\documentclass{article}
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\usepackage{hyperref}
\usepackage{amsmath}
\usepackage{tabulary}

\title{Foundations of Game Engine Development 1: Mathematics}
\author{bj}
\date{August 2020}

\begin{document}

\maketitle

\section{Vector Formulas}

{\setlength{\extrarowheight}{10pt}
\begin{tabulary}{8in}{LR}
$ ||\textbf{v}|| = \sqrt{ v_x^2  + v_y^2 + v_z^2 + ...} $ & Vector Magnitude \\
$ ||\textbf{v}|| = \sqrt{ \textbf{v} \cdot \textbf{v} } $ & \\

$ ||t\textbf{v}|| = |t| ||\textbf{v}|| $ & Scalar magnitude property \\

$ \textbf{a} \cdot \textbf{b} = a_x b_x + a_y b_y + a_x b_z + ... $ & Dot product of vectors \\ 
$ \textbf{a} \cdot \textbf{b} = ||\textbf{a}|| ||\textbf{b}|| cos\theta $ & \\
$ \textbf{a} \cdot \textbf{a} = a^2$ & Notation (indicates a scalar result) \\

\textbf{a} and \textbf{b} are orthogonal (perpindicular) if $\textbf{a} \cdot \textbf{b} = 0$ & \\

$
\textbf{a} \times \textbf{b} = \begin{bmatrix}
	a_y b_z - a_z b_y \\ 
	a_z b_x - a_x b_z \\ 
	a_x b_y - a_y b_x
	\end{bmatrix}
$ & Cross product on 3D vectors \\

$ \textbf{a} \times \textbf{b} = ||\textbf{a}|| ||\textbf{b}|| cos \theta$ &
 \\

$A = \frac{1}{2} ||(\textbf{p}_1 - \textbf{p}_0) \times (\textbf{p}_2 - \textbf{p}_0)|| $ & 
Area of a triangle defined by 3 points $\textbf{p}_0, \textbf{p}_1, \textbf{p}_2$ \\


$||\textbf{a} \times \textbf{b}|| = ||\textbf{a}|| ||\textbf{b}|| sin\theta$ & 
Magnitude of a cross product \\


$ \hat{\textbf{v}} = \frac{\textbf{v}}{||\textbf{v}||} $ & 
Normalize a vector (becomes a unit vector, members sum to 1) \\

$ \textbf{a} \times (\textbf{b} \times \textbf{c}) = 
\textbf{b}(\textbf{a} \cdot \textbf{c}) - \textbf{c}(\textbf{a} \cdot \textbf{b})$ 
& Vector Triple Product (``back minus cab'') \\

$ [ \textbf{a}, \textbf{b}, \textbf{c} ] 
= (\textbf{a} \times \textbf{b}) \cdot \textbf{c} = (\textbf{b} \times 
\textbf{c}) \cdot \textbf{a} 
= (\textbf{c} \times \textbf{a}) \cdot \textbf{b} $ & 
 Scalar triple prodct. Equals the volume of a parallelepiped formed by a,b, and c. \\
$ [ \textbf{a}, \textbf{b}, \textbf{c} ] = -[ \textbf{c}, \textbf{b}, \textbf{a} ]$ & \\

$ \textbf{a}_{||\textbf{b}} = \frac{\textbf{a} \cdot \textbf{b}}{b^2} \textbf{b} $
& Projection of a onto b (cosine/X component) \\

$ \textbf{a}_{||\textbf{b}} = \frac{1}{b^2} \begin{bmatrix}
  b_x^2 & b_x b_y & b_x b_z \\
  b_x b_y & b_y^2 & b_y b_z \\
  b_x b_z & b_y b_z & b_z^2 \\
\end{bmatrix} 
\begin{bmatrix}
a_x \\
a_y \\
a_z \\
\end{bmatrix}
$ & Projection of a onto b with 3d vectors \\

$ \textbf{a}_{\perp \textbf{b}} = \textbf{a} - \textbf{a}_{||\textbf{b}} $
& Rejection of b from a (sin/Y component) \\

$ (\textbf{a}_{||\textbf{b}})^2 + (\textbf{a}_{\perp \textbf{b}})^2 = a^2 $
& Projection/Rejection property (pythagorean theorem) \\

$ \textbf{u}_1 = \textbf{v}_1, 
\textbf{u}_2 = \textbf{v}_2 - (\textbf{v}_2)_{||\textbf{u}_1}, 
\textbf{u}_3 = \textbf{v}_3 - (\textbf{v}_3)_{||\textbf{u}_1} - (\textbf{v}_3)_{||\textbf{u}_2}, ... $
& Gram-Schmidt process (orthogonal vectors set). \\

& Common to normalize each result
(\textit{orthonormalization}) \\

$\textbf{a} \otimes \textbf{b} = \textbf{a} \textbf{b}^T = \begin{bmatrix}a_x \\ a_y \\ a_z \\ \end{bmatrix}
\begin{bmatrix}b_x & b_y & b_z \\ \end{bmatrix} $ 
& 3D Vector Outer Product \\

