Hyperbola Inverse Curve

Author

Eric W. Weisstein
April 5, 2009

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/HyperbolaInverseCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HyperbolaInverseCurve.html.

©2009 Wolfram Research, Inc. except for portions noted otherwise

General

inv=InverseCurve[{a Sec[t],b Tan[t]},{x0,y0,k},t]

Inversion Center at Center

FullSimplify[inv/.{x0->0,y0->0,b->a}]

FullSimplify[inv/.{x0->0,y0->0,a->1,b->1,k->1}]

Simplify[(1/(Cot[t]+2 Tan[t]))-((Sin[2t])/(3-Cos[2t]))]

ParametricPlot[{-((2 Cos[t])/(-3+Cos[2 t])),(1/(Cot[t]+2 Tan[t]))},{t,0,2π},PlotStyle->Red,AspectRatio->Automatic,Ticks->None]

GraphicsGrid[{Show[Function[i,InverseCurvePlot[{i Sec[t],Tan[t]},{0,0,#},{t,-Pi/2,Pi/2},PlotRange->2]]/@{-1,1}]&/@{.5,1,1.5,2}},ImageSize->400,Spacings->70]

Inversion Center at Focus

FullSimplify[inv/.{x0->Sqrt[2]a,y0->0,b->a}]

FullSimplify[inv/.{x0->Sqrt[2],y0->0,b->1,a->1,k->1}]

ParametricPlot[{((4 (Sqrt[2])-6 Cos[t])/(5-4 (Sqrt[2]) Cos[t]+Cos[2 t])),((Sin[t])/(-2 (Sqrt[2])+Cos[t]+2 Sec[t]))},{t,0,2π},PlotStyle->Red,AspectRatio->Automatic,Ticks->None]

GraphicsGrid[{Show[Function[i,InverseCurvePlot[{i Sec[t],Tan[t]},{Sqrt[2],0,#},{t,-Pi/2,Pi/2},PlotRange->3.9]]/@{-1,1}]&/@{.5,1,1.5,2}},ImageSize->400,Spacings->70]

Inversion Center at Vertex

FullSimplify[inv/.{x0->a,y0->0,b->a}]

ParametricPlot[{(1/2) (2+Cos[t]),((Cos[t] Sin[t])/(2-2 Cos[t]))},{t,0,2π},PlotStyle->Red,AspectRatio->Automatic,Ticks->None,PlotRange->{-1,1}]

GraphicsGrid[{Show[
Function[i,{
InverseCurvePlot[{i Sec[t],Tan[t]},{1,0,#},{t,-Pi/2,0},PlotRange->{{-1.2,1.2},{-1,1}}3],
InverseCurvePlot[{i Sec[t],Tan[t]},{1,0,#},{t,0,Pi/2},PlotRange->{{-1.2,1.2},{-1,1}}3]
}]/@{-1,1}]&/@{.5,1,1.5,2}},Spacings->50,ImageSize->400]//Quiet

Hyperbola with a/b==(Sqrt[3]) and Inversion Center at Vertex

FullSimplify[inv/.{x0->Sqrt[3]b,y0->0,a->Sqrt[3]b}]

ParametricPlot[{(((Sqrt[3]) (-4+Cos[t]))/(2 (-2+Cos[t]))),((Cos[t] Cot[(t/2)])/(4-2 Cos[t]))},{t,0,2π},PlotStyle->Red,AspectRatio->Automatic,Ticks->None,PlotRange->{-1,1}]

GraphicsGrid[{Show[
Function[i,{
InverseCurvePlot[{i Sqrt[3] Sec[t],Tan[t]},{Sqrt[3],0,#},{t,-Pi/2,0},PlotRange->{{-1.2,1.2},{-1,1}}3],
InverseCurvePlot[{i Sqrt[3]Sec[t],Tan[t]},{Sqrt[3],0,#},{t,0,Pi/2},PlotRange->{{-1.2,1.2},{-1,1}}3]
}]/@{-1,1}]&/@{.5,1,1.5,2}},ImageSize->400,Spacings->50]//Quiet