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1.html
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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8" />
<title>1</title>
<script type="text/x-mathjax-config">
MathJax.Hub.Config({
jax: ["input/TeX","output/HTML-CSS"],
extensions: [],
TeX: {
extensions: ["AMSmath.js","AMSsymbols.js","noErrors.js","noUndefined.js"]
},
showMathMenu: false
});
</script>
<script type="text/javascript" src="https://cdn.mathjax.org/mathjax/latest/MathJax.js">
</script>
<style>body { padding: 0; margin: 0; }
.markdown-preview-plus-view:not([data-use-github-style]) { padding: 2em; font-size: 1.2em; color: rgb(171, 178, 191); background-color: rgb(40, 44, 52); overflow: auto; }
.markdown-preview-plus-view:not([data-use-github-style]) > :first-child { margin-top: 0px; }
.markdown-preview-plus-view:not([data-use-github-style]) h1, .markdown-preview-plus-view:not([data-use-github-style]) h2, .markdown-preview-plus-view:not([data-use-github-style]) h3, .markdown-preview-plus-view:not([data-use-github-style]) h4, .markdown-preview-plus-view:not([data-use-github-style]) h5, .markdown-preview-plus-view:not([data-use-github-style]) h6 { line-height: 1.2; margin-top: 1.5em; margin-bottom: 0.5em; color: rgb(255, 255, 255); }
.markdown-preview-plus-view:not([data-use-github-style]) h1 { font-size: 2.4em; font-weight: 300; }
.markdown-preview-plus-view:not([data-use-github-style]) h2 { font-size: 1.8em; font-weight: 400; }
.markdown-preview-plus-view:not([data-use-github-style]) h3 { font-size: 1.5em; font-weight: 500; }
.markdown-preview-plus-view:not([data-use-github-style]) h4 { font-size: 1.2em; font-weight: 600; }
.markdown-preview-plus-view:not([data-use-github-style]) h5 { font-size: 1.1em; font-weight: 600; }
.markdown-preview-plus-view:not([data-use-github-style]) h6 { font-size: 1em; font-weight: 600; }
.markdown-preview-plus-view:not([data-use-github-style]) strong { color: rgb(255, 255, 255); }
.markdown-preview-plus-view:not([data-use-github-style]) del { color: rgb(124, 135, 156); }
.markdown-preview-plus-view:not([data-use-github-style]) a, .markdown-preview-plus-view:not([data-use-github-style]) a code { color: rgb(82, 139, 255); }
.markdown-preview-plus-view:not([data-use-github-style]) img { max-width: 100%; }
.markdown-preview-plus-view:not([data-use-github-style]) > p { margin-top: 0px; margin-bottom: 1.5em; }
.markdown-preview-plus-view:not([data-use-github-style]) > ul, .markdown-preview-plus-view:not([data-use-github-style]) > ol { margin-bottom: 1.5em; }
.markdown-preview-plus-view:not([data-use-github-style]) blockquote { margin: 1.5em 0px; font-size: inherit; color: rgb(124, 135, 156); border-color: rgb(75, 83, 98); border-width: 4px; }
.markdown-preview-plus-view:not([data-use-github-style]) hr { margin: 3em 0px; border-top: 2px dashed rgb(75, 83, 98); background: none; }
.markdown-preview-plus-view:not([data-use-github-style]) table { margin: 1.5em 0px; }
.markdown-preview-plus-view:not([data-use-github-style]) th { color: rgb(255, 255, 255); }
.markdown-preview-plus-view:not([data-use-github-style]) th, .markdown-preview-plus-view:not([data-use-github-style]) td { padding: 0.66em 1em; border: 1px solid rgb(75, 83, 98); }
.markdown-preview-plus-view:not([data-use-github-style]) pre, .markdown-preview-plus-view:not([data-use-github-style]) code { color: rgb(255, 255, 255); background-color: rgb(58, 63, 75); }
.markdown-preview-plus-view:not([data-use-github-style]) pre, .markdown-preview-plus-view:not([data-use-github-style]) pre.editor-colors { margin: 1.5em 0px; padding: 1em; font-size: 0.92em; border-radius: 3px; background-color: rgb(49, 54, 63); }
