155 lines
5.1 KiB
HTML
155 lines
5.1 KiB
HTML
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<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
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<html>
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<head>
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<!-- fix buggy IE8, especially for mathjax -->
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<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>An angle trisection</title>
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<link rel="stylesheet" type="text/css" media="screen" href="style.css">
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<script type="text/javascript"
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src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
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MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
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</script>
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</head>
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<body style="visibility:hidden">
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<h1>An angle trisection</h1>
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<h4>
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Construction by <a href="http://cdsmith.wordpress.com/2009/04/20/old-memories-about-trisecting-angles/">cdsmith</a>
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</h4>
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<table class="centered">
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<tr><td align="center">
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<applet code="Geometry" archive="Geometry.zip" width="600" height="450">
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<param name="background" value="ffffff">
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<param name="title" value="An angle trisection">
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<param name="e[1]" value="O;point;fixed;250,340">
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<param name="e[2]" value="A;point;fixed;450,340">
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<param name="e[3]" value="cir1;circle;radius;O,A;none;none;none;none">
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<param name="e[4]" value="B;point;circleSlider;cir1,400,0;red;red">
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<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
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<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
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<param name="e[7]" value="AB;line;connect;A,B;none;none;lightGray">
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<param name="e[8]" value="M;point;midpoint;A,B">
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<param name="e[9]" value="cA;circle;radius;A,M;none;none;green;none">
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<param name="e[10]" value="cM;circle;radius;M,A;none;none;green;none">
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<param name="e[11]" value="cB;circle;radius;B,M;none;none;green;none">
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<param name="e[12]" value="li1;line;bichord;cA,cM;none;none;none">
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<param name="e[13]" value="C;point;first;li1">
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<param name="e[14]" value="li2;line;bichord;cM,cB;none;none;none">
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<param name="e[15]" value="D;point;first;li2">
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<param name="e[16]" value="OC;line;connect;O,C;none;none;red">
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<param name="e[17]" value="OD;line;connect;O,D;none;none;red">
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<!-- angle marker -->
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<param name="e[18]" value="p1;point;fixed;285,340;none;none">
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<param name="e[19]" value="c1;circle;radius;O,p1;none;none;none;none">
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<param name="e[20]" value="l1;line;chord;OA,c1;none;none;none">
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<param name="e[21]" value="q1;point;first;l1;none;none">
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<param name="e[22]" value="l2;line;chord;O,C,c1;none;none;none">
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<param name="e[23]" value="q2;point;first;l2;none;none">
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<param name="e[24]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
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<param name="e[25]" value="l3;line;chord;OB,c1;none;none;none">
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<param name="e[26]" value="q3;point;first;l3;none;none">
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<param name="e[27]" value="s2;sector;sector;O,q2,q3;none;none;black;yellow">
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</applet>
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</td></tr>
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<tr><td>
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<b>
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Drag the point $B$ to change the angle $AOB$.<br>
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Press “r” to reset the diagram to its initial state.<br>
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The red lines $OC$ and $OD$ are approximate trisectors of the angle $AOB$.
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</b>
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</td></tr></table>
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<h2>Construction</h2>
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<p>
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To trisect the angle $AOB$ (with $OA=OB$), do:
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<ol>
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<li>
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Find the midpoint $M$ of the line segment $AB$.
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<li>
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Draw circles centered at $A$, $M$, and $B$, each of radius $\frac{1}{2}AB$,
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and mark their intersection points $C$ and $D$, as shown in the diagram
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above.
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</ol>
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The lines $OC$ and $OD$ are approximate trisectors of the angle $AOB$.
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<h2>Error Analysis</h2>
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<p>
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Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOC$,
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respectively. One may verify that
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\[
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\beta =
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\arctan\bigg(
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\frac{
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\sin\frac{\alpha}{2}
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\sin\big(\frac{\pi}{6}+\frac{\alpha}{2}\big)
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}{
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1 +
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\sin\frac{\alpha}{2}
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\cos\big(\frac{\pi}{6}+\frac{\alpha}{2}\big)
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}
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\bigg)
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=
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\frac{1}{4}\alpha + \frac{\sqrt{3}}{16} \alpha^2 - \frac{1}{16} \alpha^3
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+ O(\alpha^4).
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\]
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<p>
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This says that $\ds \beta \approx \frac{1}{4}\alpha$ when $\alpha$ is small,
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so small angles are quadrisected, rather than trisected!
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(This is clearly visible in the interactive diagram above.)
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For not-so-small angles,
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the method works reasonably well. In fact, it produces
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<em>exact trisection</em> for angles $\alpha=\pi/2$ and $\alpha=\pi$.
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<p>
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The worst error in
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the range $0 \le \alpha \le \pi$ is
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0.0214 radians = 1.23 degrees.
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This occurs at
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$\alpha=2\arctan(\sqrt{3}\pm\sqrt{2})$ which corresponds to
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angles of approximately 35 degrees and 145 degrees.
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<hr width="60%">
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<p>
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<em>This applet was created by
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<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
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using
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<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
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<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
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Applet</a>
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on June 14, 2010.
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</em>
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<p>
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<table width="100%">
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<tr>
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<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
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<td align="right" style="width:200px;">
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<a href="http://validator.w3.org/check?uri=referer">
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<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
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<a href="http://jigsaw.w3.org/css-validator/check/referer">
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<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
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</td></tr>
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</table>
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</body>
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</html>
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