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\begin{document}

\section{Пересчёт систем координат}
\subsection{Из декартовых в сферические}
\begin{align}
    \rho    & = \sqrt{x^2 + y^2 + z^2} \\
    \varphi & = \arccos{\frac{z}{\rho}} \\
    \theta  & = \atanT{y}{x}
\end{align}

\subsection{Из сферические в декартовы}
\begin{align}
    x &= \rho \cdot \sin{\varphi} \cdot \cos{\theta} \\
    y &= \rho \cdot \sin{\varphi} \cdot \sin{\theta} \\
    z &= \rho \cdot \cos{\varphi}
\end{align}

\section{Базовые преобразования}

\subsection{Перенос точки}

\begin{equation}
    T^*(\Delta x, \Delta y, \Delta z) =
    \begin{vmatrix}
        1        & 0        & 0        & 0 \\
        0        & 1        & 0        & 0 \\
        0        & 0        & 1        & 0 \\
        \Delta x & \Delta y & \Delta z & 1 \\
    \end{vmatrix}
\end{equation}

\subsection{Масштабирование точки относительно центра координат}

\begin{equation}
    S^*(Sx, Sy, Sz) =
    \begin{vmatrix}
        Sx & 0  & 0  & 0 \\
        0  & Sy & 0  & 0 \\
        0  & 0  & Sz & 0 \\
        0  & 0  & 0  & 1 \\
    \end{vmatrix}
\end{equation}

\subsection{Поворот точки вокруг оси $x$}

\begin{equation}
    R^*_x(\alpha) =
    \begin{vmatrix}
        1 & 0             & 0            & 0 \\
        0 & \cos(\alpha)  & \sin(\alpha) & 0 \\
        0 & -\sin(\alpha) & \cos(\alpha) & 0 \\
        0 & 0             & 0            & 1 \\
    \end{vmatrix}
\end{equation}

\subsection{Поворот точки вокруг оси $y$}

\begin{equation}
    R^*_y(\alpha) =
    \begin{vmatrix}
        \cos(\alpha) & 0 & -\sin(\alpha) & 0 \\
        0            & 1 & 0             & 0 \\
        \sin(\alpha) & 0 & \cos(\alpha)  & 0 \\
        0            & 0 & 0             & 1 \\
    \end{vmatrix}
\end{equation}

\subsection{Поворот точки вокруг оси $z$}

\begin{equation}
    R^*_z(\alpha) =
    \begin{vmatrix}
        \cos(\alpha)  & \sin(\alpha) & 0 & 0 \\
        -\sin(\alpha) & \cos(\alpha) & 0 & 0 \\
        0             & 0            & 1 & 0 \\
        0             & 0            & 0 & 1 \\
    \end{vmatrix}
\end{equation}

\section{Обратные операции}

\subsection{Перенос системы координат}
\begin{equation}
    T^{-1}(\Delta x, \Delta y, \Delta z) = T^*(-\Delta x, -\Delta y, -\Delta z)
\end{equation}

\subsection{Масштабирование осей системы координат}
\begin{equation}
    S^{-1}(Sx, Sy, Sz) = S^*\left(\frac{1}{Sx}, \frac{1}{Sy}, \frac{1}{Sz}\right)
\end{equation}

\subsection{Поворот системы координат вокруг осей}
\begin{equation}
    R^{-1}_i(\alpha) = R^*_i(-\alpha)
\end{equation}
\begin{where}
    $i$ -- ось системы координат ($\{x, y, z\}$).
\end{where}

