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<!doctype html>
<html lang="en">
<head>
<meta charset="utf-8">
<title>reveal.js - The HTML Presentation Framework</title>
<meta name="description" content="A framework for easily creating beautiful presentations using HTML">
<meta name="author" content="Hakim El Hattab">
<meta name="apple-mobile-web-app-capable" content="yes" />
<meta name="apple-mobile-web-app-status-bar-style" content="black-translucent" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, maximum-scale=1.0, user-scalable=no, minimal-ui">
<link rel="stylesheet" href="css/reveal.css">
<link rel="stylesheet" href="css/theme/serif.css" id="theme">
<!-- Code syntax highlighting -->
<link rel="stylesheet" href="lib/css/zenburn.css">
<link rel="stylesheet" href="css/style.css">
<link rel="stylesheet" type="text/css" href="//jsxgraph.uni-bayreuth.de/distrib/jsxgraph.css" />
<script type="text/javascript" src="//jsxgraph.uni-bayreuth.de/distrib/jsxgraphcore.js"></script>
<!-- Printing and PDF exports -->
<script>
var link = document.createElement( 'link' );
link.rel = 'stylesheet';
link.type = 'text/css';
link.href = window.location.search.match( /print-pdf/gi ) ? 'css/print/pdf.css' : 'css/print/paper.css';
document.getElementsByTagName( 'head' )[0].appendChild( link );
</script>
<!--[if lt IE 9]>
<script src="lib/js/html5shiv.js"></script>
<![endif]-->
</head>
<body>
<div class="reveal">
<!-- Any section element inside of this container is displayed as a slide -->
<div class="slides">
\(
\newcommand{\ord}[1]{\opord\left(#1\right)}
\newcommand{\abs}[1]{\lvert #1 \rvert}
\newcommand{\floor}[1]{\lfloor #1 \rfloor}
\newcommand{\ceil}[1]{\lceil #1 \rceil}
\newcommand{\opord}{\operatorname{\mathcal{O}}}
\newcommand{\fail}{\operatorname{\mathcal{F}}}
\newcommand{\flk}{\operatorname{\mathfrak{F}}}
\newcommand{\suf}{\operatorname{\sigma}}
\newcommand{\rank}{\operatorname{\mathcal{R}}}
\newcommand{\sa}{\operatorname{\mathcal{SA}}}
\newcommand{\hei}{\operatorname{\mathcal{H}}}
\newcommand{\edps}{\operatorname{\mathcal{E}}}
\newcommand{\mx}{\operatorname{\mathcal{M}}}
\newcommand{\argmax}{\operatorname{arg\,max}}
\newcommand{\cons}[1]{\left[ \: #1 \: \right]}
\newcommand{\str}[1]{\texttt{"#1"}}
\newcommand{\vec}[1]{\overrightarrow{#1}}
\)
<section>
<h1> <small>Computational Geometry</small> <br> 計算幾何 </h1>
<h3> step5 </h3>
<h3> 2017/02/09 </h3>
Slides:
https://kevin00036.github.io/ioi-lecture-geometry/#/3
</section>
<!--<section>-->
<!--<section>-->
<!--<h3> WHO AMD I?</h3>-->
<!--<ul>-->
<!--<li class="fragment"> 黃凱祺 / STEP5 / kevin00050 / 衄 </li>-->
<!--<li class="fragment"> <b><font color="blue">bcw</font></b>0x1bd2 (<font color="blue">$\beta$</font>obogei / kai-<font color="blue">$\varsigma$</font>hi / hanhan<font color="blue">$\omega$</font>) </li>-->
<!--<li class="fragment"> $\alpha \beta \gamma \delta \epsilon \ \zeta \eta \theta \iota \ \kappa \lambda \mu \nu \xi \omicron \ \pi \rho \sigma \tau \upsilon \ \phi \chi \psi \omega$</li>-->
<!--</ul>-->
<!--</section>-->
<!--</section>-->
<section>
<section>
<h3> 計算幾何能吃嗎？</h3>
<ul>
<li class="fragment"> 不能 </li>
<li class="fragment"> 用程式來計算一些幾何問題 </li>
<li class="fragment"> IOI 通常不會出 </li>
<li class="fragment"> ACM 通常一定會出 </li>
<li class="fragment"> code 通常很難寫 </li>
<li class="fragment"> 200 行是正常現象(?) </li>
<li class="fragment"> 各種機八 Case </li>
<li class="fragment"> <font color="red">WA</font> 到你不想寫... </li>
</ul>
</section>
</section>
<section>
<section>
<h3> 解析幾何 </h3>
<p> 什麼？解析度很高的幾何嗎？</p>
</section>
<section>
<h3> 解析幾何 </h3>
<ul>
<li class="fragment"> 純幾何：長度、角度、形狀、全等、相似...</li>
<li class="fragment"> <b>解析</b>幾何：座標、向量、內積、外積、行列式...難算到要人命</li>
</ul>
<p class="fragment"> 電腦上只能做四則運算... </p>
<p class="fragment"> 還是用解析幾何吧!</p>
</section>
</section>
<section>
<section>
<h3> 點 </h3>
<p> 這輕鬆! 只需要兩個數字$(x, y)$</p>
