# Latex sample document

This is what one could use to start writing his pdf in Latex. I also have an abstract section and an equation. A good template pool can be found here. An interesting link, Jacques Dafflon suggested, was this.

\documentclass{article}

\author{George Samaras}
\usepackage{graphicx}
\newcommand{\tab}{\hspace*{3em}}
\usepackage{fontspec}
\usepackage{amsmath}
\defaultfontfeatures{LetterSpace=1}
\setmainfont{Latin Modern Roman}
\begin{document}

\title{"findHalfspaces" time complexity}

\maketitle
\begin{abstract}
{\Large In this document, we show that the time complexity the function "findHalfspaces" needs to compute the halfspaces which define the polytope (given the vertices of the polytope, V) is: \begin{center}T = $\frac{V^3}{6} - \frac{V^2}{2} + \frac{V}{3}$\\\end{center}
}\end{abstract}\par
{\Large
The algorithm, in 3D case, picks each unique triplet of points - no reason
to try the same triplet more than once, as their order is irrelevant -,
construct the plane, and test the points to see if they are
on one side of the plane; if they are, then the plane is a halfspace
defining the convex hull of the points, thus the polytope.\\\\\par
}
{\Large
We utilize the linearity of the summation and we calculate three sub-summations.
$$\begin{split} \sum_{i = 0}^{V - 3} \frac{1}{2}i^2 = \\ &= \frac{1}{2}\frac{(V - 3)(V - 3 + 1) \ [ \ 2(V - 3) + 1 \ ]}{6} \\ &= \frac{1}{12} (V - 3) \ (V - 2) \ (2V - 5)\\ &= \frac{1}{12} ( V^2 - 2V - 3V + 6 ) \ (2V - 5)\\ &= \frac{1}{12} ( V^2 - 5V + 6 ) \ (2V - 5)\\ &= \frac{1}{12} ( 2V^3 - 5V^2 - 10V^2 + 25V + 12V - 30 )\\ &= \frac{1}{12} ( 2V^3 - 15V^2 + 37V - 30 )\\ &= \frac{1}{6}V^3 - \frac{15}{12}V^2 + \frac{37}{12}V - \frac{15}{6} \\\\\\ \end{split}$$
}\par
\end{document}