\break

\section{Композиция 3D преобразований}

\subsection{Поворот точки относительно линии, проходящей через начало системы координат на угол $\alpha$}

\begin{equation}
    R^{-1}_z(\theta) \cdot R^{-1}_y(\varphi) \cdot \bm{R_z(\alpha)} \cdot R^{-1}_y(-\varphi) \cdot R^{-1}_z(-\theta)
\end{equation}

\begin{align*}
    \underbrace{
        \begin{vmatrix}
            \cos(-\theta)  & \sin(-\theta) & 0 & 0 \\
            -\sin(-\theta) & \cos(-\theta) & 0 & 0 \\
            0              & 0             & 1 & 0 \\
            0              & 0             & 0 & 1 \\
        \end{vmatrix}
    }_{R^{-1}_z(\theta)}
    &
    \times
    \underbrace{
        \begin{vmatrix}
            \cos(-\varphi) & 0 & -\sin(-\varphi) & 0 \\
            0              & 1 & 0               & 0 \\
            \sin(-\varphi) & 0 & \cos(-\varphi)  & 0 \\
            0              & 0 & 0               & 1 \\
        \end{vmatrix}
    }_{R^{-1}_y(\varphi)}
\end{align*}
\begin{align*}
    \times
    \underbrace{
        \begin{vmatrix}
            \cos(\alpha)  & \sin(\alpha) & 0 & 0 \\
            -\sin(\alpha) & \cos(\alpha) & 0 & 0 \\
            0             & 0            & 1 & 0 \\
            0             & 0            & 0 & 1 \\
        \end{vmatrix}
    }_{\bm{R_z(\alpha)}}
\end{align*}
\begin{align*}
    \times
    \underbrace{
        \begin{vmatrix}
            \cos(\varphi) & 0 & -\sin(\varphi) & 0 \\
            0             & 1 & 0              & 0 \\
            \sin(\varphi) & 0 & \cos(\varphi)  & 0 \\
            0             & 0 & 0              & 1 \\
        \end{vmatrix}
    }_{R^{-1}_y(-\varphi)}
    &
    \times
    \underbrace{
        \begin{vmatrix}
            \cos(\theta)  & \sin(\theta) & 0 & 0 \\
            -\sin(\theta) & \cos(\theta) & 0 & 0 \\
            0             & 0            & 1 & 0 \\
            0             & 0            & 0 & 1 \\
        \end{vmatrix}
    }_{R^{-1}_z(-\theta)}
\end{align*}

\begin{align*}
    R^{-1}_z(\theta) \cdot R^{-1}_y(\varphi)
    &=
    \begin{vmatrix}
        \cos(-\theta)  & \sin(-\theta) & 0 & 0 \\
        -\sin(-\theta) & \cos(-\theta) & 0 & 0 \\
        0              & 0             & 1 & 0 \\
        0              & 0             & 0 & 1 \\
    \end{vmatrix}
    \times
    \begin{vmatrix}
        \cos(-\varphi) & 0 & -\sin(-\varphi) & 0 \\
        0              & 1 & 0               & 0 \\
        \sin(-\varphi) & 0 & \cos(-\varphi)  & 0 \\
        0              & 0 & 0               & 1 \\
    \end{vmatrix}
    \\ &=
    \begin{vmatrix}
        \cos(\theta) & -\sin(\theta) & 0 & 0 \\
        \sin(\theta) & \cos(\theta)  & 0 & 0 \\
        0            & 0             & 1 & 0 \\
        0            & 0             & 0 & 1 \\
    \end{vmatrix}
    \times
    \begin{vmatrix}
        \cos(\varphi)  & 0 & \sin(\varphi) & 0 \\
        0              & 1 & 0             & 0 \\
        -\sin(\varphi) & 0 & \cos(\varphi) & 0 \\
        0              & 0 & 0             & 1 \\
    \end{vmatrix}
    \\ &=
    \begin{vmatrix}
        \cos(\theta)\cos(\varphi) & -\sin(\theta) & \cos(\theta)\sin(\varphi) & 0 \\
        \sin(\theta)\cos(\varphi) & \cos(\theta)  & \sin(\theta)\sin(\varphi) & 0 \\
        -\sin(\varphi)            & 0             & \cos(\varphi)             & 0 \\
        0                         & 0             & 0                         & 1 \\
    \end{vmatrix}
\end{align*}

