\documentclass[a4paper]{article}
\usepackage{amssymb, amsmath}
\begin{document}
 
\title{Equations used in the Interactive Graphics Course}

\section{Lecture 1}

\begin{equation}
 \frac{X_w-WX_{min}}{WX_{max}-WX_{min}} = \frac{X_d-DX_{min}}{DX_{max}-DX_{min}}
\end{equation}
\newline
\begin{equation}
  X_d = \frac{(X_w-WX_{min})(DX_{max}-DX_{min})}{WX_{max}-WX_{min}} + DX_{min}
\end{equation}


\begin{equation}
[x', y', z'] =
 \left[
  \begin{array}{ c c c }
     2 & 0 & 0\\
     0 & 2 & 0 \\
     0 & 0 & 2
  \end{array} \right]
 \left[
  \begin{array}{ c }
     x \\
     y \\
     z
  \end{array} \right]
\end{equation}
\newline
\begin{equation}
 x'=2x \;,\; y'=2y \;,\; z'=2z
\end{equation}
\newline
\begin{equation}
[x', y', z', 1] = [x,y,z,1] 
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     2 & 0 & 0 & 1
  \end{array} \right]
\end{equation}
\newline
\begin{equation}
 [x,y,z,1] 
 \left[
  \begin{array}{ c c c c }
     1   & 0   & 0   & 0 \\
     0   & 1   & 0   & 0 \\
     0   & 0   & 1   & 0 \\
     t_x & t_y & t_z & 1
  \end{array} \right] = [x+t_x, y+t_y, x+t_z, 1]
\end{equation}
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     1   & 0   & 0   & 0 \\
     0   & 1   & 0   & 0 \\
     0   & 0   & 1   & 0 \\
     t_x & t_y & t_z & 1
  \end{array} \right] $has\;inversion$
 \left[
  \begin{array}{ c c c c }
     1    & 0    & 0    & 0 \\
     0    & 1    & 0    & 0 \\
     0    & 0    & 1    & 0 \\
     -t_x & -t_y & -t_z & 1
  \end{array} \right] 
$
\newline
\newline
\begin{equation}
 [x,y,z,1] 
 \left[
  \begin{array}{ c c c c }
     s_x & 0   & 0   & 0 \\
     0   & s_y & 0   & 0 \\
     0   & 0   & s_z & 0 \\
     0   & 0   & 0   & 1
  \end{array} \right] = [s_x x, s_y y, s_z z, 1]
\end{equation}
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     s_x & 0   & 0   & 0 \\
     0   & s_y & 0   & 0 \\
     0   & 0   & s_z & 0 \\
     0   & 0   & 0   & 1
  \end{array} \right] $has\;inversion$
 \left[
  \begin{array}{ c c c c }
     1/s_x & 0     & 0     & 0 \\
     0     & 1/s_y & 0     & 0 \\
     0     & 0     & 1/s_z & 0 \\
     0     & 0     & 0     & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
[x', y', z', 1] = [x,y,z,1] 
 \left[
  \begin{array}{ c c c c }
     2 & 0 & 0 & 0 \\
     0 & 2 & 0 & 0 \\
     0 & 0 & 2 & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right]
\left[
  \begin{array}{ c c c c }
     1 & 0 & 0  & 0 \\
     0 & 1 & 0  & 0 \\
     0 & 0 & 1  & 0 \\
     5 & 5 & 20 & 1
  \end{array} \right]
$
\newline
\newline
\newline
$
[x', y', z', 1] = [x,y,z,1] 
 \left[
  \begin{array}{ c c c c }
     2 & 0 & 0  & 0 \\
     0 & 2 & 0  & 0 \\
     0 & 0 & 2  & 0 \\
     5 & 5 & 20 & 1
  \end{array} \right]
$
\newline
\newline
\newline
$
R_x =
 \left[
  \begin{array}{ c c c c }
     1 & 0          & 0         & 0 \\
     0 & Cos\theta  & Sin\theta & 0 \\
     0 & -Sin\theta & Cos\theta & 0 \\
     0 & 0          & 0         & 1
  \end{array} \right]
$
\newline
\newline
\newline
$
R_y =
 \left[
  \begin{array}{ c c c c }
     Cos\theta & 0 & -Sin\theta & 0 \\
     0         & 1 &  0         & 0 \\
     Sin\theta & 0 & Cos\theta  & 0 \\
     0         & 0 & 0          & 1
  \end{array} \right]
$
\newline
\newline
\newline
$
R_z =
 \left[
  \begin{array}{ c c c c }
     Cos\theta  & Sin\theta & 0 & 0 \\
     -Sin\theta & Cos\theta & 0 & 0 \\
     0          & 0         & 1 & 0 \\
     0          & 0         & 0 & 1
  \end{array} \right]
$
\newline
\newline
\newline
\begin{tabular}{ r l }
$[x,y]$ & = $[\;rCos\phi \;,\; rSin\phi\;]$ \\
$[x_t, y_t]$ & $= [\;rCos(\phi+\theta) \;,\; rSin(\phi+\theta)\;]$ \\
 & $= [\;rCos\phi Cos\theta-rSin\phi Sin\theta \;,\; rCos\phi Sin\theta + rSin\phi Cos\theta\;]$ \\
 & $= [\;xCos\theta-ySin\theta \;,\; xSin\theta + yCos\theta\;]$ \\
 & $ = [x,y] 
 \left[
  \begin{array}{ c c }
     Cos\theta  & Sin\theta \\
     -Sin\theta & Cos\theta
  \end{array} \right] $
 \end{tabular}
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     Cos\theta  & Sin\theta & 0 & 0 \\
     -Sin\theta & Cos\theta & 0 & 0 \\
     0          & 0         & 1 & 0 \\
     0          & 0         & 0 & 1
  \end{array} \right] $has\;inversion$
 \left[
  \begin{array}{ c c c c }
     Cos\theta & -Sin\theta & 0 & 0 \\
     Sin\theta & Cos\theta  & 0 & 0 \\
     0         & 0          & 1 & 0 \\
     0         & 0          & 0 & 1
  \end{array} \right] 
$
\newline
\newline
\newline

