ÿþ<html> <head> <left><title>Approximation</title> </head> <body bgcolor="eeffee"> <hr size="4" color="#0000FF"> Go to &nbsp;<A HREF="../../../Fun_Math_by_CAD.html"> Fun_Math Content Table </A> &nbsp;<A HREF="./../Angle_Trisection.html"> Trisecting an Angle </A> <hr size="4" color="#0000FF"> <h2>Approximation </h2> <h3>1. Sum of Infinite Series </h3> <P> This approximate procedure is using the property of the following sum of the geometric series. <p> S<sub>n</sub>= 1/4 + (1/4)<sup>2</sup> + (1/4)<sup>3</sup>+ (1/4)<sup>4</sup>+ . . . + (1/4)<sup>n</sup> <p>As n increases, this series sum S<sub>n</sub> approaches 1/3 . <p>The actual trisection process is using angle bisection repeatedly . <p> For example, 1/4 = 1/2 - 1/4, (1/4)<sup>2</sup> = (1/4){(1/2) - (1/4)}, etc <p> <IMG SRC="approximation_series_tri_desc.gif" width="400" height="400" align=left> <P> <h4>Trisection Process </h4> <P>Angle to be trisected is AOB. <p>n=1 ;bisect angle AOB (point 1) <p>n=2 ;bisect angle 1OB (point 2)--Angle 2OB = 1/4 now <p>n=3 ;bisect angle 1O2 (point 3) <p>n=4 ;bisect angle 3O2 (point 4)--Angle 4Ob = 1/4 + (1/4)<sup>2</sup> <p>n=5 ;bisect angle 3O4 <p>n=6 ;bisect ... etc <p>You can see the process in <a href=Series_Trisection_anim.gif><b>animation</b></a>. <br clear=left> ******<b> <a href="approximation_series_tri_desc.dwg">approximation_series_tri_desc.dwg</a></b> ****** <p> <p><b>To create this drawing and animation: </b> <BR><b>&nbsp;&nbsp; Load <a href="gen_sect.lsp">gen_sect.lsp</a> &nbsp;&nbsp; (load "gen_sect")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> trisect_by_series </b> <br>Animation files Creation: <b>animation_series</b><p> <h3>2. von Cusa and Snellius </h3> <P>This is credited to the 15-th century mathematician, Nikolas von Cusa, and the 16-th century mathematician, Willebrod Snellius. <p> <IMG SRC="VonCusa_Snellius_desc.gif" width="400" height="400" align=left> <P> <h4>Trisection Process </h4> <P>Draw a semi-cirlce OBE. Pick a point P on BE such that PE = EO. <p>Select a point "A" on the semi-circle. AOB is the angle to be trisected. <p>Erect a line GB perpendicular to EB at B. <p>Draw a line from P through A to intersect with GB at D. <p>Find a point M on line BD such that BM = BD/3 . <p>Connect P and M. This will intersect the semi_circle at point T. <p>Line segment TO is the approximate trisecting line for angle AOB. <p>In 60 degrees case ,the error is about 9'02",which is very very good !! <p> <br clear=left> ********<b> <a href="VonCusa_Snellius_desc.dwg">VonCusa_Snellius_desc.dwg</a></b> ********* <p> <p><b>To create this drawing and animation: </b> <BR><b>&nbsp;&nbsp; Load <a href="VonCusa.lsp">VonCusa.lsp</a> &nbsp;&nbsp; (load "VonCusa")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> VonCusa </b> <h3>3. Dürer's approximation </h3> <P> In 1525 Albrecht Dürer published the following simple and very accurate approximation for angle trisection. <p> <IMG SRC="Durer_desc.gif" width="400" height="400" align=left> <P> <h4>Trisection Process </h4> <P>Angle to be trisected is AOB. <p>Let QA = QB = 1, and M1 & M2 divide line segment AB into three equal parts. <p>At M1 & M2, erect lines perpendicular to AB to intersect the arc C1 & C2 . <p>Find point D such that AD = AC1 <p>Find point E such that DE = M1D/3 <p>Draw a circle with center at A and radius AE. This circle intersect the arc at point T. <p>Line TO is the approximate trisetor of angle AOB <p> <br clear=left> **************<b> <a href="Durer_desc.dwg">Durer_desc.dwg</a></b> *************** <p> <p><b>To create this drawing and animation: </b> <BR><b>&nbsp;&nbsp; Load <a href="Durer.lsp">Durer.lsp</a> &nbsp;&nbsp; (load "Durer")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> Durer </b> <h3>4. Karajordanoff's approximation </h3> <P> This simple approximation procedure was dicovered by Karajordanoff in 1928. <p> <IMG SRC="Karajordanoff_desc.gif" width="400" height="400" align=left> <P> <h4>Trisection Process </h4> <P>Angle to be trisected is AOB. <p>two circles of radii 1 and 2 are drawn abou the angle AOB. <p>Extend a line from B passing through point C, a midchord point of AB. <p>Draw a tangent line at point A. <p>This tangent and line BC intersects at point D <p>Draw a line DT parallel to OB. T is the point this line intersects the arc AB. <p>Line TO is the approximate trisector. <p>Trisecting error for the given angle 60 degrees is about 2 minutes. <br clear=left> **********<b> <a href="Karajordanoff_desc.dwg">Karajordanoff_desc.dwg</a></b> *********** <p> <p><b>To create this drawing and animation: </b> <BR><b>&nbsp;&nbsp; Load <a href="Karajordanoff.lsp">Karajordanoff.lsp</a> &nbsp;&nbsp; (load "Karajordanoff")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> Karajordanoff </b> <h3>5. Kopf-Perron Approximation </h3> <P> This approximation method was published by Kopf in 1919, and later it was refined by Perron and d'Ocagne. <p> <IMG SRC="Kopf-Perron_desc.gif" width="400" height="400" align=left> <P> <h4>Trisection Process </h4> <P>Angle to be trisected is AOB. <p>Point5s A,B & C are on the unit circle. D is the mid point of OC, and ED is vertical to OC. <p>POint F is set such that DF is 1/3 of line length ED, and CP = CO= 1. <p>Draw an arc with its center at point F and radius equal to FB. <p>Line CA intersects this arc at point T. <p>Line TP is the approximate trisector. <p>Trisecting error for the given angle 60 degrees is about 13 minutes. <p> <br clear=left> ***********<b> <a href="Kopf-Perron_desc.dwg">Kopf-Perron_desc.dwg</a></b> ************ <p> <p><b>To create this drawing : </b> <BR><b>&nbsp;&nbsp; Load <a href="Kopf_Perron.lsp">Kopf_Perron.lsp</a> &nbsp;&nbsp; (load "Kopf_Perron")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> Kopf_Perron </b> <h3>6. D'Ocagne's Approximation </h3> <P> This extremely simple method that is surprisingly accurate for smalll angles is given by d'Ocagne.(published in 1934) <p> <p> <IMG SRC="D'Ocagne_desc.gif" width="400" height="400" align=left> <P> <h4>Trisection Process </h4> <P>Angle to be trisected is AOB. <p>Points A & B are on the unit circle. <p>Point C is the mid point of the unit radius. <p>Point M is the midpoint of the chord AB <p>The line CM is approximate trisecting line. <p>In this example case (Angle AOB = 60 degrees), the error is only 6'14",which is a very good approximation. <p> <br clear=left> ************<b> <a href="D'Ocagne_desc.dwg">D'Ocagne_desc.dwg</a></b> ************* <p> <p><b>To create this drawing : </b> <BR><b>&nbsp;&nbsp; Load <a href="D'Ocagne.lsp">D'Ocagne.lsp</a> &nbsp;&nbsp; (load "D'Ocagne")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> Docagne </b> <h3>7. Comparison of methods </h3> <p> <IMG SRC="compare_errors.gif" width="400" height="400" align=left> <p> The errors of 5 approximation methods are shown in the graph. <p>Notice that Y-axis is in logarithm scale because the error values(in seconds) range from less than 0.01 to 5000. <p>Values less than 0.1 are all set to 0.1 for display purpose. <p> <br clear=left> ************<b> <a href="compare_errors.dwg">compare_errors.dwg</a></b> ************* <p> <p> <p><b>To create this drawing : </b> <BR><b>&nbsp;&nbsp; Load <a href="compare_errors.lsp">compare_errors.lsp</a> &nbsp;&nbsp; (load "compare_errors")</b> <BR>&nbsp;&nbsp;Then from command line, type <b> compare_errors </b> <P> <h4>References </h4> <p> 1. Yates,Robert Carl : "The Trisection problem" , pp.47-56 <br> <hr size="4" color="#0000FF"> Go to &nbsp; <A HREF="../../../Fun_Math_by_CAD.html"> Fun_Math Content Table </A> &nbsp;<A HREF="./../Angle_Trisection.html"> Trisecting an Angle </A> <hr size="4" color="#0000FF"> <p align="center"></b>All questions/suggestions should be sent to <a href="mailto:takaya.iwamoto@comcast.net">Takaya Iwamoto</a></p> <p align="center">Last Updated Nov 22, 2006</p> <p align="center">Copyright 2006 Takaya Iwamoto &nbsp; All rights reserved. </Body>. </body></html>