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<h1>An angle trisection</h1>
<h4>Construction by
<a href="mailto:wayne.baker@ed.cna.nl.ca">Wayne Baker</a></h4>
<table class="centered">
<tr><td align="center">
<applet code="Geometry" archive="Geometry.zip" width="570" height="360">
<param name="background" value="ffffff">
<param name="title" value="An angle trisection">
<!-- the angle AOB -->
<param name="e[1]" value="O;point;fixed;260,310">
<param name="e[2]" value="A;point;fixed;520,310">
<param name="e[3]" value="OA;line;connect;O,A;none;none;blue">
<param name="e[4]" value="C;circle;radius;O,A;none;none;none;none">
<param name="e[5]" value="B;point;circleSlider;C,20,0;red;red">
<param name="e[6]" value="OB;line;connect;O,B;none;none;blue">
<param name="e[7]" value="z1;sector;sector;O,A,B;none;none;blue;none">
<!-- point P quadrisects the arc AB -->
<param name="e[8]" value="t1;point;angleBisector;A,O,B;none;none">
<param name="e[9]" value="t2;point;angleBisector;A,O,t1;none;none">
<param name="e[10]" value="Ot2;line;chord;O,t2,C;none;black;none">
<param name="e[11]" value="P;point;first;Ot2">
<param name="e[12]" value="AP;line;connect;A,P;none;none;green">
<!-- the arc A'OB' -->
<param name="e[13]" value="A';point;fixed;455,310">
<param name="e[14]" value="C';circle;radius;O,A';none;none;none;none">
<param name="e[15]" value="t9;line;chord;O,B,C';none;none;none">
<param name="e[16]" value="B';point;first;t9">
<param name="e[17]" value="z2;sector;sector;O,A',B';none;none;orange;none">
<!-- P' -->
<param name="e[18]" value="t3;circle;radius;A',A,P;none;none;none;none">
<param name="e[19]" value="t4;line;bichord;t3,C';none;none;none">
<param name="e[20]" value="P';point;first;t4">
<param name="e[21]" value="s1;line;chord;O,P',C;none;none;none">
<param name="e[22]" value="s2;point;first;s1;none;none">
<param name="e[23]" value="s3;line;connect;O,s2;none;none;red">
<param name="e[24]" value="A'P';line;connect;A',P';none;none;green">
<!-- angle marker -->
<param name="e[25]" value="p1;point;fixed;290,310;none;none">
<param name="e[26]" value="c1;circle;radius;O,p1;none;none;none;none">
<param name="e[27]" value="l1;line;chord;OA,c1;none;none;none">
<param name="e[28]" value="q1;point;first;l1;none;none">
<param name="e[29]" value="l2;line;chord;s3,c1;none;none;none">
<param name="e[30]" value="q2;point;first;l2;none;none">
<param name="e[31]" value="S1;sector;sector;O,q1,q2;none;none;black;orange">
<param name="e[32]" value="l3;line;chord;OB,c1;none;none;none">
<param name="e[33]" value="q3;point;first;l3;none;none">
<param name="e[34]" 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>The Basic Construction</H2>
<p>
Here is a very simple straightedge and compass
construction of an approximate angle trisector due to
<a href="mailto:wayne.baker@ed.cna.nl.ca">Wayne Baker</a>.
<p>
Let us represent the angle by
the circular arc $AB$ centered at $O$; see the diagram above.
The angle's size may be anything from 0 to 180 degrees.
To trisect, do:
<ol>
<li>
Quadrisect the angle $AOB$, that is, divide it into four
equal parts. The arc $AP$ in the diagram above represents one quarter
of the original arc $AB$. Let $L$ be the length of the chord $AP$ (shown in green).
<li>
Draw a circular arc (shown in orange)
centered at $O$ and radius 3/4 of $OA$. Mark $A'$ and $B'$ its intersections
with the rays $OA$ and $OB$, respectively.
<li>
Swing an arc (not shown) of radius $L$ centered at $A'$ and mark $P'$ its intersection with
the arc $A'B'$, as shown.
</ol>
<p>
The line $OP'$ is an approximate trisector of the angle $AOB$.
<h2>Error Analysis</h2>
<p>
Let $\alpha$ and $\beta=\tau(\alpha)$ be the sizes of the angles $AOB$ and $A'OP'$,
respectively. It is straightforward to show that
\[
\beta
= 2 \arcsin\big(\frac{4}{3}\sin\frac{\alpha}{8}\big)
= \frac{\alpha}{3} + \frac{7}{2^7\cdot3^4}\alpha^3 + O(\alpha^5)
= \frac{\alpha}{3} + \frac{7}{10368}\alpha^3 + O(\alpha^5).
\]
<!-- The first two terms of the series are the same as those
in trisect-dunham.html. The third terms are different.
b_baker := 2*arcsin(4/3*sin(a/8));
series(b_baker,a);
b_durham := a/2 - arctan(sin(a/4 - arcsin(sin(a/4)/2))*4/3);
series(b_durham, a);
-->
<p>
The error
$
\ds e(\alpha) = \tau(\alpha) - \frac{\alpha}{3}
$
is monotonically increasing in $\alpha$.