\newpage
\section{Lecture 2}
$
 A = 
 \left[
  \begin{array}{ c c c c }
     1    & 0    & 0    & 0 \\
     0    & 1    & 0    & 0 \\
     0    & 0    & 1    & 0 \\
     -C_x & -C_y & -C_z & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
v = \sqrt{d_x^2 + d_z^2}
\newline
\newline
Cos\theta = d_z / v
\newline
\newline
Sin\theta = d_x/v
\newline
\newline
B = 
 \left[
  \begin{array}{ c c c c }
     d_z/v  & 0    & d_x/v & 0 \\
     0      & 1    & 0     & 0 \\
     -d_x/v & 0    & d_z/v & 0 \\
     0      & 0    & 0     & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
v = \sqrt{d_x^2 + d_z^2}
\newline
\newline
Cos\phi = v/|d|
\newline
\newline
Sin\phi = d_y/|d|
\newline
\newline
C = 
 \left[
  \begin{array}{ c c c c }
     1 & 0       & 0       & 0 \\
     0 & v/|d|   & d_y/|d| & 0 \\
     0 & -d_y/|d|& v/|d|   & 0 \\
     0 & 0       & 0       & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
M_o = 
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 0 & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
M_p = 
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 1/f \\
     0 & 0 & 0 & 0
  \end{array} \right] 
$
\newline
\newline
\newline
$
[x,y,z,1] 
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 1/f \\
     0 & 0 & 0 & 0
  \end{array} \right] 
  = [x,y,z,z/f]
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     q_x & q_y & q_z & 0 \\
     r_x & r_y & r_z & 0 \\
     s_x & s_y & s_z & 0 \\
     C_x & C_y & C_z & 1
  \end{array} \right]
$
\newline
\newline
\newline
$
 [1,0,0,0]
 \left[
  \begin{array}{ c c c c }
     q_x & q_y & q_z & 0 \\
     r_x & r_y & r_z & 0 \\
     s_x & s_y & s_z & 0 \\
     C_x & C_y & C_z & 1
  \end{array} \right] = [q_x, q_y, q_z, 0]
$
\newline
\newline
\newline
$
 [0,0,0,1]
 \left[
  \begin{array}{ c c c c }
     q_x & q_y & q_z & 0 \\
     r_x & r_y & r_z & 0 \\
     s_x & s_y & s_z & 0 \\
     C_x & C_y & C_z & 1
  \end{array} \right] = [C_x, C_y, C_z, 1]
$
\newline
\newline
\newline
$
 [P^t_x,P^t_y,P^t_z,1] = [P_x, P_y, P_z, 1]
 \left[
  \begin{array}{ c c c c }
     u_x & v_x & w_x & 0 \\
     u_y & v_y & w_y & 0 \\
     u_z & v_z & w_z & 0 \\
     -C\cdot u & -C\cdot v & -C\cdot w & 1
  \end{array} \right] 
$
\newline
\newline
\newline