<div id="box" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box', {boundingbox: [-5, 5, 10, -5], axis:true});
board.create('point', [5,3], {name:'(5, 3)', strokecolor:'red'});
}
</script>
</section>
<section>
<h3> 向量 </h3>
<p> 同樣的$(x, y)$也可以代表一個向量</p>
<p> 兩者有什麼關聯呢? </p>
<p class="fragment"><b>位置向量</b>：從原點指到$(x, y)$的向量</p>
<div id="box2" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box2', {boundingbox: [-5, 5, 10, -5], axis:true});
var pt = board.create('point', [5,3], {name:'(5, 3)', strokecolor:'red'});
var li3 = board.create('line', [[0,0],pt], {straightFirst:false, straightLast:false, strokeWidth:2, lastArrow:true});
}
</script>
</section>
<section>
<h3> 向量運算 </h3>
<ul>
<li class="fragment"> 加法 </li>
<li class="fragment"> 減法 </li>
<li class="fragment"> 內積 : $(x_1, y_1) \cdot (x_2, y_2) = x_1y_1+x_2y_2$ </li>
<li class="fragment"> 外積 : $(x_1, y_1) \times (x_2, y_2) = x_1y_2 - x_2y_1$ </li>
<li class="fragment"> 除法? A__A</li>
</ul>
</section>
<section>
<h3> 向量運算 </h3>
<pre><code data-trim class="cpp">
#define F first
#define S second
typedef pair<double, double> pdd;
pdd operator+ (const pdd &a, const pdd &b) { return pdd(a.F+b.F, a.S+b.S);}
pdd operator- (const pdd &a, const pdd &b) { return pdd(a.F-b.F, a.S-b.S);}
pdd operator* (const pdd &a, const double &b) { return pdd(a.F*b, a.S*b);}
pdd operator/ (const pdd &a, const double &b) { return pdd(a.F/b, a.S/b);}
double dot(const pdd &a, const pdd &b) { return a.F*b.F + a.S*b.S;}
double cross(const pdd &a, const pdd &b) { return a.F*b.S - a.S*b.F;}
double abs2(const pdd &a) { return dot(a, a);}
double abs(const pdd &a) { return sqrt(dot(a, a));}
</code></pre>
</section>
</section>
<section>
<section>
<h3> 直線 </h3>
<p> 點斜式!</p>
$$ y = m x + b $$
<p> $m$ : 斜率 </p>
<p> 垂直線? G___G</p>
</section>
<section>
<h3> 直線 </h3>
<p> 點向式</p>
$$ (x, y) = (x_0, y_0) + t \cdot (d_x, d_y), \ t \in \mathbb{R}$$
<p> $\vec{P_0} = (x_0, y_0)$ : 直線上任意一點 </p>
<p> $\vec{d} = (d_x, d_y)$ : <b>方向</b>向量 </p>
<div id="box3" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box3', {boundingbox: [-5, 5, 10, -5], axis:true});
var A = [1, 1];
var B = [8, 3];
board.create('point', A, {name:'P0', strokecolor:'red'});
board.create('line',[A,B], {strokeColor:'#00ff00',strokeWidth:2});
board.create('line', [A,B], {straightFirst:false, straightLast:false, strokeWidth:2, lastArrow:true, withLabel:true, name:'d', label:{position:'top', display: 'internal'}});
}
</script>
</section>
<section>
<h3> 線段 </h3>
<p> 線段 $\overline{P_1P_2}$ </p>
$$ (x, y) = P_1 + t \cdot (P_2 - P_1), \ \color{red}{0 \le t \le 1}$$
<p> $\vec{d} = P_2 - P_1$ : <b>方向</b>向量 </p>
<div id="box4" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box4', {boundingbox: [-5, 5, 10, -5], axis:true});
var A = [1, 1];
var B = [8, 3];
board.create('point', A, {name:'P1', strokecolor:'red'});
board.create('point', B, {name:'P2', strokecolor:'red'});
board.create('segment', [A,B], {strokeWidth:2, lastArrow:true, withLabel:true, name:'d', label:{position:'top', display: 'internal'}});
}
</script>
</section>
<section>
<h3> 多邊形 </h3>
<p> $N$ 個頂點，兩兩連線繞成一圈</p>
<p> $P_1, P_2, P_3, \cdots, P_N$</p>
<p> $\overline{P_1P_2}, \overline{P_2P_3}, \cdots, \overline{P_NP_1}$ </p>
<div id="box5" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box5', {boundingbox: [-5, 5, 10, -5]});
var poly = [[0, 4], [3, 2], [3, -1], [2, 0], [-2, -2], [-3, 1]];
board.create('polygon', poly);
for(i=0; i<6; i++)
{
board.create('point', poly[i], {name:'P'+(6-i), strokecolor:'red'});
}
}
</script>
</section>
</section>
<section>
<section>
<h3> 有向直線 </h3>
$$ (x, y) = P_0 + t \cdot \color{green}{\vec{d}}, \ t \in \mathbb{R}$$
<div id="box6" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box6', {boundingbox: [-5, 5, 10, -5]});
var A = [1, 1];
var B = [8, 3];
board.create('point', A, {name:'P0', strokecolor:'red'});
board.create('line',[A,B], {strokeColor:'#00ff00',strokeWidth:2});