The worst error on the interval $0 \le \alpha \le \pi/2$ is
$e(\pi/2)$ = 0.002695 radians = 0.154 degrees.
The worst error on the interval $0 \le \alpha \le \pi$ is
$e(\pi)$ = 0.0237 radians = 1.360 degrees.
<h2>Iterative Improvement</h2>
<p>
As we see in the asymptotic expansion shown above, the
angle $\tau(\alpha)$ is slightly larger than the target value of $\alpha/3$.
Making three copies of the constructed angle, and putting them
end-to-end as in arcs $A'P'$, $P'P''$, and $P''P'''$ shown in the diagram below,
we arrive at the endpoint $P'''$ which is very slightly off the point $B'$,
and just outside the arc $A'B'$. The constructible angle $B'OP'''$ is exactly
three times the error $e(\alpha)$.
If we were able to trisect $B'OP'''$ exactly, then we
would know the error, and consequently will have achieved
the exact trisection of the original angle.
Of course the exact trisection of $B'OP'''$ is impossible in general, but we
may repeat the method outlined in the <em>Basic Construction</em> above
to obtain an <em>approximate</em> trisection of $B'OP'''$,
which will yield $ \tau\big(3\tau(\alpha) - \alpha\big) $,
and consequently an improved trisection $\tau_{\mathrm{improved}}(\alpha)$
of the original angle:
\[
\tau_{\mathrm{improved}}(\alpha) = \tau(\alpha) - \tau\big(3\tau(\alpha) - \alpha\big)
= \frac{\alpha}{3} - \frac{7^4}{2^{28}\cdot3^{13}} \alpha^9 +
O(\alpha^{11}).
\]
The error
$ \ds e_{\mathrm{improved}}(\alpha) = \frac{\alpha}{3} - \tau_{\mathrm{improved}}(\alpha)$
is monotonically increasing in $\alpha$. In particular,
$e_{\mathrm{improved}}(\pi/2) = 1.5\times 10^{-9}$ radians
$ = 8.6\times10^{-8}$ degrees.
<table class="centered">
<tr><td align="center">
<applet code="Geometry" archive="Geometry.zip" width="570" height="360">
<param name=background value="ffffff">
<param name=title value="An angle trisection">
<!-- the angle AOB -->
<param name="e[1]" value="O;point;fixed;260,310">
<param name="e[2]" value="A;point;fixed;520,310">
<param name="e[3]" value="OA;line;connect;O,A">
<param name="e[4]" value="C;circle;radius;O,A;none;none;none;none">
<param name="e[5]" value="B;point;circleSlider;C,50,0">
<param name="e[6]" value="OB;line;connect;O,B">
<param name="e[7]" value="z1;sector;sector;O,A,B;none;none;blue;none">
<!-- point P quadrisects the arc AB -->
<param name="e[8]" value="t1;point;angleBisector;A,O,B;none;none">
<param name="e[9]" value="t2;point;angleBisector;A,O,t1;none;none">
<param name="e[10]" value="Ot2;line;chord;O,t2,C;none;black;none">
<param name="e[11]" value="P;point;first;Ot2">
<param name="e[12]" value="OP;line;connect;O,P;none;none;none">
<!-- the arc A'OB' -->
<param name="e[13]" value="A';point;fixed;455,310">
<param name="e[14]" value="C';circle;radius;O,A';none;none;none;none">
<param name="e[15]" value="t9;line;chord;O,B,C';none;none;none">
<param name="e[16]" value="B';point;first;t9">
<param name="e[17]" value="z2;sector;sector;O,A',B';none;none;orange;none">
<!-- P' -->
<param name="e[18]" value="t3;circle;radius;A',A,P;none;none;none;none">
<param name="e[19]" value="t4;line;bichord;t3,C';none;none;none">
<param name="e[20]" value="P';point;first;t4">
<param name="e[21]" value="s1;line;chord;O,P',C;none;none;none">
<param name="e[22]" value="s2;point;first;s1;none;none">
<param name="e[23]" value="s3;line;connect;O,s2;none;none;red">
<!-- P'' -->
<param name="e[24]" value="t5;circle;radius;P',A,P;none;none;none;none">
<param name="e[25]" value="t6;line;bichord;t5,C';none;none;none">
<param name="e[26]" value="P'';point;first;t6">
<!-- P'''
Note the trailing spaces after P'''. These become a part of the label!
-->
<param name="e[27]" value="t7;circle;radius;P'',A,P;none;none;none;none">
<param name="e[28]" value="t8;line;bichord;t7,C';none;none;none">
<param name="e[29]" value="P''' ;point;first;t8">
<param name="e[30]" value="OP''';line;connect;O,P''' ;none;none;black">
<param name="e[31]" value="u1;sector;sector;O,B',P''' ;none;none;none;magenta">
</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 $AOB$.<br>
The arcs $P'P''$ and $P''P'''$ are copies of $A'P'$. The endpoint $P'''$
is just slightly off the point $B'$.<br>
The (very small and nearly indiscernible)
angle $B'OP'''$ is three times the trisection error.
</b>
</td></tr></table>
<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
May 31, 2010.
</em>
<p>
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<td valign="top">Go to <a href="index.html#trisections">list of trisections</a></td>
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