\newpage
\section{Tutorial 2}
$
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & -10 & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     0.8 & 0 & 0 & 0 \\
     0 & 0.8 & 0 & 0 \\
     0 & 0 & 0.8 & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 10 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     0.8 & 0 & 0 & 0 \\
     0 & 0.8 & 0 & 0 \\
     0 & 0 & 0.8 & 0 \\
     0 & 0 & -8 & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 10 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     0.8 & 0 & 0 & 0 \\
     0 & 0.8 & 0 & 0 \\
     0 & 0 & 0.8 & 0 \\
     0 & 0 & 2 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & -10 & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     0.8 & 0 & 0 & 0 \\
     0 & 0.8 & 0 & 0 \\
     0 & 0 & 0.8 & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     0.97 & -0.26 & 0 & 0 \\
     0.26 & 0.97 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 10 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     0.8 & 0 & 0 & 0 \\
     0 & 0.8 & 0 & 0 \\
     0 & 0 & 0.8 & 0 \\
     0 & 0 & -8 & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     0.97 & -0.26 & 0 & 0 \\
     0.26 & 0.97 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 10 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     0.776 & -0.208 & 0 & 0 \\
     0.208 & 0.776 & 0 & 0 \\
     0 & 0 & 0.8 & 0 \\
     0 & 0 & 2 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     0.8 & 0 & 0 & 0 \\
     0 & 0.8 & 0 & 0 \\
     0 & 0 & 0.8 & 0 \\
     0 & 0 & 2 & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0.5 \\
     0 & 0 & 0 & 0
  \end{array} \right] 
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     0.8 & 0 & 0 & 0 \\
     0 & 0.8 & 0 & 0 \\
     0 & 0 & 0.8 & 0.4 \\
     0 & 0 & 2 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
 \left[
  \begin{array}{ c c c c }
     0 & 0 & -1 & 0 \\
     0 & 1 & 0 & 0 \\
     -1 & 0 & 0 & 0 \\
     -10 & -10 & 50 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
\newpage
\section{Lecture 8}
\vspace*{0.15in}
\centering
\begin{tabular}{ | c | c | c | c |}
\hline
    & $\mathbf{P_0}$ & $\mathbf{P_1}$ & $\mathbf{P_2}$  \\
\hline
$\mu$ & $0$ & $1/2$ & $1$ \\
\hline
\end{tabular}
\vspace*{0.15in}

\centering
\begin{tabular}{ | c | c | c | c |}
\hline
    & $\mathbf{P_0}$ & $\mathbf{P_1}$ & $\mathbf{P_2}$  \\
\hline
$\mu$ & $0$ & $1$ & $1/2$ \\
\hline
\end{tabular}

\vspace*{0.15in}

\centering
\begin{tabular}{ | c | c | c | c |}
\hline
    & $\mathbf{P_0}$ & $\mathbf{P_1}$ & $\mathbf{P_2}$  \\
\hline
$\mu$ & $1/2$ & $0$ & $1$ \\
\hline
\end{tabular}
\vspace*{0.15in}

\centering
\begin{tabular}{ | c | c | c | c |}
\hline
    & $\mathbf{P_0}$ & $\mathbf{P_1}$ & $\mathbf{P_2}$  \\
\hline
$\mu$ & $0$ & $1$ & $1/4$ \\
\hline
\end{tabular}
\vspace*{0.15in}

\section{Lecture 9}
$
 \mathbf P(\mu,\nu) = [\mu^3 \; \mu^2 \; \mu \; 1]
 \left[
  \begin{array}{ c c c c }
     \mathbf a & \mathbf b & \mathbf c & \mathbf d \\
     \mathbf b & \mathbf e & \mathbf f & \mathbf g \\
     \mathbf c & \mathbf f & \mathbf h & \mathbf i \\
     \mathbf d & \mathbf g & \mathbf i & \mathbf j
  \end{array} \right] 
 \left[
  \begin{array}{ c }
     \nu^3 \\
     \nu^2 \\
     \nu \\
      1
  \end{array} \right] 
$

\section{Lecture 18}
$
\left[
  \begin{array}{ c c c c }
     u_x & u_y & u_z & 0 \\
     v_x & v_y & v_z & 0 \\
     w_x & w_y & w_z & 0 \\
     C_x & C_y & C_z & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$\left[
  \begin{array}{ c c c c }
     u_x & u_y & u_z & 0 \\
     v_x & v_y & v_z & 0 \\
     w_x & w_y & w_z & 0 \\
     C_x & C_y & C_z & 1
  \end{array} \right] 
 \left[
  \begin{array}{ c c c c }
     u_x & v_x & w_x & 0 \\
     u_y & v_y & w_y & 0 \\
     u_z & v_z & w_z & 0 \\
     -C\cdot u & -C\cdot v & -C\cdot w & 1
  \end{array} \right] =
\left[
  \begin{array}{ c c c c }
     1 & 0 & 0 & 0 \\
     0 & 1 & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$ T_i^{i-1} =
\left[
  \begin{array}{ c c c c }
     u_x^i & u_y^i & u_z^i & 0 \\
     v_x^i & v_y^i & v_z^i & 0 \\
     w_x^i & w_y^i & w_z^i & 0 \\
     C_x^i & C_y^i & C_z^i & 1
  \end{array} \right] 
$
\newline
\newline
\newline
$
T_n^0 = \prod_{i=n}^0 T_i^{i-1}
$
\newline
\newline
\newline
$R_{Euler} =\left[
  \begin{array}{ c c c c }
     Cos(\theta) & -Sin(\theta) & 0 & 0 \\
     Sin(\theta) & Cos(\theta) & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right] 
\left[
  \begin{array}{ c c c c }
     Cos(\phi) & 0 & -Sin(\phi) & 0 \\
     0 & 1 & 0 & 0 \\
     Sin(\phi) & 0 & Cos(\phi) & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right] 
  \left[
  \begin{array}{ c c c c }
     Cos(\psi) & -Sin(\psi) & 0 & 0 \\
     Sin(\psi) & Cos(\psi) & 0 & 0 \\
     0 & 0 & 1 & 0 \\
     0 & 0 & 0 & 1
  \end{array} \right] 
$
\end{document}