board.create('line', [A,B], {straightFirst:false, straightLast:false, strokeWidth:2, lastArrow:true, withLabel:true, name:'d', label:{position:'top', display: 'internal'}});
}
</script>
</section>
<section>
<h3> 有向直線 </h3>
<p> 直線<font color="blue">左</font>邊(<font color="blue">正</font>方向) :
$\color{green}{\vec{d}} \times \vec{P_0P} \color{blue}{>} 0$</p>
<p> 直線<font color="red">右</font>邊(<font color="red">負</font>方向) :
$\color{green}{\vec{d}} \times \vec{P_0P} \color{red}{<} 0$</p>
<div id="box7" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box7', {boundingbox: [-5, 5, 10, -5]});
var A = [1, 1];
var B = [8, 3];
var C = [1, 4];
var D = [6, -2];
board.create('point', A, {name:'P0', strokecolor:'red'});
board.create('point', C, {name:'P', strokecolor:'red'});
board.create('point', D, {name:'P', strokecolor:'red'});
board.create('line',[A,B], {strokeColor:'#00ff00',strokeWidth:2});
var l = board.create('segment', [A,B], {lastArrow:true, withLabel:true, name:'d', label:{position:'top', display: 'internal'}, strokeColor:'green'});
var l2 = board.create('segment', [A,C], {lastArrow:true});
var l3 = board.create('segment', [A,D], {lastArrow:true, strokeColor:'red'});
ineq = board.create('inequality', [l]);
ineq = board.create('inequality', [l], {inverse: true, fillColor: 'blue'});
}
</script>
</section>
<section>
<h3> 有向角度 </h3>
<p> 與轉的方向有關!</p>
<p> 逆時針(往<font color="blue">左</font>邊轉) : <font color="blue">正</font>角度 </p>
<p> 順時針(往<font color="red">右</font>邊轉) : <font color="red">負</font>角度 </p>
$$ \vec{BA} \times \vec{BC} = |BA||BC| \sin(\angle ABC)$$
$$ \vec{BA} \cdot \vec{BC} = |BA||BC| \cos(\angle ABC)$$
<div id="box8" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box8', {boundingbox: [-5, 5, 10, -5]});
var A = [1, 1];
var B = [8, 3];
var C = [1, 4];
var D = [6, -2];
board.create('point', A, {name:'B', strokecolor:'red'});
board.create('point', B, {name:'A', strokecolor:'red'});
board.create('point', C, {name:'C', strokecolor:'red'});
board.create('point', D, {name:'C', strokecolor:'red'});
var l = board.create('segment', [A,B], {lastArrow:true, strokeColor:'green'});
var l2 = board.create('segment', [A,C], {lastArrow:true});
var l3 = board.create('segment', [A,D], {lastArrow:true, strokeColor:'red'});
var a1 = board.create('angle', [B,A,C], {strokeColor:'blue', fillColor:'blue', radius:1, withLabel:false});
var a2 = board.create('angle', [D,A,B], {strokeColor:'red', fillColor:'red', radius:1, withLabel:false});
}
</script>
</section>
<section>
<h3> 判斷方向 </h3>
$$\mathrm{ori}(P_1, P_2, P_3) =
\mathrm{sign}(\vec{P_1P_2} \times \vec{P_1P_3})$$
<p>
<ul>
<li class="fragment"> $1$ : $P_3$ 在 $\vec{P_1P_2}$ <font color="blue">左</font>邊</li>
<li class="fragment"> $-1$ : $P_3$ 在 $\vec{P_1P_2}$ <font color="red">右</font>邊</li>
<li class="fragment"> $0$ : $P_1, P_2, P_3$ 三點共線</li>
</ul>
</p>
<div id="box9" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box9', {boundingbox: [-5, 5, 10, -5]});
var A = [1, 1];
var B = [8, 3];
var C = [1, 4];
var D = [6, -2];
board.create('point', A, {name:'P1', strokecolor:'red'});
board.create('point', B, {name:'P2', strokecolor:'red'});
board.create('point', C, {name:'P3', strokecolor:'red'});
board.create('point', D, {name:'P3', strokecolor:'red'});
var l = board.create('segment', [A,B], {lastArrow:true, strokeColor:'green'});
var l2 = board.create('segment', [A,C], {lastArrow:true});
var l3 = board.create('segment', [A,D], {lastArrow:true, strokeColor:'red'});
var a1 = board.create('angle', [B,A,C], {strokeColor:'blue', fillColor:'blue', radius:1, withLabel:false});
var a2 = board.create('angle', [D,A,B], {strokeColor:'red', fillColor:'red', radius:1, withLabel:false});
}
</script>
</section>
<section>
<h3> 有向面積 </h3>
<p> 三角形的<b>有向</b>面積</p>
$$ \Delta ABC = \frac{1}{2} \vec{BA} \times \vec{BC}$$
<div id="box10" class="jxgbox" style="width:300px; height:200px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box10', {boundingbox: [-5, 5, 10, -5]});
