feat: Start implementing Rostamian's pages (#39)
Began with incenter.html, the first one alphabetically. Needed one new point construction method, and a new option to see what was going on. Got the planar diagrams on that page working. The next step on #36 will be to get 3D diagrams as the theorem on this page generalizes to 3D. That will be a bigger task, so merging this now. Reviewed-on: #39 Co-authored-by: Glen Whitney <glen@studioinfinity.org> Co-committed-by: Glen Whitney <glen@studioinfinity.org>
This commit is contained in:
parent
e7361f94a7
commit
c99b51dafa
21 changed files with 3549 additions and 37 deletions
154
public/rostamian/trisect-cdsmith.html
Normal file
154
public/rostamian/trisect-cdsmith.html
Normal file
|
|
@ -0,0 +1,154 @@
|
|||
<!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 by <a href="http://cdsmith.wordpress.com/2009/04/20/old-memories-about-trisecting-angles/">cdsmith</a>
|
||||
</h4>
|
||||
|
||||
<table class="centered">
|
||||
<tr><td align="center">
|
||||
<applet code="Geometry" archive="Geometry.zip" width="600" height="450">
|
||||
<param name="background" value="ffffff">
|
||||
<param name="title" value="An angle trisection">
|
||||
|
||||
<param name="e[1]" value="O;point;fixed;250,340">
|
||||
<param name="e[2]" value="A;point;fixed;450,340">
|
||||
<param name="e[3]" value="cir1;circle;radius;O,A;none;none;none;none">
|
||||
<param name="e[4]" value="B;point;circleSlider;cir1,400,0;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="AB;line;connect;A,B;none;none;lightGray">
|
||||
<param name="e[8]" value="M;point;midpoint;A,B">
|
||||
<param name="e[9]" value="cA;circle;radius;A,M;none;none;green;none">
|
||||
<param name="e[10]" value="cM;circle;radius;M,A;none;none;green;none">
|
||||
<param name="e[11]" value="cB;circle;radius;B,M;none;none;green;none">
|
||||
<param name="e[12]" value="li1;line;bichord;cA,cM;none;none;none">
|
||||
<param name="e[13]" value="C;point;first;li1">
|
||||
<param name="e[14]" value="li2;line;bichord;cM,cB;none;none;none">
|
||||
<param name="e[15]" value="D;point;first;li2">
|
||||
|
||||
<param name="e[16]" value="OC;line;connect;O,C;none;none;red">
|
||||
<param name="e[17]" value="OD;line;connect;O,D;none;none;red">
|
||||
|
||||
<!-- angle marker -->
|
||||
<param name="e[18]" value="p1;point;fixed;285,340;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,C,c1;none;none;none">
|
||||
<param name="e[23]" value="q2;point;first;l2;none;none">
|
||||
<param name="e[24]" value="s1;sector;sector;O,q1,q2;none;none;black;orange">
|
||||
<param name="e[25]" value="l3;line;chord;OB,c1;none;none;none">
|
||||
<param name="e[26]" value="q3;point;first;l3;none;none">
|
||||
<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 lines $OC$ and $OD$ are approximate trisectors of the angle $AOB$.
|
||||
</b>
|
||||
</td></tr></table>
|
||||
|
||||
<h2>Construction</h2>
|
||||
|
||||
<p>
|
||||
To trisect the angle $AOB$ (with $OA=OB$), do:
|
||||
|
||||
<ol>
|
||||
|
||||
<li>
|
||||
Find the midpoint $M$ of the line segment $AB$.
|
||||
|
||||
<li>
|
||||
Draw circles centered at $A$, $M$, and $B$, each of radius $\frac{1}{2}AB$,
|
||||
and mark their intersection points $C$ and $D$, as shown in the diagram
|
||||
above.
|
||||
|
||||
</ol>
|
||||
|
||||
The lines $OC$ and $OD$ are approximate trisectors of the angle $AOB$.
|
||||
|
||||
<h2>Error Analysis</h2>
|
||||
|
||||
<p>
|
||||
Let $\alpha$ and $\beta$ be the sizes of the angles $AOB$ and $AOC$,
|
||||
respectively. One may verify that
|
||||
\[
|
||||
\beta =
|
||||
\arctan\bigg(
|
||||
\frac{
|
||||
\sin\frac{\alpha}{2}
|
||||
\sin\big(\frac{\pi}{6}+\frac{\alpha}{2}\big)
|
||||
}{
|
||||
1 +
|
||||
\sin\frac{\alpha}{2}
|
||||
\cos\big(\frac{\pi}{6}+\frac{\alpha}{2}\big)
|
||||
}
|
||||
\bigg)
|
||||
=
|
||||
\frac{1}{4}\alpha + \frac{\sqrt{3}}{16} \alpha^2 - \frac{1}{16} \alpha^3
|
||||
+ O(\alpha^4).
|
||||
\]
|
||||
|
||||
<p>
|
||||
This says that $\ds \beta \approx \frac{1}{4}\alpha$ when $\alpha$ is small,
|
||||
so small angles are quadrisected, rather than trisected!
|
||||
(This is clearly visible in the interactive diagram above.)
|
||||
For not-so-small angles,
|
||||
the method works reasonably well. In fact, it produces
|
||||
<em>exact trisection</em> for angles $\alpha=\pi/2$ and $\alpha=\pi$.
|
||||
|
||||
<p>
|
||||
The worst error in
|
||||
the range $0 \le \alpha \le \pi$ is
|
||||
0.0214 radians = 1.23 degrees.
|
||||
This occurs at
|
||||
$\alpha=2\arctan(\sqrt{3}\pm\sqrt{2})$ which corresponds to
|
||||
angles of approximately 35 degrees and 145 degrees.
|
||||
|
||||
<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 14, 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>
|
||||
Loading…
Add table
Add a link
Reference in a new issue