186 lines
6.6 KiB
HTML
186 lines
6.6 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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Free Jamison, <i>Trisection Approximation</i>, American Mathematical Monthly,
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vol. 61, no. 5, May 1954, pp. 334–336.
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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="700" 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;200,200">
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<param name="e[2]" value="A;point;fixed;200,350">
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<param name="e[3]" value="C1;circle;radius;O,A;none;none;lightGray;none">
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<param name="e[4]" value="B;point;circleSlider;C1,280,0;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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<!-- the points F, D, C -->
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<param name="e[7]" value="x1;point;angleBisector;A,O,B;none;none">
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<param name="e[8]" value="x2;point;angleBisector;x1,O,B;none;none">
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<param name="e[9]" value="x3;point;angleBisector;x1,O,x2;none;none">
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<param name="e[10]" value="F;point;cutoff;O,x2,O,A">
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<param name="e[11]" value="D;point;cutoff;O,x3,O,A">
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<param name="e[12]" value="C;point;extend;F,O,F,O">
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<!-- the lines FC, CE, OE -->
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<param name="e[13]" value="FC;line;connect;F,C;none;none;lightGray">
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<param name="e[14]" value="CD;line;connect;C,D;none;none;green">
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<param name="e[15]" value="E;point;extend;C,D,C,F">
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<param name="e[16]" value="DE;line;connect;D,E;none;none;green">
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<param name="e[17]" value="OE;line;connect;O,E;none;none;red">
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<param name="e[18]" value="p1;point;fixed;225,200;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;OE,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;yellow">
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<param name="e[25]" value="p2;point;fixed;225,200;none;none">
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<param name="e[26]" value="c2;circle;radius;O,p2;none;none;none;none">
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<param name="e[27]" value="l3;line;chord;OE,c2;none;none;none">
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<param name="e[28]" value="q3;point;first;l3;none;none">
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<param name="e[29]" value="l4;line;chord;OB,c2;none;none;none">
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<param name="e[30]" value="q4;point;first;l4;none;none">
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<param name="e[31]" value="s2;sector;sector;O,q3,q4;none;none;black;orange">
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<!-- needed for the error analysis, not the construction -->
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<param name="e[32]" value="OD;line;connect;O,D;none;none;lightGray">
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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 $AOB$
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(but stay on the right half of the circle).<br>
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Press “r” to reset the diagram to its initial state.<br>
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The red line $OE$ 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>The construction</h2>
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<p>
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This construction, due to Free Jamison
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(see the reference at the top of this page)
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is a more accurate variant of the construction described in
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<a href="trisect-jamison.html">a simpler construction</a>.
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<p>
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Consider the circular arc $AB$ centered at $O$, shown in the diagram above.
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Assume the angle $AOB$ is between 0 and 180 degrees.
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To trisect $AOB$, do:
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<ol>
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<li> Pick the points $F$ and $D$ on the arc $BA$ such that
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arc $BF$ = 2/8 of the arc $BA$ and
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arc $BD$ = 3/8 of the arc $BA$.
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<li> Extend $FO$ to intersect the circle at a point $C$.
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<li> Draw the line $CD$ and extend it to a point $E$ such that $DE$ equals the
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circle's diameter.
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</ol>
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<p>
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The line $OE$ 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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<p>
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Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $EOB$, respectively.
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The angle $FOD$ equals $\alpha/8$ by the construction, therefore the
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angle $FCD$, which is half the central angle $FOD$, is equal to
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$\alpha/16$.
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The triangle $DOC$ is isosceles, therefore the angle $ODC$ also equals $\alpha/16$.
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<p>
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In the triangle $OED$, let $x$ and $y$ be the sizes of the angles
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$OED$ and $EOD$, respectively. Since the sum $x+y$ of the triangle's internal
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angles equals the triangle's
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external angle $ODC$, we have $x+y = \alpha/16$. Let us note, however,
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that the angle $y$ equals $DOB$ minus $EOB$. Thus $y = 3\alpha/8 - \beta$,
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whence $x = \beta - 5\alpha/16$.
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<p>
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In the triangle $OED$, the side $DE$ is twice the side $OD$ by the construction,
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therefore the law of sines gives $\sin y = 2 \sin x$. Consequently,
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$\sin(3\alpha/8 - \beta) = 2 \sin(\beta - 5\alpha/16)$.
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Solving this for $\beta$ we arrive at:
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\[
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\beta
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= \frac{5}{16} \alpha + \arctan \frac{\sin(a/16)}{2+\cos(a/16)}
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= \frac{1}{3} \alpha - \frac{1}{2^{12}\cdot3^4} \alpha^3 + O(\alpha^5)
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= \frac{1}{3} \alpha - \frac{1}{331776} \alpha^3 + O(\alpha^5).
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\]
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<p>
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We see that the trisection error $e(\alpha) = \alpha/3 - \beta$ is given by:
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\[
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e(\alpha) = \frac{1}{48}\alpha - \arctan \frac{\sin(a/16)}{2+\cos(a/16)}.
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\]
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(This formula is also given in Jamison's article.)
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The function $e(a)$ 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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$e(\pi/2)$ = 0.0000117 radians = 0.00067 degrees.
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The worst error on the interval $0 \le \alpha \le \pi$ is
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$e(\pi)$ = 0.000093756 radians = 0.00537 degrees.
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Quite impressive!
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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> on
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July 22, 2002.
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<br>Cosmetic revisions on June 7, 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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