var A = [1, -3];
var B = [8, 3];
var C = [1, 4];
board.create('point', A, {name:'B', strokecolor:'red'});
board.create('point', B, {name:'A', strokecolor:'red'});
board.create('point', C, {name:'C', strokecolor:'red'});
var l = board.create('segment', [A,B], {lastArrow:true});
var l2 = board.create('segment', [A,C], {lastArrow:true});
var l3 = board.create('segment', [B,C], {});
var poly = board.create('polygon', [A,B,C], {fillColor:'blue'});
var t = board.create('text', [3, 3, '+'], {fontSize: 36});
}
</script>
</section>
<section>
<h3> 有向面積 </h3>
<p> <b>凸多邊形</b>的有向面積</p>
<p> 切割成$N$個三角形!</p>
$$ \mathrm{Area}(P_1P_2\cdots P_N) = \Delta P_1AP_2 + \Delta P_2AP_3
+ \cdots + \Delta P_NAP_1$$
<div id="box11" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box11', {boundingbox: [-5, 5, 5, -5]});
var A = [0, 0];
var poly = [[3, 0], [2, 3], [-1, 3], [-3, -1], [0, -2]];
board.create('point', A, {name:'A'});
board.create('polygon', poly);
for(i=0; i<5; i++)
{
board.create('point', poly[i], {name:'P'+(i+1)});
board.create('segment', [poly[i],poly[(i+1)%5]], {lastArrow:true});
board.create('segment', [A, poly[i]], {lastArrow:true});
}
}
</script>
</section>
<section>
<h3> 有向面積 </h3>
<p> <b>凹</b>多邊形的有向面積</p>
<p> 照切!</p>
<p> <b>有向</b>面積的好處 : 正負會抵消掉</p>
<div id="box12" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box12', {boundingbox: [-5, 5, 5, -5]});
var A = [0, 0];
var poly = [[3, 0], [2, 3], [-1, 3], [-3, -1], [0, -2]];
board.create('point', A, {name:'A'});
board.create('polygon', poly);
for(i=0; i<5; i++)
{
board.create('point', poly[i], {name:'P'+(i+1)});
board.create('segment', [poly[i],poly[(i+1)%5]], {lastArrow:true});
board.create('segment', [A, poly[i]], {lastArrow:true});
}
}
</script>
</section>
<section>
<h3> 有向面積 </h3>
<p> 任意多邊形的有向面積</p>
<p> 乾脆把中心點$A$選成原點，那麼$\vec{AP_i}$就變成$P_i$位置向量</p>
$$ \mathrm{Area}(P_1P_2\cdots P_N) = \sum_{i=1}^{N} \Delta P_iOP_{i+1}
= \frac{1}{2} \sum_{i=1}^{N} P_i \times P_{i+1}$$
<p>(定義$P_{N+1} = P_1$)</p>
<p>時間複雜度 : $O(N)$</p>
</section>
<section>
<p> 真的嗎...^_^;;</p>
<div id="box13" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box13', {boundingbox: [-5, 5, 5, -5]});
var poly2 = [];
for(i=0; i<20; i++)
{
var x = Math.random()*9-4.5;
var y = Math.random()*9-4.5;
poly2.push([x, y]);
}
board.create('polygon', poly2, {fillColor: 'yellow'});
}
</script>
</section>
<section>
<h3> 簡單多邊形 </h3>
<ul>
<li class="fragment"> 邊不相交 </li>
</ul>
<br>
<div id="box14" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box14', {boundingbox: [-5, 5, 5, -5]});
var poly = [[3, 0], [0, 3], [1, 1], [-1, 3], [-1, -2], [0, 1], [1, -4], [1, 0]];
board.create('polygon', poly);
}
</script>
</section>
<section>
<h3> 有向面積 </h3>
<p> <b>簡單多邊形</b>的有向面積</p>
$$ \mathrm{Area}(P_1P_2\cdots P_N) = \sum_{i=1}^{N} \Delta P_iOP_{i+1}
= \frac{1}{2} \sum_{i=1}^{N} P_i \times P_{i+1}$$
<p>(定義$P_{N+1} = P_1$)</p>
<p>時間複雜度 : $O(N)$</p>
</section>
</section>
<section>
<section>
<h3> 線段相交 </h3>
<div class="ques">
給定兩條線段 $\overline{P_1P_2}$ 和 $\overline{P_3P_4}$，請問他們有交點嗎？
</div>
<div id="box15" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box15', {boundingbox: [-5, 5, 5, -5]});
var P1 = board.create('point', [-2, -1], {name: 'P1'});
var P2 = board.create('point', [2, 3], {name: 'P2'});
var P3 = board.create('point', [-3, 2], {name: 'P3'});
var P4 = board.create('point', [2, -2], {name: 'P4'});
board.create('segment', [P1, P2]);
board.create('segment', [P3, P4]);
}
</script>
</section>
<section>
<h3> 直線相交 </h3>
先考慮這個問題:
<div class="ques">
給定兩條<b>直線</b> $\overline{P_1P_2}$ 和 $\overline{P_3P_4}$，請問他們有交點嗎？
</div>
<div id="box16" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box16', {boundingbox: [-5, 5, 5, -5]});
var P1 = board.create('point', [-2, -1], {name: 'P1'});
var P2 = board.create('point', [2, 3], {name: 'P2'});
var P3 = board.create('point', [-3, 2], {name: 'P3'});