.markdown-preview-plus-view:not([data-use-github-style]) kbd { color: rgb(255, 255, 255); border-width: 1px 1px 2px; border-style: solid; border-color: rgb(75, 83, 98) rgb(75, 83, 98) rgb(62, 68, 81); border-image: initial; background-color: rgb(58, 63, 75); }
.markdown-preview-plus-view:not([data-use-github-style]) ul.contains-task-list li.task-list-item { position: relative; list-style-type: none; }
.markdown-preview-plus-view:not([data-use-github-style]) ul.contains-task-list li.task-list-item input.task-list-item-checkbox { position: absolute; transform: translateX(-100%); width: 30px; }
.markdown-preview-plus-view[data-use-github-style] { font-family: "Helvetica Neue", Helvetica, "Segoe UI", Arial, freesans, sans-serif; line-height: 1.6; word-wrap: break-word; padding: 30px; font-size: 16px; color: rgb(51, 51, 51); background-color: rgb(255, 255, 255); overflow: auto; }
.markdown-preview-plus-view[data-use-github-style] > :first-child { margin-top: 0px !important; }
.markdown-preview-plus-view[data-use-github-style] > :last-child { margin-bottom: 0px !important; }
.markdown-preview-plus-view[data-use-github-style] a:not([href]) { color: inherit; text-decoration: none; }
.markdown-preview-plus-view[data-use-github-style] .absent { color: rgb(204, 0, 0); }
.markdown-preview-plus-view[data-use-github-style] .anchor { position: absolute; top: 0px; left: 0px; display: block; padding-right: 6px; padding-left: 30px; margin-left: -30px; }
.markdown-preview-plus-view[data-use-github-style] .anchor:focus { outline: none; }
.markdown-preview-plus-view[data-use-github-style] h1, .markdown-preview-plus-view[data-use-github-style] h2, .markdown-preview-plus-view[data-use-github-style] h3, .markdown-preview-plus-view[data-use-github-style] h4, .markdown-preview-plus-view[data-use-github-style] h5, .markdown-preview-plus-view[data-use-github-style] h6 { position: relative; margin-top: 1em; margin-bottom: 16px; font-weight: bold; line-height: 1.4; }
.markdown-preview-plus-view[data-use-github-style] h1 .octicon-link, .markdown-preview-plus-view[data-use-github-style] h2 .octicon-link, .markdown-preview-plus-view[data-use-github-style] h3 .octicon-link, .markdown-preview-plus-view[data-use-github-style] h4 .octicon-link, .markdown-preview-plus-view[data-use-github-style] h5 .octicon-link, .markdown-preview-plus-view[data-use-github-style] h6 .octicon-link { display: none; color: rgb(0, 0, 0); vertical-align: middle; }
.markdown-preview-plus-view[data-use-github-style] h1:hover .anchor, .markdown-preview-plus-view[data-use-github-style] h2:hover .anchor, .markdown-preview-plus-view[data-use-github-style] h3:hover .anchor, .markdown-preview-plus-view[data-use-github-style] h4:hover .anchor, .markdown-preview-plus-view[data-use-github-style] h5:hover .anchor, .markdown-preview-plus-view[data-use-github-style] h6:hover .anchor { padding-left: 8px; margin-left: -30px; text-decoration: none; }
.markdown-preview-plus-view[data-use-github-style] h1:hover .anchor .octicon-link, .markdown-preview-plus-view[data-use-github-style] h2:hover .anchor .octicon-link, .markdown-preview-plus-view[data-use-github-style] h3:hover .anchor .octicon-link, .markdown-preview-plus-view[data-use-github-style] h4:hover .anchor .octicon-link, .markdown-preview-plus-view[data-use-github-style] h5:hover .anchor .octicon-link, .markdown-preview-plus-view[data-use-github-style] h6:hover .anchor .octicon-link { display: inline-block; }
