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Re: Help needed on how plot a stereographic projection

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• Subject: [mg130980] Re: Help needed on how plot a stereographic projection
• From: "Eduardo M. A. M.Mendes" <emammendes at gmail.com>
• Date: Fri, 31 May 2013 03:38:58 -0400 (EDT)
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Hello

Many many thanks.

David has pointed out the examples on the stereographic projection.  I
bought David's package a couple of years ago and I keep forgetting the many
nice functions given there.

Presentations will give a nice plot, I am sure of it.  

Cheers

Ed


-----Original Message-----
From: Murray Eisenberg [mailto:murray at math.umass.edu] 
Sent: Thursday, May 30, 2013 3:15 PM
To: mathgroup
Subject: [mg130980] Re: Help needed on how plot a stereographic projection

You may wish to take a look at David Park's "Presentations" add-on:

  http://home.comcast.net/~djmpark/DrawGraphicsPage.html

This includes, among a host of useful things, a built-in stereographic
projection function as well as built-in functions for displaying graphics
objects defined directly in terms of complex numbers (i.e. without your
having to tear them apart in to real and imaginary part) and complex
functions, including the Riemann sphere.

Some examples of what's possible with complex numbers and complex functions,
among other things (and aside from those in the extensive Presentations
documentation), are available at 

  http://www.st-andrews.ac.uk/~pl10/c/djmpark/

(you can view the pdf versions there without needing the package) and in the
paper "Visualizing Complex Functions with the Presentations Application" 

  http://www.mathematica-journal.com/issue/v11i2/EisenbergPark.html

written by David and me. That paper, which you may view in static form in
the pdf version, includes an example of using stereographic projection to
lift a complex function to the Riemann sphere.

On May 30, 2013, at 6:14 AM, Eduardo M. A. M. Mendes <emammendes at gmail.com>
wrote:

> 
> Although I have been using Mathematica for more than year, I feel that I
haven't barely scratched the surface of what Mathematica can do.  
> 
> The following example gives the result that I need but the outcome is ugly
and slow.
> 
> ClearAll[stereographicProjection];
> 
> stereographicProjection::usage="stereographicProjection[complexnumber] 
> will return the stereoprojection of a complex point considering the 
> Riemann sphere";
> 
> SyntaxInformation[stereographicProjection]={"ArgumentsPattern"->{_}};
> 
> stereographicProjection[complexnumber_]:=
> Module[{a1,a2,a3},
> If[ComplexExpand[Abs[complexnumber]]==Infinity,
> a1=0;a2=0;a3=1,
> =
> a1=ComplexExpand[Re[complexnumber]]/(1+ComplexExpand[Abs[complexnumber
> ]]^2);
> 
> a2=ComplexExpand[Im[complexnumber]]/(1+ComplexExpand[Abs[complexnumber
> ]]^2);
> =
> a3=ComplexExpand[Abs[complexnumber]]^2/(1+ComplexExpand[Abs[complexnum
> ber]]^2)];
> {a1,a2,a3}
> ]
> 
> tab3=Table[stereographicProjection[(s+1)/(s^2 (s-1))/.{s-> I 
> \[Omega]}],{\[Omega],-1000,1000,0.1}];
> 
> =
> Show[ContourPlot3D[x^2+y^2+(z-1/2)^2==(1/2)^2,{x,-1,1},{y,-1,1},{z,0,1
> },Mesh->Automatic,AxesLabel-> = 
> {"x","y","z"},BoxRatios->{1,1,1/2},ImageSize-> = 
> Large],ListPointPlot3D[tab3,PlotStyle->Directive[PointSize[Large],Mage
> nta],ImageSize-> = 
> Large],ListPointPlot3D[{stereographicProjection[-1]},PlotStyle->Direct
> ive[PointSize[0.02],Red]],ImageSize-> Large]
> 
> a) Is there another way of getting the same plot?
> b) How to get the points of tab3 connected? 
> c) How to change the opacity of the sphere?
> 
> 
> Improvements, suggestion and critiscims are welcome.  
> 
> Many thanks
> 
> Ed
> 

---
Murray Eisenberg                                    murray at math.umass.edu
Mathematics & Statistics Dept.       
Lederle Graduate Research Tower            phone 413 549-1020 (H)
University of Massachusetts                               413 545-2838 (W)
710 North Pleasant Street                         fax   413 545-1801
Amherst, MA 01003-9305

• References:
  • Help needed on how plot a stereographic projection
    • From: "Eduardo M. A. M. Mendes" <emammendes@gmail.com>