\href{https://www.wolframalpha.com/input/?i=%7B%7Bcos(-theta),sin(-theta),0,0%7D,%7B-sin(-theta),cos(-theta),0,0%7D,%7B0,0,1,0%7D,%7B0,0,0,1%7D%7D*%7B%7Bcos(-phi),0,-sin(-phi),0%7D,%7B0,1,0,0%7D,%7Bsin(-phi),0,cos(-phi),0%7D,%7B0,0,0,1%7D%7D}{WolframAlpha}

\begin{align*}
    R^{-1}_y(-\varphi) \cdot R^{-1}_z(-\theta)
    &=
    \begin{vmatrix}
        \cos(\varphi) & 0 & -\sin(\varphi) & 0 \\
        0             & 1 & 0              & 0 \\
        \sin(\varphi) & 0 & \cos(\varphi)  & 0 \\
        0             & 0 & 0              & 1 \\
    \end{vmatrix}
    \times
    \begin{vmatrix}
        \cos(\theta)  & \sin(\theta) & 0 & 0 \\
        -\sin(\theta) & \cos(\theta) & 0 & 0 \\
        0             & 0            & 1 & 0 \\
        0             & 0            & 0 & 1 \\
    \end{vmatrix}
    \\ &=
    \begin{vmatrix}
        \cos(\varphi)\cos(\theta) & \cos(\varphi)\sin(\theta) & -\sin(\varphi) & 0 \\
        -\sin(\theta)             & \cos(\theta)              & 0              & 0 \\
        \sin(\varphi)\cos(\theta) & \sin(\varphi)\sin(\theta) & \cos(\varphi)  & 0 \\
        0                         & 0                         & 0              & 1 \\
    \end{vmatrix}
\end{align*}

\href{https://www.wolframalpha.com/input/?i=%7B%7Bcos(phi),0,-sin(phi),0%7D,%7B0,1,0,0%7D,%7Bsin(phi),0,cos(phi),0%7D,%7B0,0,0,1%7D%7D*%7B%7Bcos(theta),sin(theta),0,0%7D,%7B-sin(theta),cos(theta),0,0%7D,%7B0,0,1,0%7D,%7B0,0,0,1%7D%7D}{WolframAlpha}

\begin{align*}
    & \left[ R^{-1}_z(\theta) \cdot R^{-1}_y(\varphi) \right] \cdot \bm{R_z(\alpha)} =
    \\ &=
    \begin{vmatrix}
        \cos(\theta)\cos(\varphi) & -\sin(\theta) & \cos(\theta)\sin(\varphi) & 0 \\
        \sin(\theta)\cos(\varphi) & \cos(\theta)  & \sin(\theta)\sin(\varphi) & 0 \\
        -\sin(\varphi)            & 0             & \cos(\varphi)             & 0 \\
        0                         & 0             & 0                         & 1 \\
    \end{vmatrix}
    \times
    \begin{vmatrix}
        \cos(\alpha)  & \sin(\alpha) & 0 & 0 \\
        -\sin(\alpha) & \cos(\alpha) & 0 & 0 \\
        0             & 0            & 1 & 0 \\
        0             & 0            & 0 & 1 \\
    \end{vmatrix}
    \\ &=
    \begin{vmatrix}
        \cos(\theta)\cos(\varphi)\cos(\alpha) + \sin(\theta)\sin(\alpha) & \cos(\theta)\cos(\varphi)\sin(\alpha) - \sin(\theta)\cos(\alpha) & \cos(\theta)\sin(\varphi) & 0 \\
        \sin(\theta)\cos(\varphi)\cos(\alpha) - \cos(\theta)\sin(\alpha) & \sin(\theta)\cos(\varphi)\sin(\alpha) + \cos(\theta)\cos(\alpha) & \sin(\theta)\sin(\varphi) & 0 \\
        -\sin(\varphi)\cos(\alpha)                                       & -\sin(\varphi)\sin(\alpha)                                       & \cos(\varphi)             & 0 \\
        0                                                                & 0                                                                & 0                         & 1 \\
    \end{vmatrix}
\end{align*}

