Sándor Kabai
Modeling pseudosphere with cones
A tractrix is the curve traced by a point connected to another point by a segment (leash) of fixed length as the second point moves along a straight line. The pseudosphere is the solid obtained by revolving the tractrix about its asymptote (the aforementioned straight line). In this Demonstration, the pseudosphere is approximated by a series of cones formed by revolving the leash about the asymptote.
Modeling pseudosphere with cones
A tractrix is the curve traced by a point connected to another point by a segment (leash) of fixed length as the second point moves along a straight line. The pseudosphere is the solid obtained by revolving the tractrix about its asymptote (the aforementioned straight line). In this Demonstration, the pseudosphere is approximated by a series of cones formed by revolving the leash about the asymptote.
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Manipulate[Module[{tat,v1,v2,plane,p1,p2,c1,cc},tat=ParametricPlot3D[{(k-Tanh[k]),Sech[k],0},{k,-3,3}][[1]];v1={Red,Thickness[0.006],Line[{{-2,0,0},{2,0,0}}]};v2={Green,Thickness[0.008],Specularity[1],Line[{{(t-Tanh[t]),Sech[t],0},{(t-Tanh[t])+Sqrt[1-Sech[t]^2],0,0}}]};plane={RGBColor[1,0.8,0.7],Polygon[{{-2,-0.3,0},{2,-0.3,0},{2,1.3,0},{-2,1.3,0}}]};p1=Sphere[{(t-Tanh[t]),Sech[t],0},.04];p2=Sphere[{(t-Tanh[t])+Sqrt[1-Sech[t]^2],0,0},.04];c1=Graphics3D[Cone[{{(t-Tanh[t]),0,0},{(t-Tanh[t])+Sqrt[1-Sech[t]^2],0,0}},Sech[t]]][[1]];cc=Graphics3D[Table[Cone[{{sep(ip-Tanh[ip]),0,0},{sep(ip-Tanh[ip])+Sqrt[1-Sech[ip]^2],0,0}},Sech[ip]],{i,noc}]][[1]];Graphics3D[{v1,v2,tat,plane,p1,p2,If[c1s,c1,{}],cc},BoxedFalse,SphericalRegionTrue,PlotRange->{{-2,3},{-1.4,1.9},{-1,1}},ViewPoint{0,0,5},ImageSize{550,300},ViewAngle0.09]],{{t,1.3,"move leash"},0.01,2,.01,Appearance"Labeled"},{{c1s,False,"show cone"},{True,False}},{{p,0.25,"pitch"},0.1,0.5,.01,Appearance"Labeled"},{{noc,4,"number of cones"},0,12,1,RadioButton},{{sep,1,"separate cones"},1,2,.01,Appearance"Labeled"},TrackedSymbols->{t,c1s,p,noc,sep}]
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