var P4 = board.create('point', [2, -2], {name: 'P4'});
board.create('line', [P1, P2], {strokeColor:'lightgreen'});
board.create('line', [P3, P4], {strokeColor:'lightgreen'});
board.create('segment', [P1, P2]);
board.create('segment', [P3, P4]);
}
</script>
</section>
<section>
<h3> 直線相交 </h3>
<p> 好像還蠻簡單的... </p>
<ul>
<li class="fragment"> 兩直線不平行 : 肯定有交點! </li>
<li class="fragment"> 兩直線平行 : 不相交 or 完全重合</li>
</ul>
</section>
<section>
<h3> 直線求交點 </h3>
<ul style="line-height: 150%">
<li class="fragment"> $ \vec{d_1} = \vec{P_1P_2}, \vec{d_2} = \vec{P_3P_4}$ </li>
<li class="fragment"> 交點 $ x = P_1 + t\vec{d_1} = P_3 + s\vec{d_2}$ </li>
<li class="fragment"> $ t\vec{d_1} = \vec{P_1P_3} + s\vec{d_2}$ </li>
<li class="fragment"> $ t\vec{d_1} \times \vec{d_2} = \vec{P_1P_3} \times \vec{d_2}$ </li>
<li class="fragment"> $\displaystyle t = \frac{\vec{P_1P_3} \times \vec{d_2}}{\vec{d_1} \times \vec{d_2}}$ </li>
<br>
<li class="fragment"> $\vec{d_1} \times \vec{d_2} \neq 0$ (為什麼?)</li>
</ul>
<div id="box16.5" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box16.5', {boundingbox: [-5, 5, 5, -5]});
var P1 = board.create('point', [-2, -1], {name: 'P1'});
var P2 = board.create('point', [2, 3], {name: 'P2'});
var P3 = board.create('point', [-3, 2], {name: 'P3'});
var P4 = board.create('point', [2, -2], {name: 'P4'});
var ln1 = board.create('segment', [P1, P2], {lastArrow:true, withLabel:true, label:{position:'top'}, name:'d1'});
var ln2 = board.create('segment', [P3, P4], {lastArrow:true, withLabel:true, label:{position:'top'}, name:'d2'});
var Pp = board.create('intersection', [ln1, ln2], {name:'x', strokeColor:'green'});
}
</script>
</section>
<section>
<h3> 線段求交點 </h3>
<ul>
<li class="fragment"> 類似的方法可以求出 $s$ </li>
<li class="fragment"> $0 \le t \le 1$ and $0 \le s \le 1$ : 兩<b>線段</b>有交點!</li>
</ul>
</section>
<section>
<h3> 另一種方法 </h3>
<ul>
<li class="fragment"> 如果 $\overline{P_1P_2}$ 和 $\overline{P_3P_4}$ 有交點的話 ... </li>
<li class="fragment"> $P_3$ 和 $P_4$ 在 $\vec{P_1P_2}$ 的<b>異側</b> </li>
<li class="fragment"> $P_1$ 和 $P_2$ 在 $\vec{P_3P_4}$ 的<b>異側</b> </li>
<li class="fragment"> $\mathrm{ori}(P_1, P_2, P_3) \cdot \mathrm{ori}(P_1, P_2, P_4) < 0$ </li>
<li class="fragment"> $\mathrm{ori}(P_3, P_4, P_1) \cdot \mathrm{ori}(P_3, P_4, P_2) < 0$ </li>
</ul>
<div id="box17" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box17', {boundingbox: [-5, 5, 5, -5]});
var P1 = board.create('point', [-2, -1], {name: 'P1'});
var P2 = board.create('point', [2, 3], {name: 'P2'});
var P3 = board.create('point', [-3, 2], {name: 'P3'});
var P4 = board.create('point', [2, -2], {name: 'P4'});
board.create('segment', [P1, P2]);
board.create('segment', [P3, P4]);
}
</script>
</section>
<section>
<h3> 慘了了 </h3>
<div id="box18" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box18', {boundingbox: [-5, 5, 5, -5]});
var P1 = board.create('point', [-2, -1], {name: 'P1'});
var P2 = board.create('point', [2, 3], {name: 'P2'});
var P3 = board.create('point', [-1, 0], {name: 'P3'});
var P4 = board.create('point', [2, -2], {name: 'P4'});
board.create('segment', [P1, P2]);
board.create('segment', [P3, P4]);
}
</script>
</section>
<section>
<h3> 線段相交 </h3>
<ul>
<li class="fragment"> 兩線段無交點 $\Leftrightarrow P_3, P_4$ 在 $\vec{P_1P_2}$ 同側
<font color="red">或</font> $P_1, P_2$ 在 $\vec{P_3P_4}$ 同側</li>
<li class="fragment"> 萬一兩線段平行呢?</li>
<li class="fragment"> 延伸的直線不重合 : 無交點</li>
<li class="fragment"> 延伸的直線重合 : 一維的判斷交點</li>
</ul>
<div id="box19" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box19', {boundingbox: [-5, 5, 5, -5]});
var P1 = board.create('point', [-2, 0], {name: 'P1'});
var P2 = board.create('point', [0, 0], {name: 'P2'});
var P3 = board.create('point', [1, 0], {name: 'P3'});
var P4 = board.create('point', [4, 0], {name: 'P4'});
board.create('segment', [P1, P2]);
board.create('segment', [P3, P4]);
}
</script>
</section>
</section>
<section>
<section>