.markdown-preview-plus-view[data-use-github-style] h1 tt, .markdown-preview-plus-view[data-use-github-style] h2 tt, .markdown-preview-plus-view[data-use-github-style] h3 tt, .markdown-preview-plus-view[data-use-github-style] h4 tt, .markdown-preview-plus-view[data-use-github-style] h5 tt, .markdown-preview-plus-view[data-use-github-style] h6 tt, .markdown-preview-plus-view[data-use-github-style] h1 code, .markdown-preview-plus-view[data-use-github-style] h2 code, .markdown-preview-plus-view[data-use-github-style] h3 code, .markdown-preview-plus-view[data-use-github-style] h4 code, .markdown-preview-plus-view[data-use-github-style] h5 code, .markdown-preview-plus-view[data-use-github-style] h6 code { font-size: inherit; }
.markdown-preview-plus-view[data-use-github-style] h1 { padding-bottom: 0.3em; font-size: 2.25em; line-height: 1.2; border-bottom: 1px solid rgb(238, 238, 238); }
.markdown-preview-plus-view[data-use-github-style] h1 .anchor { line-height: 1; }
.markdown-preview-plus-view[data-use-github-style] h2 { padding-bottom: 0.3em; font-size: 1.75em; line-height: 1.225; border-bottom: 1px solid rgb(238, 238, 238); }
.markdown-preview-plus-view[data-use-github-style] h2 .anchor { line-height: 1; }
.markdown-preview-plus-view[data-use-github-style] h3 { font-size: 1.5em; line-height: 1.43; }
.markdown-preview-plus-view[data-use-github-style] h3 .anchor { line-height: 1.2; }
.markdown-preview-plus-view[data-use-github-style] h4 { font-size: 1.25em; }
.markdown-preview-plus-view[data-use-github-style] h4 .anchor { line-height: 1.2; }
.markdown-preview-plus-view[data-use-github-style] h5 { font-size: 1em; }
.markdown-preview-plus-view[data-use-github-style] h5 .anchor { line-height: 1.1; }
.markdown-preview-plus-view[data-use-github-style] h6 { font-size: 1em; color: rgb(119, 119, 119); }
.markdown-preview-plus-view[data-use-github-style] h6 .anchor { line-height: 1.1; }
.markdown-preview-plus-view[data-use-github-style] p, .markdown-preview-plus-view[data-use-github-style] blockquote, .markdown-preview-plus-view[data-use-github-style] ul, .markdown-preview-plus-view[data-use-github-style] ol, .markdown-preview-plus-view[data-use-github-style] dl, .markdown-preview-plus-view[data-use-github-style] table, .markdown-preview-plus-view[data-use-github-style] pre { margin-top: 0px; margin-bottom: 16px; }
.markdown-preview-plus-view[data-use-github-style] hr { height: 4px; padding: 0px; margin: 16px 0px; background-color: rgb(231, 231, 231); border: 0px none; }
.markdown-preview-plus-view[data-use-github-style] ul, .markdown-preview-plus-view[data-use-github-style] ol { padding-left: 2em; }
.markdown-preview-plus-view[data-use-github-style] ul.no-list, .markdown-preview-plus-view[data-use-github-style] ol.no-list { padding: 0px; list-style-type: none; }
.markdown-preview-plus-view[data-use-github-style] ul ul, .markdown-preview-plus-view[data-use-github-style] ul ol, .markdown-preview-plus-view[data-use-github-style] ol ol, .markdown-preview-plus-view[data-use-github-style] ol ul { margin-top: 0px; margin-bottom: 0px; }
.markdown-preview-plus-view[data-use-github-style] li > p { margin-top: 16px; }
.markdown-preview-plus-view[data-use-github-style] dl { padding: 0px; }
.markdown-preview-plus-view[data-use-github-style] dl dt { padding: 0px; margin-top: 16px; font-size: 1em; font-style: italic; font-weight: bold; }
.markdown-preview-plus-view[data-use-github-style] dl dd { padding: 0px 16px; margin-bottom: 16px; }
.markdown-preview-plus-view[data-use-github-style] blockquote { padding: 0px 15px; color: rgb(119, 119, 119); border-left: 4px solid rgb(221, 221, 221); }
.markdown-preview-plus-view[data-use-github-style] blockquote > :first-child { margin-top: 0px; }
.markdown-preview-plus-view[data-use-github-style] blockquote > :last-child { margin-bottom: 0px; }
.markdown-preview-plus-view[data-use-github-style] table { display: block; width: 100%; overflow: auto; word-break: keep-all; }
