i)^TJ(v_i)+\lambda I_{N\times N})^{-1} J(v_i)^T F(v_i) \]
\form#31:$ J(v) $
\form#32:\[ J(v) = \frac{d}{dv}F(v) = \left[ \begin{array}{ccccc} \frac{\partial F_1(v)}{\partial v_1} & \frac{\partial F_1(v)}{\partial v_2} & \frac{\partial F_1(v)}{\partial v_3} & ... & \frac{\partial F_1(v)}{\partial v_N} \\ \frac{\partial F_2(v)}{\partial v_1} & \frac{\partial F_2(v)}{\partial v_2} & \frac{\partial F_2(v)}{\partial v_3} & ... & \frac{\partial F_2(v)}{\partial v_N} \\ \frac{\partial F_3(v)}{\partial v_1} & \frac{\partial F_3(v)}{\partial v_2} & \frac{\partial F_3(v)}{\partial v_3} & ... & \frac{\partial F_3(v)}{\partial v_N} \\ . & . & . & & . \\ . & . & . & & . \\ . & . & . & & . \\ \frac{\partial F_M(v)}{\partial v_1} & \frac{\partial F_M(v)}{\partial v_2} & \frac{\partial F_M(v)}{\partial v_3} & ... & \frac{\partial F_M(v)}{\partial v_N} \\ \end{array} \right] \]
\form#33:$ \lambda $
\form#34:$ F $
\form#35:$ J $
\form#36:$ \lambda I_{N \times N} $
\form#37:\[ S_{top}(v) = \begin{cases} true & if~F(v)<\epsilon \\ false & else \end{cases} \]
\form#38:$ R $
\form#39:\[ \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ . \\ . \\ . \\ x_N \\ \end{array} \right] \stackrel{transformate}{\rightarrow} \left[ \begin{array}{c} \frac{x_1 \times R}{L} \\ \frac{x_2 \times R}{L} \\ \frac{x_3 \times R}{L} \\ . \\ . \\ . \\ \frac{x_N \times R}{L} \\ \end{array} \right] \\ \]
\form#40:$ L=\sqrt{x_1^2 + x_2^2 + x_3^2 + ... + x_N^2 } $
\form#41:$ L $
\form#42:$ f $
\form#43:\[ \left[ \begin{array}{c} x_1 \\ x_2 \\ x_3 \\ . \\ . \\ . \\ x_N \\ \end{array} \right] \stackrel{transformate}{\rightarrow} \left[ \begin{array}{c} \frac{-x_1 \times f}{x_N} \\ \frac{-x_2 \times f}{x_N} \\ \frac{-x_3 \times f}{x_N} \\ . \\ . \\ . \\ -f \\ \end{array} \right] \\ \]
\form#44:$ x_N = -f $
\form#45:$ L=\sqrt{x_1^2+x_2^2+...+x_N^2} $
\form#46:\[ \frac{R}{L^3} \times \left[ \begin{array}{ccccc} L^2-x_1^2 & -x_1x_2 & -x_1x_3 & ... & -x_1x_N \\ -x_2x_1 & L^2-x_2^2 & -x_2x_3 & ... & -x_2x_N \\ -x_3x_1 & -x_3x_2 & L^2-x_3^2 & ... & -x_3x_N \\ . & . & . & & . \\ . & . & . & & . \\ . & . & . & & . \\ -x_Nx_1 & -x_Nx_2 & -x_Nx_3 & ... & L^2-x_N^2 \\ \end{array} \right] \]
\form#47:\[ R \times \left[ \begin{array}{c} \frac{x_1}{L} \\ \frac{x_2}{L} \\ \frac{x_3}{L} \\ . \\ . \\ . \\ \frac{x_N}{L} \\ \end{array} \right] \]
\form#48:\[ f \times \left[ \begin{array}{ccccc} \frac{-1}{x_N} & 0 & 0 & ... & \frac{1}{x_N^2} \\ 0 & \frac{-1}{x_N} & 0 & ... & \frac{1}{x_N^2} \\ 0 & 0 & \frac{-1}{x_N} & ... & \frac{1}{x_N^2} \\ . & . & . & & . \\ . & . & . & & . \\ . & . & . & & . \\ 0 & 0 & 0 & ... & 0 \\ \end{array} \right] \]
\form#49:\[ f \times \left[ \begin{array}{c} \frac{-x_1}{x_N} \\ \frac{-x_2}{x_N} \\ \frac{-x_3}{x_N} \\ . \\ . \\ . \\ -1 \\ \end{array} \right] \]
\form#50:\[ v_{output} = H(h_1, G(g_1, g_2, g_3, F(f_1, v_{input}))) \]
\form#51:$ v_{input}(x,y,z), v_{output} $
\form#52:$ y $
\form#53:\[ m_{jacobian} = \frac{\partial H(h_1, G(g_1, g_2, g_3, F(f_1, v_{input})))} {\partial G(g_1, g_2, g_3, F(f_1, v_{input})) } \frac{\partial G(g_1, g_2, g_3, F(f_1, v_{input}))} {\partial F(f_1, v_{input}) } \frac{\partial F(f_1, v_{input})} {\partial v_{input} } \frac{\partial v_{input}} {\partial y} \]
\form#54:\[ \frac{\partial v_{input}}{\partial y} = \left[ \begin{array}{c} 0 \\ 1 \\ 0 \\ \end{array} \right] \]
\form#55:\[ v_{output} = H(h_1, h_2, G(g_1, g_2, g_3, F(f_1, v_{input}))) \]
\form#56:$ f_1, g_1, g_2, g_3, h_1, h_2 $
\form#57:$ F, G, H $
\form#58:\[ M_{jacobian} = \frac{\partial H(h_1, h_2, G(g_1, g_2, g_3, F(f_1, v_{input})))} {\partial G(g_1, g_2, g_3, F(f_1, v_{input})) } \frac{\partial G(g_1, g_2, g_3, F(f_1, v_{input})) } {\partial F(f_1, v_{input}) } \frac{\partial F(f_1, v_{input}) } {\partial v_{input} } \frac{\partial v_{input} } {\partial y } \]
\form#59:\[ v_{output} = I(i_1,i_2, H(h_1,h_2, G(g_1,g_2,g_3, F(f_1, v_{input})))) \]
\form#60:$ f_1, g_1,g_2,g_3, h_1,h_2, i_1,i_2 $
\form#61:$ F, G, H, I $
\form#62:$ g_2 $
\form#63:\[ M_{jacobian} = \frac{\partial I(i_1,i_2, H(h_1,h_2, G(g_1,g_2,g_3, F(f_1, v_{input}))))} {\partial H(h_1,h_2, G(g_1,g_2,g_3, F(f_1, v_{input}))) } \frac{\partial H(h_1,h_2, G(g_1,g_2,g_3, F(f_1, v_{input}))) } {\partial G(g_1,g_2,g_3, F(f_1, v_{input})) } \frac{\partial G(g_1,g_2,g_3, F(f_1, v_{input})) } {\partial g_2 } \]
\form#64:$ f_1, g_1, g_2, g_3, h_1, h_2, i_1, i_2 $
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