<h3> 多邊形內外判斷</h3>
<div class="ques">
給定一個<b>簡單多邊形</b>和一個點 $A$，請問 $A$ 在該多邊形的內部還是外部？
</div>
<div id="box20" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box20', {boundingbox: [-5, 5, 5, -5]});
var A = [-0.5, 2];
var poly = [[3, 0], [0, 3], [1, 1], [-1, 3], [-1, -2], [0, 1], [1, -4], [1, 0]];
board.create('point', A, {name:'A', strokeColor:'green'});
board.create('polygon', poly);
}
</script>
</section>
<section>
<h3> 多邊形內外判斷</h3>
<ul>
<li class="fragment"> 想辦法往外走，不能越過多邊形邊界，走得到外面就是外部 </li>
<li class="fragment"> 時間複雜度 : $O(?)$ </li>
</ul>
</section>
<section>
<h3> 多邊形內外判斷</h3>
<ul>
<li class="fragment"> 開外掛穿牆(?) </li>
<li class="fragment"> 往一個方向一直走 </li>
<li class="fragment"> 每穿過一次牆，內部變外部，外部變內部 </li>
<li class="fragment"> 走到無限遠處... 這下總是外部了吧! </li>
<li class="fragment"> 計算總共穿幾次牆 </li>
<li class="fragment"> 偶數次 : 原本的點在<b>外部</b> </li>
<li class="fragment"> 奇數次 : 原本的點在<b>內部</b> </li>
</ul>
<div id="box21" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box21', {boundingbox: [-5, 5, 5, -5]});
var A = [-0.5, 2];
var B = [6, 3];
var poly = [[3, 0], [0, 3], [1, 1], [-1, 3], [-1, -2], [0, 1], [1, -4], [1, 0]];
board.create('point', A, {name:'A', strokeColor:'green'});
board.create('polygon', poly);
board.create('segment', [A,B]);
}
</script>
</section>
<section>
<h3> 多邊形內外判斷</h3>
<ul>
<li class="fragment"> 要怎麼計算穿牆次數呢？ </li>
<li class="fragment"> 線段相交! </li>
<li class="fragment"> 把多邊形的 $N$ 條邊都檢查一次</li>
<li class="fragment"> 時間複雜度 : $O(N)$</li>
</ul>
</section>
<section>
<h3> 慘了了</h3>
<ul>
<li class="fragment"> 萬一剛好交到邊、頂點上呢？ </li>
<li class="fragment"> 可能會重複算到次數 </li>
<li class="fragment"> 隨機取一個方向射線，交到邊、頂點的機率很低 (為什麼?) </li>
<li class="fragment"> 座標都是整數(範圍 $W$ )的話，外面的點取 $(W, W+1)$，肯定不會交到邊點 </li>
</ul>
<div id="box22" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box22', {boundingbox: [-5, 5, 5, -5]});
var A = [-0.5, 2];
var B = [2, 7];
var poly = [[3, 0], [0, 3], [1, 1], [-1, 3], [-1, -2], [0, 1], [1, -4], [1, 0]];
board.create('point', A, {name:'A', strokeColor:'green'});
board.create('polygon', poly);
board.create('segment', [A,B]);
}
</script>
</section>
<section>
<h3> 多邊形內外判斷</h3>
<ul>
<li class="fragment"> 如果是<b>凸</b>多邊形呢？ </li>
<li class="fragment"> 中間找一個點 $A$，連線到每個頂點 </li>
<li class="fragment"> 把一圈分成 $N$ 個角度區間 </li>
<li class="fragment"> 查詢時，先二分搜出在哪個區間，再檢查是否在三角形內</li>
<li class="fragment"> 時間複雜度 : 預處理 $O(N)$，查詢$O(\log N)$</li>
</ul>
<div id="box23" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box23', {boundingbox: [-5, 5, 5, -5]});
var A = [0, 0];
var poly = [[3, 0], [2, 3], [-1, 3], [-3, -1], [0, -2]];
board.create('point', A, {name:'A'});
board.create('point', [2,1], {name:'P'});
board.create('polygon', poly);
for(i=0; i<5; i++)
{
board.create('point', poly[i], {name:'P'+(i+1)});
board.create('segment', [A, poly[i]], {lastArrow:true});
}
board.create('segment', [A, 'P'], {lastArrow:true, strokeColor:'yellow'});
board.create('polygon', [A,poly[0],poly[1]], {fillColor:'blue'});
}
</script>
</section>
</section>
<section>
<section>
<h3> 凸包 </h3>
<div class="ques">
給定平面上一個點集，請找出包含這個點集的最小面積<b>凸</b>多邊形。
</div>
<div id="box24" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box24', {boundingbox: [-5, 5, 5, -5]});
var poly = [[3, 0], [1, 0], [1, 2], [-1, 2], [-1, 0], [-3, 0], [-3, -2], [3, -2]];
var poly2 = [[3, 0], [1, 2], [-1, 2], [-3, 0], [-3, -2], [3, -2]];
board.create('polygon', poly);
board.create('polygon', poly2, {fillColor: 'yellow'});
}
</script>
</section>
<section>
<h3> <b>"凸"</b>包 </h3>
<p class="fragment">字不要寫錯啊!</p>
<p class="fragment">不然就會變這樣...</p>
<img src="sanp15.png" class="fragment"/>
</section>
<section>
<h3> "凸"多邊形? </h3>
<ul>
<li class="fragment"> 內角皆小於 $180^\circ$ </li>
<li class="fragment"> 兩點連線包含在多邊形內 </li>
</ul>
<div id="box25" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box25', {boundingbox: [-5, 5, 5, -5]});
var poly2 = [[3, 0], [1, 2], [-1, 2], [-3, 0], [-3, -2], [3, -2]];
