File:Circle2cardioid.png
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[edit]DescriptionCircle2cardioid.png |
English: Conformal mapping from circle to cardioid Polski: Przekształcenie koła w kardioidę |
Date | |
Source |
Own work by uploader ( using Maxima and Gnuplot ) with help of many people This plot was created with Gnuplot by n. |
Author | Adam majewski |
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Long description
[edit]Cardioid is an image of circle under complex map [1]. It can be seen in nature[2]
where :
- is a circle with center at the origin and radius=1/2
- Cardioid is a boundary or period=1 hyperbolic component of Mandelbrot set
- is a complex map from circle to cardioid
or in other words :
Explicit equation of this map is :
Compare with inverse multiplier map [3]:
Maxima source code
[edit]/* batch file for maxima conformal mapping from : the circle with center=0 and radius=1/2 given by equation : abs(2*z)=1 where : z:x+y*%i=r*cos(t)+%i*r*sin(t) = onto cardioid ( boundary of main hyperbolic component of Mandelbrot set is given by equation: c:w-w*w; based on :Conformal Mappings And The Area Of The Mandelbrot Set by David Allingham page 18 http://www.eng.warwick.ac.uk/staff/doa/reports/allingham-thesis1995.pdf archive copy at the Wayback Machine */ /* conformal map */ f(w):=w-w*w; iMax:100; /* number of points to draw */ dt:1/iMax; /* circle D={w:abs(2w)=1 } where w=l(t) t is angle in turns ; 1 turn = 360 degree = 2*Pi radians */ l(t):=%e^(%i*t*2*%pi)/2; /* point to point method of drawing */ t:0; /* angle in turns */ /* compute first point of curve, create list and save point to this list */ /* point of unit circle w:l(t); */ w:rectform(ev(l(t), numer)); circleList:makelist (w, j, 1, 1); for i:1 thru iMax step 1 do block (t:t+dt, w:rectform(ev(l(t), numer)), circleList:cons(w,circleList) ); /* convert one list into 2 lists */ xxCircle:map(realpart, circleList); yyCircle:map(imagpart, circleList); /* conformal mapping from circle to cardioid */ cardioidList: map(f, circleList); /* convert one list into 2 lists */ xxCardioid:map(realpart, cardioidList); yyCardioid:map(imagpart, cardioidList); load(draw); /* Mario Rodríguez Riotorto http://www.telefonica.net/web2/biomates/maxima/gpdraw/index.html archive copy at the Wayback Machine */ draw(file_name = "a", pic_width=1000, pic_height= 500, terminal = 'png, columns = 2, gr2d(title = " circle D={w:abs(2*w)=1 } ", points_joined =true, color = red, point_type = 0, points(xxCircle,yyCircle)), gr2d(title = "cardioid {c: c = w - w*w } ", points_joined =true, color = blue, point_type = 0, points(xxCardioid,yyCardioid)) );
References
[edit]- ↑ 3D-XplorMath \ Conformal Maps \ a*z^b+b*z
- ↑ Cardioid as a shadow of ring. Foto by Sebastian Tkacz
- ↑ Boundary of components of Mandelbrot set computed using boundary equations
Licensing
[edit]I, the copyright holder of this work, hereby publish it under the following licenses:
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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.http://www.gnu.org/copyleft/fdl.htmlGFDLGNU Free Documentation Licensetruetrue |
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Date/Time | Thumbnail | Dimensions | User | Comment | |
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current | 11:01, 4 October 2008 | 1,000 × 500 (7 KB) | Soul windsurfer (talk | contribs) | {{Information |Description=circle is not unit circle |Source= |Date= |Author= |Permission= |other_versions= }} | |
10:03, 4 October 2008 | 1,000 × 500 (7 KB) | Soul windsurfer (talk | contribs) | {{Information |Description={{en|1=Conformal mapping from circle to cardioid}} |Source=Own work by uploader |Author=Adam majewski |Date=4.10.2008 |Permission= |other_versions= }} <!--{{ImageUpload|full}}--> |
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