97 lines
6.1 KiB
HTML
97 lines
6.1 KiB
HTML
<!DOCTYPE html>
|
|
<html lang="en">
|
|
<head>
|
|
<meta charset="utf-8" />
|
|
<meta name="author" content="Colin Hughes" />
|
|
<meta name="description" content="A website dedicated to the fascinating world of mathematics and programming" />
|
|
<meta name="keywords" content="programming,mathematics,problems,puzzles" />
|
|
|
|
<title>Problem 228 - Project Euler</title>
|
|
<link rel="shortcut icon" href="http://projecteuler.net/favicon.ico" />
|
|
<link rel="stylesheet" type="text/css" href="style_main.css" />
|
|
<link rel="stylesheet" type="text/css" href="style_light.css" />
|
|
<script type="text/x-mathjax-config">
|
|
MathJax.Hub.Config({
|
|
jax: ["input/TeX", "output/HTML-CSS"],
|
|
tex2jax: {
|
|
inlineMath: [ ["$","$"], ["\\(","\\)"] ],
|
|
displayMath: [ ["$$","$$"], ["\\[","\\]"] ],
|
|
processEscapes: true
|
|
},
|
|
"HTML-CSS": { availableFonts: ["TeX"] }
|
|
});
|
|
</script>
|
|
|
|
<script type="text/javascript" src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML">
|
|
</script>
|
|
</head>
|
|
|
|
<body>
|
|
<div id="container">
|
|
|
|
<div id="nav" class="noprint">
|
|
<ul>
|
|
<li><a href="about" title="About" accesskey="h">About</a></li>
|
|
<li><a href="register" title="Register" accesskey="1">Register</a></li>
|
|
<li id="current"><a href="problems" title="Problems" accesskey="2">Problems</a></li>
|
|
<li><a href="login" title="Login" accesskey="3">Login</a></li>
|
|
</ul>
|
|
</div>
|
|
<div id="info_panel"><a href="rss2_euler.xml"><img src="images/icon_rss.png" alt="RSS Feed" title="RSS Feed" /></a><a href="secure=b6875"><img src="images/icon_lock.png" alt="Use secure connection" title="Use secure connection" /></a></div>
|
|
|
|
<div id="logo" class="noprint">
|
|
<img src="images/pe_banner_light.png" alt="Project Euler .net" />
|
|
</div>
|
|
|
|
<div id="content">
|
|
<div style="text-align:center;" class="print"><img src="images/pe_banner.png" alt="projecteuler.net" style="border:none;" /></div>
|
|
<h2>Minkowski Sums</h2><div class="info" style="cursor:help;width:200px;margin-bottom:10px;"><h3>Problem 228</h3><span style="width:300px;color:#666;">Published on Saturday, 17th January 2009, 05:00 am; Solved by 813</span></div>
|
|
<div class="problem_content" role="problem">
|
|
<p>Let <var>S</var><sub>n</sub> be the regular <var>n</var>-sided polygon – or <i>shape</i> – whose vertices
|
|
|
|
<var>v</var><sub><var>k</var></sub> (<var>k</var> = 1,2,…,<var>n</var>) have coordinates:</p>
|
|
<table>
|
|
<tr>
|
|
<td width="40"></td>
|
|
<td><var>x</var><sub><var>k</var></sub>   =  
|
|
cos( <sup>2<var>k</var>-1</sup>/<sub><var>n</var></sub> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />180° )</td>
|
|
</tr><tr>
|
|
<td width="40"></td>
|
|
<td><var>y</var><sub><var>k</var></sub>   =
|
|
sin( <sup>2<var>k</var>-1</sup>/<sub><var>n</var></sub> <img src='images/symbol_times.gif' width='9' height='9' alt='×' border='0' style='vertical-align:middle;' />180° )</td>
|
|
</tr>
|
|
</table>
|
|
|
|
<p>Each <var>S</var><sub><var>n</var></sub> is to be interpreted as a filled shape consisting of all points on the perimeter and in the interior.</p>
|
|
|
|
<p>The <i>Minkowski sum</i>, <var>S</var>+<var>T</var>, of two shapes <var>S</var> and <var>T</var> is the result of
|
|
|
|
adding every point in <var>S</var> to every point in <var>T</var>, where point addition is performed coordinate-wise:
|
|
|
|
(<var>u</var>, <var>v</var>) + (<var>x</var>, <var>y</var>) = (<var>u</var>+<var>x</var>, <var>v</var>+<var>y</var>).</p>
|
|
|
|
<p>For example, the sum of <var>S</var><sub>3</sub> and <var>S</var><sub>4</sub> is the six-sided shape shown in pink below:</p>
|
|
|
|
<div style="text-align:center;">
|
|
<img src="project/images/p_228.png" alt="picture showing S_3 + S_4" /></div>
|
|
|
|
<p>How many sides does <var>S</var><sub>1864</sub> + <var>S</var><sub>1865</sub> + … + <var>S</var><sub>1909</sub> have?</p>
|
|
</div><br />
|
|
<br /></div>
|
|
|
|
|
|
|
|
|
|
|
|
<div id="footer" class="noprint">
|
|
<a href="copyright">Project Euler Copyright Information</a>
|
|
<!--/Creative Commons License--><!-- <rdf:RDF xmlns="http://web.resource.org/cc/" xmlns:dc="http://purl.org/dc/elements/1.1/" xmlns:rdf="http://www.w3.org/1999/02/22-rdf-syntax-ns#" xmlns:rdfs="http://www.w3.org/2000/01/rdf-schema#">
|
|
<Work rdf:about="">
|
|
<license rdf:resource="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/" />
|
|
<dc:type rdf:resource="http://purl.org/dc/dcmitype/Text" />
|
|
</Work>
|
|
<License rdf:about="http://creativecommons.org/licenses/by-nc-sa/2.0/uk/"><permits rdf:resource="http://web.resource.org/cc/Reproduction"/><permits rdf:resource="http://web.resource.org/cc/Distribution"/><requires rdf:resource="http://web.resource.org/cc/Notice"/><requires rdf:resource="http://web.resource.org/cc/Attribution"/><prohibits rdf:resource="http://web.resource.org/cc/CommercialUse"/><permits rdf:resource="http://web.resource.org/cc/DerivativeWorks"/><requires rdf:resource="http://web.resource.org/cc/ShareAlike"/></License></rdf:RDF> -->
|
|
</div>
|
|
</div>
|
|
<div style="height:1px;"> </div></body>
|
|
</html> |