<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN">
<html>
<head>
<!-- fix buggy IE8, especially for mathjax -->
<meta http-equiv="X-UA-Compatible" content="IE=EmulateIE7">
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>An angle trisection</title>
<link rel="stylesheet" type="text/css" media="screen" href="style.css">
<script type="text/javascript"
src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML,http://userpages.umbc.edu/~rostamia/mathjax-config.js">
MathJax.Hub.Queue( function() {document.body.style.visibility="visible"} );
</script>
</head>
<body style="visibility:hidden">
<h1>An angle trisection</h1>
<h4>Construction due to
<a href="mailto:avniu66@hotmail.com">Avni Pllana</a></h4>
<table class="centered">
<tr><td align="center">
<applet code="Geometry" archive="Geometry.zip" width="450" height="260">
<param name="background" value="ffffff">
<param name="title" value="An angle trisection">
<param name="e[1]" value="O;point;fixed;210,225">
<param name="e[2]" value="A;point;fixed;410,225">
<param name="e[3]" value="cir1;circle;radius;O,A;none;none;none;none">
<param name="e[4]" value="B;point;circleSlider;cir1,0,0;red;red">
<param name="e[5]" value="OA;line;connect;O,A;none;none;blue">
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
<param name="e[7]" value="arcAB;sector;sector;O,A,B;none;none;blue;none">
<param name="e[8]" value="pt1;point;angleBisector;A,O,B;none;none">
<param name="e[9]" value="C;point;cutoff;O,pt1,O,A">
<param name="e[10]" value="li1;line;connect;O,C;none;none;lightGray">
<param name="e[11]" value="M;point;midpoint;O,C">
<param name="e[12]" value="pt3;point;angleBisector;A,O,C;none;none">
<param name="e[13]" value="D;point;cutoff;O,pt3,O,A">
<param name="e[14]" value="li2;line;connect;O,D;none;none;lightGray">
<param name="e[15]" value="N;point;midpoint;M,D">
<param name="e[16]" value="MD;line;connect;M,D;none;none;cyan">
<param name="e[17]" value="li3;line;cutoff;O,N,O,A;none;none;red">
<!-- angle marker -->
<param name="e[18]" value="p1;point;fixed;240,225;none;none">
<param name="e[19]" value="c1;circle;radius;O,p1;none;none;none;none">
<param name="e[20]" value="l1;line;chord;OA,c1;none;none;none">
<param name="e[21]" value="q1;point;first;l1;none;none">
<param name="e[22]" value="l2;line;chord;O,N,c1;none;none;none">
<param name="e[23]" value="q2;point;first;l2;none;none">
<param name="e[24]" value="l3;line;chord;OB,c1;none;none;none">
<param name="e[25]" value="q3;point;first;l3;none;none">
<param name="e[26]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
<param name="e[27]" value="s2;sector;sector;O,q2,q3;none;none;black;yellow">
</applet>
</td></tr>
<tr><td>
<b>
Drag the point $B$ to change the angle $AOB$.<br>
Press “r” to reset the diagram to its initial state.<br>
The red line is an approximate trisector of the angle $AOB$.
</b>
</td></tr></table>
<h2>Construction</h2>
<p>
This approximate trisection, due to Avni Pllana, was announced
<a href="http://mathforum.org/kb/message.jspa?messageID=1084688">in a
message</a>
in the <code>geometry.puzzles</code> newsgroup on July 23, 2003.
Scroll to the bottom of that page to view the related discussion thread.
<p>
Consider the angle $AOB$ given by the circular arc $AB$ centered at $O$,
as shown in the diagram above.
<ol>
<li>
Pick points $C$ and $D$ on the arc $AB$ so that $OC$ bisects the angle $AOB$
and $OD$ bisects the angle $AOC$.
<li>
Let $M$ be the midpoint of the line segment $OC$.
<li>
Let $N$ be the midpoint of the line segment $MD$.
</ol>
The line $ON$ is an approximate trisector of the angle $AOB$.
<h2>Error Analysis</h2>
<p>
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AON$,
respectively. One may verify that:
\[
\beta
= \arctan \frac
{\sin\frac{\alpha}{2} + 2\sin\frac{\alpha}{4}}
{\cos\frac{\alpha}{2} + 2\cos\frac{\alpha}{4}}
= \frac{1}{3}\alpha - \frac{1}{2^6\cdot3^4} \alpha^3 + O(\alpha^5)
= \frac{1}{3}\alpha - \frac{1}{5184} \alpha^3 + O(\alpha^5).
\]
<em>Hint:</em> Represent the points as complex numbers
in the polar form $re^{i\theta}$.
<p>
The error
$
\ds e(\alpha) = \frac{\alpha}{3} - \beta
$
increases monotonically with $\alpha$.
The worst error on the interval $0 \le \alpha \le \pi/2$ is
$e(\pi/2)$ = 0.000757 radians = 0.0434 degrees.
The worst error on the interval $0 \le \alpha \le \pi$ is
$e(\pi)$ = 0.00630 radians = 0.361 degrees.
That's quite good for such a simple construction.
<h2>An interesting coincidence</h2>
<p>
The angle $\beta$ constructed by this method coincides <em>exactly</em>
with that of <a href="trisect-jamison.html">Lindberg's construction</a>,
where $\beta$ is given as:
\[
\beta
= \frac{1}{4} \alpha + \arctan
\frac{\sin\frac{\alpha}{4}}{2+\cos\frac{\alpha}{4}}.
\]
One way to verify that the seemingly different expressions
for $\beta$ are in fact identical,
is to compare their derivatives. In both cases we have:
\[
\frac{d\beta}{d\alpha} =
\frac{3(1 + \cos\frac{\alpha}{4})}{2(5 + 4\cos\frac{\alpha}{4})}.
\]
<hr width="60%">
<p>
<em>This applet was created by
<a href="http://userpages.umbc.edu/~rostamia">Rouben Rostamian</a>
using
<a href="http://aleph0.clarku.edu/~djoyce/home.html">David Joyce</a>'s
<a href="http://aleph0.clarkU.edu/~djoyce/java/Geometry/Geometry.html">Geometry
Applet</a>
on June 10, 2010.
</em>
<p>
<table width="100%">
<tr>
<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
<td align="right" style="width:200px;">
<a href="http://validator.w3.org/check?uri=referer">
<img src="/~rostamia/images/valid-html401.png" class="noborder" width="88" height="31" alt="Valid HTML"></a>
<a href="http://jigsaw.w3.org/css-validator/check/referer">
<img src="/~rostamia/images/valid-css.png" class="noborder" width="88" height="31" alt="Valid CSS"></a>
</td></tr>
</table>
</body>
</html>