\href{https://www.wolframalpha.com/input/?i=%7B%7BCos%5B%CE%B8%5DCos%5B%CF%95%5D,-Sin%5B%CE%B8%5D,Cos%5B%CE%B8%5DSin%5B%CF%95%5D,0%7D,%7BCos%5B%CF%95%5DSin%5B%CE%B8%5D,Cos%5B%CE%B8%5D,Sin%5B%CE%B8%5DSin%5B%CF%95%5D,0%7D,%7B-Sin%5B%CF%95%5D,0,Cos%5B%CF%95%5D,0%7D,%7B0,0,0,1%7D%7D*%7B%7Bcos(alpha),sin(alpha),0,0%7D,%7B-sin(alpha),cos(alpha),0,0%7D,%7B0,0,1,0%7D,%7B0,0,0,1%7D%7D}{WolframAlpha}

\begin{align*}
    & \left[ R^{-1}_z(\theta) \cdot R^{-1}_y(\varphi) \cdot \bm{R_z(\alpha)} \right] \cdot \left[ R^{-1}_y(-\varphi) \cdot R^{-1}_z(-\theta) \right] =
    \\ &=
    \begin{vmatrix}
        \cos(\theta)\cos(\varphi)\cos(\alpha) + \sin(\theta)\sin(\alpha) & \cos(\theta)\cos(\varphi)\sin(\alpha) - \sin(\theta)\cos(\alpha) & \cos(\theta)\sin(\varphi) & 0 \\
        \sin(\theta)\cos(\varphi)\cos(\alpha) - \cos(\theta)\sin(\alpha) & \sin(\theta)\cos(\varphi)\sin(\alpha) + \cos(\theta)\cos(\alpha) & \sin(\theta)\sin(\varphi) & 0 \\
        -\sin(\varphi)\cos(\alpha)                                       & -\sin(\varphi)\sin(\alpha)                                       & \cos(\varphi)             & 0 \\
        0                                                                & 0                                                                & 0                         & 1 \\
    \end{vmatrix}
    \\ &\times
    \begin{vmatrix}
        \cos(\varphi)\cos(\theta) & \cos(\varphi)\sin(\theta) & -\sin(\varphi) & 0 \\
        -\sin(\theta)             & \cos(\theta)              & 0              & 0 \\
        \sin(\varphi)\cos(\theta) & \sin(\varphi)\sin(\theta) & \cos(\varphi)  & 0 \\
        0                         & 0                         & 0              & 1 \\
    \end{vmatrix}
    \\ &=
    \begin{vmatrix}
        r_{11} & r_{12} & r_{13} & r_{14} \\
        r_{21} & r_{22} & r_{23} & r_{24} \\
        r_{31} & r_{32} & r_{33} & r_{34} \\
        r_{41} & r_{42} & r_{43} & r_{44} \\
    \end{vmatrix}
\end{align*}

\begin{align*}
    r_{11}
    &= \left[ \cos(\theta)\cos(\varphi)\cos(\alpha) + \sin(\theta)\sin(\alpha) \right] \cdot \left[\cos(\varphi)\cos(\theta)\right] \\
    &+ \left[ \cos(\theta)\cos(\varphi)\sin(\alpha) - \sin(\theta)\cos(\alpha) \right] \cdot \left[-\sin(\theta)\right] \\
    &+ \left[\cos(\theta)\sin(\varphi)\right] \cdot \left[\sin(\varphi)\cos(\theta)\right] \\
    \\
    r_{12}
    &= \left[ \cos(\theta)\cos(\varphi)\cos(\alpha) + \sin(\theta)\sin(\alpha) \right] \cdot \left[ \cos(\varphi)\sin(\theta) \right] \\
    &+ \left[ \cos(\theta)\cos(\varphi)\sin(\alpha) - \sin(\theta)\cos(\alpha) \right] \cdot \left[\cos(\theta)\right] \\
    &+ \left[\cos(\theta)\sin(\varphi)\right] \cdot \left[\sin(\varphi)\sin(\theta)\right] \\
    \\
    r_{13}
    &= \left[ \cos(\theta)\cos(\varphi)\cos(\alpha) + \sin(\theta)\sin(\alpha) \right] \cdot \left[-\sin(\varphi)\right] \\
    &+ \left[\cos(\theta)\sin(\varphi)\right] \cdot \left[\cos(\varphi)\right] \\
    \\
    r_{14} &= 0 \\
\end{align*}