.markdown-preview-plus-view[data-use-github-style] table th { font-weight: bold; }
.markdown-preview-plus-view[data-use-github-style] table th, .markdown-preview-plus-view[data-use-github-style] table td { padding: 6px 13px; border: 1px solid rgb(221, 221, 221); }
.markdown-preview-plus-view[data-use-github-style] table tr { background-color: rgb(255, 255, 255); border-top: 1px solid rgb(204, 204, 204); }
.markdown-preview-plus-view[data-use-github-style] table tr:nth-child(2n) { background-color: rgb(248, 248, 248); }
.markdown-preview-plus-view[data-use-github-style] img { max-width: 100%; box-sizing: border-box; }
.markdown-preview-plus-view[data-use-github-style] .emoji { max-width: none; }
.markdown-preview-plus-view[data-use-github-style] span.frame { display: block; overflow: hidden; }
.markdown-preview-plus-view[data-use-github-style] span.frame > span { display: block; float: left; width: auto; padding: 7px; margin: 13px 0px 0px; overflow: hidden; border: 1px solid rgb(221, 221, 221); }
.markdown-preview-plus-view[data-use-github-style] span.frame span img { display: block; float: left; }
.markdown-preview-plus-view[data-use-github-style] span.frame span span { display: block; padding: 5px 0px 0px; clear: both; color: rgb(51, 51, 51); }
.markdown-preview-plus-view[data-use-github-style] span.align-center { display: block; overflow: hidden; clear: both; }
.markdown-preview-plus-view[data-use-github-style] span.align-center > span { display: block; margin: 13px auto 0px; overflow: hidden; text-align: center; }
.markdown-preview-plus-view[data-use-github-style] span.align-center span img { margin: 0px auto; text-align: center; }
.markdown-preview-plus-view[data-use-github-style] span.align-right { display: block; overflow: hidden; clear: both; }
.markdown-preview-plus-view[data-use-github-style] span.align-right > span { display: block; margin: 13px 0px 0px; overflow: hidden; text-align: right; }
.markdown-preview-plus-view[data-use-github-style] span.align-right span img { margin: 0px; text-align: right; }
.markdown-preview-plus-view[data-use-github-style] span.float-left { display: block; float: left; margin-right: 13px; overflow: hidden; }
.markdown-preview-plus-view[data-use-github-style] span.float-left span { margin: 13px 0px 0px; }
.markdown-preview-plus-view[data-use-github-style] span.float-right { display: block; float: right; margin-left: 13px; overflow: hidden; }
.markdown-preview-plus-view[data-use-github-style] span.float-right > span { display: block; margin: 13px auto 0px; overflow: hidden; text-align: right; }
.markdown-preview-plus-view[data-use-github-style] code, .markdown-preview-plus-view[data-use-github-style] tt { padding: 0.2em 0px; margin: 0px; font-size: 85%; background-color: rgba(0, 0, 0, 0.04); border-radius: 3px; }
.markdown-preview-plus-view[data-use-github-style] code::before, .markdown-preview-plus-view[data-use-github-style] tt::before, .markdown-preview-plus-view[data-use-github-style] code::after, .markdown-preview-plus-view[data-use-github-style] tt::after { letter-spacing: -0.2em; content: " "; }
.markdown-preview-plus-view[data-use-github-style] code br, .markdown-preview-plus-view[data-use-github-style] tt br { display: none; }
.markdown-preview-plus-view[data-use-github-style] del code { text-decoration: inherit; }
.markdown-preview-plus-view[data-use-github-style] pre > code { padding: 0px; margin: 0px; font-size: 100%; word-break: normal; white-space: pre; background: transparent; border: 0px; }
.markdown-preview-plus-view[data-use-github-style] .highlight { margin-bottom: 16px; }
