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\documentclass[aspectratio=169,hyperref={pdfencoding=auto,psdextra}]{beamer}
\usepackage[utf8]{inputenc}
\usepackage{CJKutf8}
\usepackage{ulem}
\usepackage{graphicx}
\usepackage{fancyvrb}
\usepackage{amsmath}
\usepackage{multicol}
\usetheme{Malmoe}
\usecolortheme{default}
\begin{CJK*}{UTF8}{gbsn}
\title{「初中数学都不会」}
\subtitle{——Computational Geometrics}
\author{Chris Xiong}
\date{2017-08-18}
\begin{document}
\frame{\titlepage}
\begin{frame}
\frametitle{「初中数学都不会」}
\framesubtitle{Outline}
\begin{itemize}
\item Basics
\begin{itemize}
\item Vectors
\item Distances
\item Area
\item Intersections
\item Transformations
\end{itemize}
\item Convex hull
\item Extra fun
\item Templates
\item 殴打宇宙智障
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Basics}
\framesubtitle{Vectors}
\begin{itemize}
\item Computational Gemoetrics in ICPC=\\
\hspace{1cm}40\% Junior high school maths+\\
\hspace{1cm}30\% Senior high school maths+\\
\hspace{1cm}20\% Brainstorming+\\
\hspace{1cm}10\% College maths
\item $\mathbf{u}$, $\mathbf{v}$, $\vec{u}$, $\vec{v}$
\item $\vec{u}+\vec{v}$,$\vec{u}-\vec{v}$,$s\vec{u}$,$\vec{u}\cdot\vec{v}$,$\vec{u}\times\vec{v}$
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Basics}
\framesubtitle{Distances}
\begin{itemize}
\item point 2 point
\item point 2 line
\pause
\item polygon 2 polygon???
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Basics}
\framesubtitle{Area}
\begin{itemize}
\item Circle???
\item Triangle?
\begin{itemize}
\item \sout{$ah/2$}
\item Heron's formula
\item Cross product
\end{itemize}
\item Polygon???
\begin{itemize}
\item Slicing
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Basics}
\framesubtitle{Intersections}
\begin{itemize}
\item point \& line
\item point \& segment
\item line \& line
\item line \& segment
\item segment \& segment
\item circle \& line
\item circle \& circle
\pause
\item \sout{convex} polygon \& point \pause
\begin{itemize}
\item \sout{randomizing} (polyhedron)
\item ray casting
\item winding number
\end{itemize}
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Basics}
\framesubtitle{Transformations}
\begin{itemize}
\item Scaling
\item Rotation
\item ${\displaystyle {\begin{bmatrix}s_x&0\\0&s_y\end{bmatrix}}}$
\item ${\displaystyle {{\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}}}}$
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Convex hull}
\framesubtitle{WTF???}
\begin{itemize}
\item Smallest convex polygon (set) that contains a set of points.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Convex hull}
\framesubtitle{Algorithms}
\begin{itemize}
\item Brute force: gift wrapping
\item Graham scan
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Convex hull}
\framesubtitle{Gift wrapping}
\begin{itemize}
\item The left most point must be in the resulting set
\item Starting from that point, select the next point such that
all points are \textbf{to the right} of the newly formed line.
\item Time complexity: $O(nh)$
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Convex hull}
\framesubtitle{Graham scan}
\begin{itemize}
\item Find an extreme point and sort the remaining point according
to polar angle.
\item Start with a stack with two items, the extreme point and the
first point in the sorted list.
\item ($P_0$ denotes the top element of the stack, $P_1$ denotes the
element under $P_0$) For each point left in the list ($T$), pop the
stack until $\overrightarrow{P_1P_0}$ and $\overrightarrow{P_0T}$ forms
\textbf{a left turn}. Then push $T$ onto the stack.
\item Time complexity: $O(n log n)$ (with sorting)
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Convex hull}
\framesubtitle{Example: NEERC 08 A}
\end{frame}
\begin{frame}
\frametitle{Extra fun}
\framesubtitle{Example: PetrSU SC 04 C}
Arc-arc intersection
\includegraphics[scale=0.5]{c1.png}
\end{frame}
\begin{frame}
\frametitle{Extra fun}
\framesubtitle{Example: PetrSU SC 04 C}
\includegraphics[scale=0.36]{c2.png}
\end{frame}
\begin{frame}
\frametitle{Extra fun}
\framesubtitle{Example: MultiUniversity Traning 20170815 H}
\end{frame}
\begin{frame}
\frametitle{Extra fun}
\framesubtitle{Example: Asia GuangzhouRC 14 H}
\end{frame}
\begin{frame}
\frametitle{Extra fun}
\framesubtitle{Example: "Angry birds"}
One of the few problems of computational geometrics in ICPC
that involves college maths.
\end{frame}
\begin{frame}
\frametitle{Extra fun}
\framesubtitle{CG. Combined with other algorithms}
\end{frame}
\begin{frame}
\frametitle{Extra fun}
\framesubtitle{Rotating calipers}
Solves:
\begin{itemize}
\item Max/min width of convex polygon
\item Max/min distance between two convex polygons
\item Antipodal points/Farthest pairs
\item Union/intersection of convex polygons
\item Min area/perimeter obb
\item etc.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Templates}
\framesubtitle{Critical for Computational Geometrics (maybe)}
\begin{itemize}
\item Basic operations
\item Basic intersections
\item Circle-line intersection
\item Circle-triangle intersection
\item Circle-circle intersection
\item 3D vector operations
\item Half-plane intersection
\item Cloest/Farthest Pair
\item etc.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{殴打宇宙智障}
\framesubtitle{光辉事迹+1}
\begin{itemize}
\item 建完图后把图清空然后跑网络流\pause
\item 熟悉你的模板\pause
\item 多校睡觉\pause
\item 队伍组成
\item 代码手？\pause
\item 违反中央八项规定\pause
\item 口技
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{殴打宇宙智障}
\framesubtitle{后事}
\begin{itemize}
\item 21日最后一次讲课
\item 8月31日计科退宿舍
\item 如有不同以宇宙智障为准
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{殴打宇宙智障}
\framesubtitle{没来得及讲的东西}
\begin{itemize}
\item 概率（解方程/dp）、高斯消元
\item 网络流
\item 好像没什么了？
\item Good luck to ya.
\end{itemize}
\end{frame}
\end{CJK*}
\end{document}
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