board.create('polygon', poly2, {fillColor: 'yellow'});
var A = board.create('point', [-2, -1]);
var B = board.create('point', [-1, 1]);
var l = board.create('segment', [A, B]);
}
</script>
</section>
<section>
<h3> 拿條橡皮筋綁起來！ </h3>
<div style="float:right">
<ul>
<li class="fragment"> 先找一個必定在凸包上的點 <br>(最左下角的點一定是！)</li>
<li class="fragment"> 每次找最靠右邊的，加入凸包</li>
<li class="fragment"> 時間複雜度 : $O(N^2)$</li>
</ul>
</div>
<div id="box26" style="width:300px; height:300px; float:left">
<img src="https://upload.wikimedia.org/wikipedia/commons/b/bc/ConvexHull.png">
</div>
</section>
<section>
<h3> Monotone Chain </h3>
<ul>
<li class="fragment"> 每次都重新排序，太辛苦了!</li>
<li class="fragment"> 一次從左到右排序好，一個一個點慢慢加</li>
<li class="fragment"> 每多加一個點時，檢查看會不會影響凸包</li>
</ul>
</section>
<section>
<h3> Monotone Chain </h3>
<p> 檢查新增的點與目前凸包最後2個點的夾角:</p>
<ul>
<li class="fragment"> $< 180^\circ$ : 加入凸包</li>
<li class="fragment"> $\ge 180^\circ$ : 中間那個點刪掉</li>
<li class="fragment"> 用外積判斷夾角</li>
<li class="fragment"> 凸包根本沒有2個點? 那直接加入</li>
</ul>
<div id="box27" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box27', {boundingbox: [-5, 5, 5, -5]});
var A = board.create('point', [-4, 1], {name:'A'});
var B = board.create('point', [-3, 0], {name:'B'});
var C = board.create('point', [-1, 1], {name:'C'});
board.create('segment', [A,B]);
board.create('segment', [B,C], {dash:true});
var D = board.create('point', [0, -1], {name:'A'});
var E = board.create('point', [1, 0], {name:'B'});
var F = board.create('point', [3, -2], {name:'C'});
board.create('segment', [D,E]);
board.create('segment', [E,F], {dash:true});
board.create('line', [[-0.5,-6],[-0.5,6]], {dash:true, strokeColor:'lightgreen'});
}
</script>
</section>
<section>
<h3> Monotone Chain </h3>
<img src="http://www.csie.ntnu.edu.tw/~u91029/Andrew'sMonotoneChain1.png">
</section>
<section>
<h3> Monotone Chain </h3>
<ul>
<li class="fragment"> 用 stack 維護目前凸包 </li>
<li class="fragment"> 每個點會被加入 1 次</li>
<li class="fragment"> 每個點<b>最多</b>被刪除 1 次</li>
<li class="fragment"> 時間複雜度 : $O(N \log N) + O(N)$</li>
<li class="fragment"> 這樣找到的是<b>下</b>凸包</li>
<li class="fragment"> 從右到左排序再做一次，接起來才是完整的凸包</li>
</ul>
</section>
<section>
<h3> 加權平均 </h3>
<ul style:"float:right">
<li class="fragment"> 兩個點 $P_1, P_2$ 的 <b>加權平均</b> 長怎樣？</li>
<li class="fragment"> $P = \color{red}t \cdot P_1 + \color{red}{(1-t)} \cdot P_2, \color{red}{0 \le t \le 1}$</li>
<li class="fragment"> 落在線段 $P_1 P_2$ ！</li>
<li class="fragment"> 三個點 $P_1, P_2$ 的 <b>加權平均</b> 長怎樣？</li>
<li class="fragment"> $P = \color{green}{\alpha} \cdot P_1 + \color{blue}{\beta} \cdot P_2 + \color{purple}{\gamma}\cdot P_3$<br>
$\color{green}{\alpha}, \color{blue}{\beta}, \color{purple}{\gamma} \ge 0, \color{green}{\alpha} + \color{blue}{\beta} + \color{purple}{\gamma} = 1$</li>
<li class="fragment"> 落在三角形 $\Delta P_1 P_2 P_3$ ！</li>
</ul>
<div id="box101" class="jxgbox" style="width:300px; height:300px; float:left"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box101', {boundingbox: [-5, 5, 5, -5]});
var P1 = board.create('point', [-2, 3], {name: 'P1'});
var P2 = board.create('point', [2, 3], {name: 'P2'});
var P3 = board.create('point', [-2, 0], {name: 'P1'});
var P4 = board.create('point', [2, 0], {name: 'P2'});
var P5 = board.create('point', [1, -3], {name: 'P3'});
board.create('segment', [P1, P2]);
board.create('polygon', [P3, P4, P5]);
}
</script>
</section>
<section>
<h3> 加權平均 </h3>
<ul style:"float:right">
<li class="fragment"> $N$ 個點 $P_1, P_2$ 的 <b>加權平均</b> 長怎樣？</li>
<li class="fragment"> $P = \sum_{i=1}^N \color{green}{\alpha_i} \cdot P_i$ <br>
$\color{green}{\alpha_i} \ge 0, \sum_{i=1}^N \color{green}{\alpha_i} = 1$</li>
<li class="fragment"> 落在 $P_1 \sim P_N$ 的<b>凸包</b> ！</li>
<li class="fragment"> 事實上凸包有一個等價定義：<br>$N$ 個點的所有加權平均形成的集合</li>
<li class="fragment"> 這種加權平均又稱為 <br> <b>凸組合 (Convex combination)</b></li>
</ul>