\begin{align*}
    r_{21}
    &= \left[ \sin(\theta)\cos(\varphi)\cos(\alpha) - \cos(\theta)\sin(\alpha) \right] \cdot \left[\cos(\varphi)\cos(\theta)\right] \\
    &+ \left[ \sin(\theta)\cos(\varphi)\sin(\alpha) + \cos(\theta)\cos(\alpha) \right] \cdot \left[-\sin(\theta)\right] \\
    &+ \left[\sin(\theta)\sin(\varphi)\right] \cdot \left[\sin(\varphi)\cos(\theta)\right] \\
    \\
    r_{22}
    &= \left[ \sin(\theta)\cos(\varphi)\cos(\alpha) - \cos(\theta)\sin(\alpha) \right] \cdot \left[\cos(\varphi)\sin(\theta)\right] \\
    &+ \left[ \sin(\theta)\cos(\varphi)\sin(\alpha) + \cos(\theta)\cos(\alpha) \right] \cdot \left[\cos(\theta)\right] \\
    &+ \left[\sin(\theta)\sin(\varphi)\right] \cdot \left[\sin(\varphi)\sin(\theta)\right] \\
    \\
    r_{23}
    &= \left[ \sin(\theta)\cos(\varphi)\cos(\alpha) - \cos(\theta)\sin(\alpha) \right] \cdot \left[-\sin(\varphi)\right] \\
    &+ \left[\sin(\theta)\sin(\varphi)\right] \cdot \left[\cos(\varphi)\right] \\
    \\
    r_{24} &= 0 \\
    \\
\end{align*}

\begin{align*}
    r_{31}
    &= \left[ -\sin(\varphi)\cos(\alpha) \right] \cdot \left[ \cos(\varphi)\cos(\theta) \right] \\
    &+ \left[ -\sin(\varphi)\sin(\alpha) \right] \cdot \left[ -\sin(\theta) \right] \\
    &+ \left[ \cos(\varphi) \right] \cdot \left[ \sin(\varphi)\cos(\theta) \right] \\
    &= -\sin(\varphi)\cos(\alpha)\cos(\varphi)\cos(\theta) + \sin(\varphi)\sin(\alpha)\sin(\theta) + \cos(\varphi)\sin(\varphi)\cos(\theta) \\
    &= \sin(\varphi)\left[ \sin(\alpha)\sin(\theta) - \left( \cos(\alpha)-1 \right) \cos(\varphi)\cos(\theta) \right] \\
    \\
    r_{32}
    &= \left[ -\sin(\varphi)\cos(\alpha) \right] \cdot \left[ \cos(\varphi)\sin(\theta) \right] \\
    &+ \left[ -\sin(\varphi)\sin(\alpha) \right] \cdot \left[ \cos(\theta) \right] \\
    &+ \left[ \cos(\varphi) \right] \cdot \left[ \sin(\varphi)\sin(\theta) \right] \\
    &= -\sin(\varphi)\cos(\alpha)\cos(\varphi)\sin(\theta) - \sin(\varphi)\sin(\alpha)\cos(\theta) + \cos(\varphi)\sin(\varphi)\sin(\theta) \\
    &= -\sin(\varphi)\left[ \left( \cos(\alpha)-1 \right) \cos(\varphi)\sin(\theta) + \sin(\alpha)\cos(\theta) \right] \\
    \\
    r_{33}
    &= \left[ -\sin(\varphi)\cos(\alpha) \right] \cdot \left[ -\sin(\varphi) \right] \\
    &+ \left[ \cos(\varphi) \right] \cdot \left[ \cos(\varphi) \right] \\
    &= \sin^2(\varphi)\cos(\alpha) + \cos^2(\varphi) \\
    \\
    r_{34} &= 0 \\
    \\
\end{align*}

\begin{align*}
    r_{41} &= 0 \\
    r_{42} &= 0 \\
    r_{43} &= 0 \\
    r_{44} &= 1 \\
\end{align*}

\end{document}