```Date: May 31, 2013 3:18 AM
Author: emammendes@gmail.com
Subject: Re: Help needed on how plot a stereographic projection

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 onlything I can is to rotate the figure.b)      Is there a way to change the line color (tab3) from cold (blue) tohot (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@gmail.com] Sent: Thursday, May 30, 2013 4:03 PMTo: Eduardo M. A. M. MendesCc: MathGroupSubject: Re: Help needed on how plot a stereographic projection Use of ComplexExpand on real values (output of Re, Im, or Abs) isunnecessary.ClearAll[stereographicProjection];stereographicProjection::usage =   "stereographicProjection[complexnumber] will return the stereoprojectionof \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@gmail.com> wrote:HelloAlthough I have been using Mathematica for more than year, I feel that Ihaven't barely scratched the surface of what Mathematica can do.The following example gives the result that I need but the outcome is uglyand slow.ClearAll[stereographicProjection];stereographicProjection::usage="stereographicProjection[complexnumber] willreturn the stereoprojection of a complex point considering the Riemannsphere";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],ImageSize-> =Large],ListPointPlot3D[{stereographicProjection[-1]},PlotStyle->Directive[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 thanksEd
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