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Re: Help needed on how plot a stereographic projection

• To: mathgroup at smc.vnet.net
• Subject: [mg130981] Re: Help needed on how plot a stereographic projection
• From: "Eduardo M. A. M.Mendes" <emammendes at gmail.com>
• Date: Fri, 31 May 2013 03:39:18 -0400 (EDT)
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• References: <20130530101424.F340969EC@smc.vnet.net> <CAEtRDSf8_7Q2P9ZPa35BnC0=a6-aV9PdB4dHLe-sP2ma3r7B1w@mail.gmail.com>

Hello

 

Many many thanks.

 

A couple of questions if I may:

 

a)      Does the ToolTip command means that if the mouse over -1 the msg
"Project of -1"will show up?  It does not seem to work for me.  The only
thing I can is to rotate the figure.

b)      Is there a way to change the line color (tab3) from cold (blue) to
hot (read) as  goes from 0 to infinity and 0 to -infinity? 

c)       Is there a way to get grid lines on the planes x-y,y-z,z-x?

d)      Another way, faster, to generate tab3.

 

Once more, thank you.  

 

Ed

 

 

From: Bob Hanlon [mailto:hanlonr357 at gmail.com] 
Sent: Thursday, May 30, 2013 4:03 PM
To: Eduardo M. A. M. Mendes
Cc: MathGroup
Subject: [mg130981] Re: Help needed on how plot a stereographic projection

 

Use of ComplexExpand on real values (output of Re, Im, or Abs) is
unnecessary.


ClearAll[stereographicProjection];


stereographicProjection::usage = 
  "stereographicProjection[complexnumber] will return the stereoprojection
of \
a complex point considering the Riemann sphere";


SyntaxInformation[stereographicProjection] = {"ArgumentsPattern" -> {_}};


stereographicProjection[complexnumber_] :=
 Module[
  {abs2 = Abs[complexnumber]^2},
  If[abs2 == Infinity,
   {0, 0, 1},
   {Re[complexnumber]/(1 + abs2),
    Im[complexnumber]/(1 + abs2),
    abs2/(1 + abs2)}]]


tab3 = Table[
    stereographicProjection[
     (s + 1)/(s^2 (s - 1)) /. {s -> I w}],
    {w, -1000, 1000, 0.1}] // Quiet;


Show[
 ParametricPlot3D[
  {Cos[p] Sin[t], Sin[p] Sin[t], 1 + Cos[t]}/2,
  {p, 0, 2 Pi}, {t, 0, Pi},
  PlotStyle -> Opacity[0.5],
  Mesh -> Automatic],
 Graphics3D[{
   Darker[Magenta],
   AbsoluteThickness[3],
   Tooltip[Line[tab3],
    "Projection of (s+1)/(s^2 (s-1))"],
   Red,
   PointSize[.02],
   Tooltip[Point[{stereographicProjection[-1]}],
    "Projection of -1"]}],
 ImageSize -> Large,
 AxesLabel -> {"x", "y", "z"},
 PlotRange -> {{-1, 1}, {-1, 1}, {0, 1}},
 BoxRatios -> {1, 1, 1/2}]

 

 

Bob Hanlon

 

 

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

Hello

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[complexnumber]]^
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],Magenta],I
mageSize-> =
Large],ListPointPlot3D[{stereographicProjection[-1]},PlotStyle->Directive[Po
intSize[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

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