.markdown-preview-plus-view[data-use-github-style] .highlight pre, .markdown-preview-plus-view[data-use-github-style] pre { padding: 16px; overflow: auto; font-size: 85%; line-height: 1.45; background-color: rgb(247, 247, 247); border-radius: 3px; }
.markdown-preview-plus-view[data-use-github-style] .highlight pre { margin-bottom: 0px; word-break: normal; }
.markdown-preview-plus-view[data-use-github-style] pre { word-wrap: normal; }
.markdown-preview-plus-view[data-use-github-style] pre code, .markdown-preview-plus-view[data-use-github-style] pre tt { display: inline; max-width: initial; padding: 0px; margin: 0px; overflow: initial; line-height: inherit; word-wrap: normal; background-color: transparent; border: 0px; }
.markdown-preview-plus-view[data-use-github-style] pre code::before, .markdown-preview-plus-view[data-use-github-style] pre tt::before, .markdown-preview-plus-view[data-use-github-style] pre code::after, .markdown-preview-plus-view[data-use-github-style] pre tt::after { content: normal; }
.markdown-preview-plus-view[data-use-github-style] kbd { display: inline-block; padding: 3px 5px; font-size: 11px; line-height: 10px; color: rgb(85, 85, 85); vertical-align: middle; background-color: rgb(252, 252, 252); border-width: 1px; border-style: solid; border-color: rgb(204, 204, 204) rgb(204, 204, 204) rgb(187, 187, 187); border-image: initial; border-radius: 3px; box-shadow: rgb(187, 187, 187) 0px -1px 0px inset; }
.markdown-preview-plus-view[data-use-github-style] a { color: rgb(51, 122, 183); }
.markdown-preview-plus-view[data-use-github-style] pre, .markdown-preview-plus-view[data-use-github-style] code { color: inherit; }
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<body class="markdown-preview-plus-view">
<h1>Computational Geometry: Course 1</h1>
<h1>Course information</h1>
<p>URL: <a href="http://homepages.ulb.ac.be/~slanger/cg18">http://homepages.ulb.ac.be/~slanger/cg18</a></p>
<h2>Summary</h2>
<p>Course is based on how to solve computational problems on computers. Big parts of
the course are:</p>
<ul>
<li>Definitions
<ul>
<li>Models of computation</li>
<li>Objects</li>
</ul>
</li>
<li>Convexity</li>
<li>Convex Hull</li>
<li>Lower Bounds</li>
<li>Polygons</li>
<li>Triangles</li>
<li>Planar graphs</li>
<li>Geometric data structures</li>
<li>Proximity
<ul>
<li>Voronoi</li>
<li>Delaunay</li>
</ul>
</li>
<li>Line arrangements</li>
<li>Linear programming</li>
<li>Duality</li>
</ul>
<h2>Schedule</h2>
<ul>
<li>Mon 8h-10h</li>
<li>Thurs 10h-12h</li>
</ul>
<h2>Assignments</h2>
<ul>
<li>Homework per course (10%)
<ul>
<li>In javascript</li>
<li>p5.js</li>
</ul>
</li>
<li>Project (50%) + Presentation (40%)</li>
<li>2nd session: Written exam in august that replaces presentation</li>
</ul>
<h1>First chapter: Introduction</h1>
<h2>Objects to manipulate</h2>
<ul>
<li>Points</li>
<li>Line</li>
<li>Segment</li>
<li>Triangle</li>
<li>Disks</li>
<li>Polygons…</li>
</ul>
<p>We can see that points create most of the objects that we manipulate and with disks we introduce the concept of distance (because it’s represented by the central point and a radius)</p>
<h2>Models of computation</h2>
<ul>
<li>Turing Machine</li>
<li>RAM Model <span class="math"><script type="math/tex">\to</script></span> integers</li>
<li><span class="math"><script type="math/tex">\lambda</script></span>-calculus</li>
</ul>
<p>The RAM model uses integers so it can be difficult to represent irrational numbers (like Pythagoras).</p>
<ul>
<li>Ruler & Compass</li>
</ul>
<p>This model uses Euclid operations to create whatever geometric constructions it exists (with a ruler and compass).</p>
<ul>
<li>Real-RAM (abstract, no machine can represent real numbers)</li>
</ul>
