153 lines
5.2 KiB
HTML
153 lines
5.2 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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William R. Raiford, <i>An approximate trisection</i>,
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American Mathematical Monthly,
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vol. 68, no. 9, Nov 1961, p. 917.
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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="450" height="400">
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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;210,365">
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<param name="e[2]" value="A;point;fixed;410,365">
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<param name="e[3]" value="pt0;point;fixed;410,0;none;none">
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<param name="e[4]" value="li0;line;connect;A,pt0;none;none;green">
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<param name="e[5]" value="cir1;circle;radius;O,A;none;none;none;none">
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<param name="e[6]" value="B;point;circleSlider;cir1,0,300;red;red">
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<param name="e[7]" value="OA;line;connect;O,A;none;none;blue">
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<param name="e[8]" value="OB;line;connect;O,B;none;none;blue">
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<param name="e[9]" value="arcAB;sector;sector;O,A,B;none;none;blue;none">
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<param name="e[10]" value="pt1;point;angleBisector;A,O,B;none;none">
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<param name="e[11]" value="C;point;cutoff;O,pt1,O,A">
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<param name="e[12]" value="OC;line;connect;O,C;none;none;lightGray">
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<param name="e[13]" value="li1;line;connect;B,C;none;none;lightGray">
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<param name="e[14]" value="li2;line;extend;B,C,B,C;none;none;lightGray">
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<param name="e[15]" value="T;point;intersection;li0,li2">
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<param name="e[16]" value="OT;line;connect;O,T;none;none;red">
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<!-- angle marker -->
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<param name="e[17]" value="p1;point;fixed;240,385;none;none">
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<param name="e[18]" value="c1;circle;radius;O,p1;none;none;none;none">
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<param name="e[19]" value="l1;line;chord;OA,c1;none;none;none">
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<param name="e[20]" value="q1;point;first;l1;none;none">
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<param name="e[21]" value="l2;line;chord;O,T,c1;none;none;none">
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<param name="e[22]" value="q2;point;first;l2;none;none">
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<param name="e[23]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
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<!-- angle marker -->
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<param name="e[24]" value="l3;line;chord;OB,c1;none;none;none">
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<param name="e[25]" value="q3;point;first;l3;none;none">
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<param name="e[26]" 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 line $OT$ is an approximate trisector 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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The construction described in the article cited at the top of the page,
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is quite straightforward. Consider the angle $AOB$ represented by the
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circular arc $AB$ centered at $O$, as shown in the diagram above.
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To trisect $AOB$ do:
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<ol>
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<li>
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Erect a perpendicular to $OA$ at $A$ (shown in green).
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<li>
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Construct the bisector $OC$ of the angle $AOB$.
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<li>
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Connect $B$ to $C$ and extend to intersect the green line at a point $T$.
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</ol>
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The line $OT$ is an approximate trisector 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 $AOT$,
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respectively. One may verify that
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\[
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\beta
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= \arctan \Big( \sin\alpha - (1 - \cos\alpha)
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\cot \big( \frac{3}{4}\alpha \big) \Big)
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= \frac{1}{3}\alpha + \frac{1}{2^3\cdot3^4} \alpha^3 + O(\alpha^5)
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= \frac{1}{3}\alpha + \frac{1}{648} \alpha^3 + O(\alpha^5).
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\]
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<p>
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The error
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$
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\ds e(\alpha) = \beta - \frac{\alpha}{3}
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$
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is monotonically increasing in $\alpha$.
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The worst error on the interval $0 \le \alpha \le \pi/2$ is
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$e(\pi/2)$ = 0.0063 radians = 0.361 degrees.
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The worst error on the interval $0 \le \alpha \le \pi$ is
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$e(\pi)$ = 0.06 radians = 3.435 degrees.
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<p>
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<span class="name">Raiford</span>, whose affiliation is given as IBM,
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states that he has calculated
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the error in increments of one degree in an IBM 709. Computers
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were novelties when that article was published.
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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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