<div id="box102" class="jxgbox" style="width:300px; height:300px; float:left"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box102', {boundingbox: [-5, 5, 5, -5]});
var P1 = board.create('point', [-2, 3], {name:'P1'});
var P2 = board.create('point', [4, 0], {name:'P2'});
var P3 = board.create('point', [3, -1], {name:'P3'});
var P4 = board.create('point', [-1, 4], {name:'P4'});
var P5 = board.create('point', [1, 2], {name:'P5'});
var P6 = board.create('point', [1, 1], {name:'P6'});
var P7 = board.create('point', [-2, -2], {name:'P7'});
board.create('polygon', [P1, P7, P3, P2, P4]);
}
</script>
</section>
<section>
<h3> 另一種定義 </h3>
<ul style:"float:right">
<li class="fragment"> 如此一來可以定義更高維度的凸包！<br>
$P = \sum_{i=1}^N \color{green}{\alpha_i} \cdot P_i$ <br>
$\color{green}{\alpha_i} \ge 0, \sum_{i=1}^N \color{green}{\alpha_i} = 1$</li>
<li class="fragment"> 回過頭來看剛才說過的<b>凸多邊形</b>條件:<br>
<i>任兩點連線段，還在多邊形內部</i><br>
可以推廣成：<br><b>任取一些點的凸組合(加權平均)都還在內部</b>
<br> 或者說 <br>
<b>凸包還是自己!!</b>
</li>
<li class="fragment">凸多邊形的很多好性質可以從這裡推導！</li>
</ul>
<div id="box103" class="jxgbox" style="width:300px; height:300px; float:left">
<img src="https://i.stack.imgur.com/PFkhV.png">
</div>
</section>
</section>
<section>
<section>
<h3> 掃描線 </h3>
<p>拿一條垂直線，從左掃到右</p>
<p class="fragment"> 實作上就只是把點照 $x$ 座標排序 </p>
<div id="box29" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box29', {boundingbox: [-5, 5, 5, -5]});
var poly2 = [];
for(i=0; i<20; i++)
{
var x = Math.random()*9-4.5;
var y = Math.random()*9-4.5;
poly2.push([x, y]);
board.create('point', poly2[i], {withLabel:false});
}
var P = board.create('point', [-5,-6]);
var Q = board.create('point', [-5,6]);
var li = board.create('line', [P, Q]);
var f = function(){
P.moveTo([5, -6], 2000);
Q.moveTo([5, 6], 2000);
P.moveTo([-5, -6], 1);
Q.moveTo([-5, 6], 1);
setTimeout(f, 3000);
};
f();
}
</script>
</section>
<section>
<h3> 矩形覆蓋次數 </h3>
<div class="prob">
給定平面上 $N$ 個矩形和 $M$ 個點，問每個點被包含在<b>幾個</b>矩形中？
</div>
<div id="box30" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box30', {boundingbox: [-5, 5, 5, -5]});
var poly2 = [];
for(i=0; i<10; i++)
{
var x = Math.random()*9-4.5;
var y = Math.random()*9-4.5;
poly2.push([x, y]);
board.create('point', poly2[i], {withLabel:false});
}
for(i=0;i<5;i++)
{
var x1 = Math.random()*9-4.5;
var y1 = Math.random()*9-4.5;
var x2 = Math.random()*9-4.5;
var y2 = Math.random()*9-4.5;
board.create('polygon', [[x1,y1],[x1,y2],[x2,y2],[x2,y1]], {vertices:{visible:false}});
}
}
</script>
</section>
<section>
<h3> 矩形覆蓋次數 </h3>
<p>從左掃到右，會經歷一些<b>事件</b></p>
<ul>
<li class="fragment"> 矩形的左邊界</li>
<li class="fragment"> 矩形的右邊界</li>
<li class="fragment"> 查詢點</li>
</ul>
<p class="fragment">兩個事件之間，什麼都不會發生!</p>
<div id="box31" class="jxgbox" style="width:300px; height:300px;"></div>
<script type="text/javascript">
{
var board = JXG.JSXGraph.initBoard('box31', {boundingbox: [-5, 5, 5, -5]});
var poly2 = [];
var xco = [];
for(i=0; i<10; i++)
{
var x = Math.random()*9-4.5;
var y = Math.random()*9-4.5;
poly2.push([x, y]);
xco.push(x);
board.create('point', poly2[i], {withLabel:false});
}
for(i=0;i<5;i++)
{
var x1 = Math.random()*9-4.5;
var y1 = Math.random()*9-4.5;
var x2 = Math.random()*9-4.5;
var y2 = Math.random()*9-4.5;
xco.push(x1);
xco.push(x2);
board.create('polygon', [[x1,y1],[x1,y2],[x2,y2],[x2,y1]], {vertices:{visible:false}});
}
xco.sort(function(x, y){return x-y;});
console.log(xco);
sz = xco.length;
var pos = sz-1;
var AA = board.create('point', [-5, -6]);
var AB = board.create('point', [-5, 6]);
var li = board.create('line', [AA, AB]);
setInterval(function(){
pos = (pos + 1) % sz;
AA.moveTo([xco[pos], -6]);
AB.moveTo([xco[pos], 6]);
}, 500);
}
</script>
</section>
<section>
<h3> 矩形覆蓋次數 </h3>
<ul>
<li class="fragment"> 掃描線 : 1 維 </li>
<li class="fragment"> 維護每個 $y$ 座標被多少矩形覆蓋</li>
<li class="fragment"> 線段樹! </li>
<li class="fragment"> 左邊界 : $y$ 區間 +1 </li>
<li class="fragment"> 右邊界 : $y$ 區間 -1 </li>