$ =
\begin{bmatrix}
a_x b_z & a_x b_y & a_x b_z \\
a_y b_x & a_y b_y & a_y b_z \\
a_z b_x & a_z b_y & a_z b_z \\
\end{bmatrix}
$ & \\

$ \textbf{v} = (v_x, v_y, v_z, 0)$ & Homogenous direction vector \\
$ \textbf{\textit{p}} = (v_x, v_y, v_z, 1)$ & Homogenous position vector \\

\end{tabulary} % Vector formulas table
}

\section{Matrix Formulas}

{\setlength{\extrarowheight}{10pt}
 
\begin{tabulary}{8in}{LR}
$ \textbf{A}_{n,m}\textbf{B}_{m,p} = \textbf{C}_{n,p} $ & Matrix mult result size; 
also cols A must equal rows B \\

$ \textbf{M}_{i,j}^T = \textbf{M}_{j,i} $ & Matrix transpose \\

$ \textbf{M} = [\textbf{a} \textbf{b} \textbf{c}]$ & Column vector representation \\
$ \textbf{M} = \begin{bmatrix}
	X & a & b \\
	a & X & c \\
	b & c & X
	\end{bmatrix}
 $ & Symmetric matrix (X is be any value) \\
 
 $\textbf{M}^T = \textbf{M}$ & \\
 
 $ \textbf{M} = \begin{bmatrix}
	0 & a & b \\
	-a & 0 & c \\
	-b & -c & 0
	\end{bmatrix}$ &
 Antisymmetric matrix or skew-symmetric matrix 
 (diagonals must be 0) \\

$\textbf{M}^T = -\textbf{M}$ & \\ 

$ \textbf{M}\textbf{v} = v_x \textbf{a} + v_y \textbf{b} + v_z \textbf{c}$ &
Column vector mult notation (a,b,c cols of M) \\

$ \begin{vmatrix}a & b \\ c & d \end{vmatrix} = ad - bc$ & 2D Matrix determinant \\

$ det(\textbf{M}) = \sum\limits_{i=0}^{n-1} M_{ik}(-1)^{i+k}|\textbf{M}_{\bar{ik}}| $ 
& Recursive determinant (each $|M_{ik}|$ is the sub matrix excluding the current col) \\

 & k is chosen as any row; good when row is mostly zeros\\

$ C_{ij}(\textbf{M}) = (-1)^{i+j}|\textbf{M}_{\bar{ij}}|$ & Cofactor matrix definition \\
$ \textbf{M}^{-1} = \frac{1}{det(\textbf{M})} \textbf{C}^T(\textbf{M})$ 
& Inverse; $C^T(M) =$ adjugate matrix \\

$\textbf{A}^{-1} = \frac{1}{A_{00}A_{11} - A_{01}A_{10}} 
\begin{bmatrix}
A_{11} & -A_{01} \\
-A_{10} & A_{00}
\end{bmatrix}$ & 2D Matrix inverse \\

$\textbf{B}^{-1} = \frac{1}{det(\textbf{B})} 
\begin{bmatrix}
B_{11}B_{22} - B_{12}B_{21} & B_{02}B_{21} - B_{01}B_{22} & B_{01}B_{12} - B_{02}B_{11} \\
B_{12}B_{20} - B_{10}B_{22} & B_{00}B_{22} - B_{02}B_{20} & B_{02}B_{10} - B_{00}B_{12} \\
B_{10}B_{21} - B_{11}B_{20} & B_{01}B_{20} - B_{00}B_{21} & B_{00}B_{11} - B_{01}B_{10}
\end{bmatrix} $ & 3D Matrix inverse \\

$ \textbf{M}^{-1} = \frac{1}{[\textbf{a},\textbf{b},\textbf{c}]} 
\begin{bmatrix}
\textbf{b} \times \textbf{c} \\
\textbf{c} \times \textbf{a} \\
\textbf{a} \times \textbf{b} \\
\end{bmatrix}
$ & 3D matrix inverse (a,b,c cols of M) \\

$\textbf{M}^{-1} = \frac{1}{\textbf{s} \cdot \textbf{v} + \textbf{t} \cdot \textbf{u}}
\begin{bmatrix}
\textbf{b} \times \textbf{v} + y\textbf{t} | -\textbf{b} \cdot \textbf{t} \\
\textbf{v} \times \textbf{a} - x\textbf{t} | \textbf{a} \cdot \textbf{t} \\
\textbf{d} \times \textbf{u} + w\textbf{s} | -\textbf{d} \cdot \textbf{s} \\
\textbf{u} \times \textbf{c} - z\textbf{s} | \textbf{c} \cdot \textbf{s}
\end{bmatrix}
$ & 4D Matrix inverse. a,b,c,d = 3D column vectors of M. \\

& x,y,z,w = last row of M, x = $a \times b$, t = $c \times d$,
u = $ya -xb$, v = $wc -zd$ \\

\end{tabulary} % Matrix formulas table
}

\input{transformations.tex}
\input{quaternions.tex}
\input{geometry.tex}
\input{grassman.tex}

\end{document}