156 lines
5.3 KiB
HTML
156 lines
5.3 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>Construction attributed to A. G. O<br>
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From page 133 of<br>
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Underwood Dudley, <i>The Trisectors</i>, 2nd edition, 1996.
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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="260">
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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;210,225">
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<param name="e[2]" value="A;point;fixed;420,225">
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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;cir1,0,0;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="arcAB;sector;sector;O,A,B;none;none;blue;none">
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<param name="e[8]" value="pt1;point;angleBisector;A,O,B;none;none">
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<param name="e[9]" value="C;point;cutoff;O,pt1,O,A">
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<param name="e[10]" value="OC;line;connect;O,C;none;none;lightGray">
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<param name="e[11]" value="D;point;fixed;280,225">
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<param name="e[12]" value="E;point;cutoff;OB,O,D">
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<param name="e[13]" value="G;point;cutoff;OC,O,D">
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<param name="e[14]" value="F;point;extend;A,O,O,D">
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<param name="e[15]" value="OF;line;connect;O,F;none;none;lightGray">
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<param name="e[16]" value="sec1;sector;sector;O,D,E;none;none;orange">
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<param name="e[17]" value="sec2;sector;sector;O,E,F;none;none;lightGray">
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<param name="e[18]" value="li1;line;chord;F,G,cir1;none;none;none">
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<param name="e[19]" value="T;point;last;li1">
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<param name="e[20]" value="FT;line;connect;F,T;none;none;lightGray">
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<param name="e[21]" value="OT;line;connect;O,T;none;none;red">
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<!-- angle markers -->
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<param name="e[22]" value="p1;point;fixed;240,225;none;none">
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<param name="e[23]" value="c1;circle;radius;O,p1;none;none;none;none">
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<param name="e[24]" value="l1;line;chord;OA,c1;none;none;none">
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<param name="e[25]" value="q1;point;first;l1;none;none">
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<param name="e[26]" value="l2;line;chord;OT,c1;none;none;none">
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<param name="e[27]" value="q2;point;first;l2;none;none">
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<param name="e[28]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
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<param name="e[29]" value="l3;line;chord;OB,c1;none;none;none">
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<param name="e[30]" value="q3;point;first;l3;none;none">
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<param name="e[31]" 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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We wish to trisect the given angle $AOB$ represented by the circular arc
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$AB$ centered at $O$, as shown in the diagram above.
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<ol>
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<li>
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Draw the bisector $OC$ of the angle $AOB$.
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<li>
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Draw the circular arc $DE$ centered at $O$ so that $OD = \frac{1}{3} OA$.
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Let $G$ be where the line $OC$ intersects the arc $DE$.
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<li>
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Locate $F$ on the extension of $OA$ so that $OF=OD$.
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<li>
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Connect $FG$ and extend to the intersection point $T$ with
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the arc $AB$.
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</ol>
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The line $OT$ (shown in red) 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. It is straightforward to show that
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\[
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\beta
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= \frac{\alpha}{4} + \arcsin\Big(
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\frac{1}{3}\sin\frac{1}{4}\alpha \Big)
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= \frac{1}{3}\alpha - \frac{1}{2^4\cdot3^4} \alpha^3 + O(\alpha^7)
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= \frac{1}{3}\alpha - \frac{1}{1296} \alpha^3 + O(\alpha^7).
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\]
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The term after $\alpha^3$ is $\alpha^7$. That's not a typo.
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<p>
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The error
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$
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\ds e(\alpha) = \frac{\alpha}{3} - \beta
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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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$e(\pi/2) =$ 0.003 radians = 0.171 degrees.
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The worst error on the interval $0 \le \alpha \le \pi$ is
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$e(\pi)$ = 0.024 radians = 1.367 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 July 26, 2002.<br>
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Cosmetic revisions on June 13, 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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