<p>There are some problems that proves the importance of those 2 models, some problems are easy to solve in one model and viceversa.</p>
<ul>
<li><a href="https://en.wikipedia.org/wiki/Squaring_the_circle">Squaring the circle</a></li>
<li><a href="https://en.wikipedia.org/wiki/Angle_trisection">Angle trisection</a></li>
<li><a href="https://en.wikipedia.org/wiki/Doubling_the_cube">Doubling the cube</a></li>
</ul>
<p>With the Real-RAM Model we can do those operations:<span class="math"><script type="math/tex">+</script></span>, <span class="math"><script type="math/tex">-</script></span>, <span class="math"><script type="math/tex">\times</script></span>, /, <span class="math"><script type="math/tex">\sqrt{}</script></span>, <span class="math"><script type="math/tex">\sin</script></span>, <span class="math"><script type="math/tex">\cos</script></span>, <span class="math"><script type="math/tex">\pi</script></span>. <span class="math"><script type="math/tex">\to</script></span> the square root calculation is expensive. If we add the floor function, we can solve <span class="math"><script type="math/tex">NP</script></span>-complete problems in polynomial time (the model would be too powerful).</p>
<h2>Main operations that we are going to use</h2>
<ul>
<li>Norm: <span class="math"><script type="math/tex">|a| = \sqrt{a_x^2 + a_x^2}</script></span></li>
<li>Scalar product: <span class="math"><script type="math/tex">(a_x, a_y)\dot{} (b_x, b_y) = a_xb_x + a_yb_y = |a||b|\sin{\theta}</script></span>, where <span class="math"><script type="math/tex">\theta</script></span> is the angle between the vectors.</li>
<li>Cross product: <span class="math"><script type="math/tex">a \times b = \begin{bmatrix}
1_x & 1_y & 1_z \\
a_x & a_y & a_z \\
b_x & b_y & b_z \\
\end{bmatrix}</script></span></li>
</ul>
<p>/!\ Important: <span class="math"><script type="math/tex">|a\times b| = |a||b|sin{\theta}</script></span></p>
<p><span class="math"><script type="math/tex">a\times b = 1_z \begin{vmatrix} a_x & a_y \\ b_x & b_y \end{vmatrix} = (a_xb_y - a_yb_x)1_z</script></span></p>
<p>To find the sinus of an angle now we can just calculate it with all the equivalences. Given that we have 3 points u, v and w.</p>
<span class="math"><script type="math/tex; mode=display">a = v-u\\
b = w-u\\
a\times b = v_xw_y - u_xw_y - v_xu_y + u_xu_y - v_yw_x + v_yu_x + u_yw_x - u_yu_x = \begin{vmatrix} u_x & u_y & 1\\
v_x & v_y & 1\\
w_x & w_y & 1 \end{vmatrix}</script></span>
<h1>Convexity</h1>
<p>Having group C:</p>
<span class="math"><script type="math/tex; mode=display">C\ \text{is convex iff}\ \ \ \forall a,b\in C [ab] \subseteq C\\
[ab] = \{\lambda a + (1-\lambda)b : 0 \leq \lambda \leq 1\}</script></span>
<p><strong>Convex combination</strong>:</p>
<span class="math"><script type="math/tex; mode=display"> \forall P_1, P_2, \dots, P_n = \{\lambda_1P_1+\lambda_2P_2+\dots+\lambda_nP_n : \lambda_1 + \dots + \lambda_n = 1\ \ \lambda_i \geq 0\}
</script></span>
<p><strong>Convex Hull</strong>:</p>
<p>The Convex Hull of <span class="math"><script type="math/tex">S = \{P_1, \dots, P_n\}</script></span> is the smallest convex set that contains S. We can also say that the Convex Hull of a set S is the intersection of all convex sets that contain S:</p>
<span class="math"><script type="math/tex; mode=display">CH(S) = \bigcap_{\substack{C\ \text{convex} \\ S \subseteq C}} C
</script></span>
<p><img src="https://miro.medium.com/max/1354/1*F4IUmOJbbLMJiTgHxpoc7Q.png" alt="Convex Hull"></p>
<h1>HOMEWORK:</h1>
<p>Click on 3 points and then have a script that says if it is a right turn or a left turn.</p>
</body>
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