Rectangular Hyperbola

Author

Eric W. Weisstein
June 6, 2014

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

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

©2014 Wolfram Research, Inc. except for portions noted otherwise

Equation

Parametric {Cosh[t],Sinh[t]}

ParametricPlot[{Cosh[t],Sinh[t]},{t,-10,10},PlotStyle->Red,
AspectRatio->Automatic,PlotRange->5{{-1,1},{-1,1}},ImageSize->Small]

ParametricPlot[{Cosh[t],Sinh[t]},{t,0,10},PlotStyle->Red,
AspectRatio->Automatic,PlotRange->5{{-1,1},{-1,1}},ImageSize->Small]

Parametric {Sec[t],Tan[t]}

ParametricPlot[{Sec[t],Tan[t]},{t,-π,π},PlotStyle->Red,
AspectRatio->Automatic,PlotRange->5{{-1,1},{-1,1}},ImageSize->Small,Exclusions->{-1,1}Pi/2]

Note weird (and slow) behavior; see https://bugs.wolfram.com/show?number=209073

Manipulate[ParametricPlot[{Sec[t],Tan[t]},{t,0,x},PlotStyle->Red,
AspectRatio->Automatic,PlotRange->5{{-1,1},{-1,1}},Exclusions->{-1,1}Pi/2,ImageSize->Small],{{x,π},.0001,2π}]

Manipulate[ParametricPlot[{Sec[t],Tan[t]},{t,0,x},PlotStyle->Red,
AspectRatio->Automatic,PlotRange->5{{-1,1},{-1,1}},ImageSize->Small],{{x,π},.0001,2π}]

Range[-4,4,1]Pi/2

ParametricPlot[{Sec[t],Tan[t]},{t,0,.01},ImageSize->Small]

Range[-4,4,1]Pi/2

ParametricPlot[{Sec[t],Tan[t]},{t,0,.01},Exclusions->Range[-4,4,1]Pi/2]

{ParametricPlot[{Sec[t],Tan[t]},{t,-π/2,π/2},PlotStyle->Red,
AspectRatio->Automatic,PlotRange->5{{-1,1},{-1,1}},ImageSize->Small],
ParametricPlot[{Sec[t],Tan[t]},{t,π/2+.0001,3π/2},PlotStyle->Red,
AspectRatio->Automatic,PlotRange->5{{-1,1},{-1,1}},ImageSize->Small]}

Polar

Solve[Tan[θ]==Tan[t]/Sec[t],t]

Solve[

FullSimplify[(Sqrt[Sec[t]^2+Tan[t]^2])/.t->ArcSin[Tan[q]]]

PolarPlot[(Sqrt[Sec[2 q]]),{q,0,2π}]

FullSimplify[(a b)/Sqrt[b^2 Cos[q]^2-a^2 Sin[q]^2]/.b->a,a>0&&-π<q<π]

PolarPlot[(1/(Sqrt[Cos[2 q]])),{q,0,2π}]

Graphic

a=1; b=1; c=Sqrt[a^2+b^2];
f1={-c,0};
f2={c,0};
o={0,0};
pts={f1,"(Subscript[F, 1])",f2,"(Subscript[F, 2])",o,"O"};

Off[ParametricPlot::pptr];
Show[GraphicsArray[{
PolarPlot[{-Sqrt[Sec[2t]],Sqrt[Sec[2t]]},{t,-.8,.8},
AspectRatio->Automatic,Ticks->None,
Epilog->{PerpSquare[{{{-1,1},{1,-1}},{{-1,-1},{1,1}}},.25,1],PointLabels[pts,.03,{.1,.15}],Dashing[{.04,.04}],Line[3{{-1,1},{1,-1}}],
Line[3{{-1,-1},{1,1}}]},
PlotRange->{{-3,3},{-3,3}},DisplayFunction->Identity],
Plot[1/x,{x,-5,5},AspectRatio->Automatic,
PlotRange->{{-3,3},{-3,3}},Ticks->None,
DisplayFunction->Identity]
}]]

e=Sqrt[2];
Show[Graphics[{
PolarPlot[(e^2-1)/(1-e Cos[θ]),{θ,-Pi/4,Pi/4},Ticks->None,DisplayFunction->Identity][[1]],
PolarPlot[(e^2-1)/(1-e Cos[θ]),{θ,Pi/4,7Pi/4},Ticks->None,DisplayFunction->Identity][[1]],
{PointSize[.04],Point[{0,0}],Point[{-e,0}]},
Line[{{-4,0},{1,0}}]
}],
DisplayFunction->$DisplayFunction,AspectRatio->Automatic,
PlotRange->{{-3.414,.5},{-1.5,1.5}}]

Hyperbola[a_,b_,opts___]:=ImplicitPlot[
{x^2/a^2-y^2/b^2==1, x==y a/b, x==-y a/b},{x,-2,2},AspectRatio->Automatic,Ticks->None,PlotStyle->{{Dashing[{1,0}],Thickness[.015]}
{Dashing[{0.05,0.02}]},{Dashing[{0.05,0.02}]}},
opts];

Hyperbola[]:=Hyperbola[1,1];

hyp[{x_,s_}]:={x,s Sqrt[x^2-1]}

a=1; b=1; c=Sqrt[a^2+b^2];
f1={-c,0};
f2={c,0};
o={0,0};
tri=hyp/@{{1.2,1},{-1.3,-1},{-1.5,1}};
pts={f1,"(Subscript[F, 1])",f2,"(Subscript[F, 2])",o,"O",Orthocenter[tri],"H",
	tri[[1]],"A",tri[[2]],"B",tri[[3]],"C"};

Show[Graphics[{
Triangle[tri],
{GrayLevel[.5],Altitudes[tri]},
PointLabels[pts,.04,{.1,.15}],
NinePointCircle[tri],
PerpSquare[{{{-1,1},{1,-1}},{{-1,-1},{1,1}}},.2,1],
Hyperbola[1,1,DisplayFunction->Identity][[1]]
}],DisplayFunction->$DisplayFunction,Axes->True,Ticks->False,
AspectRatio->Automatic]

a=1; b=1; c=Sqrt[a^2+b^2];
f1={-c,0};
f2={c,0};
o={0,0};
tri=hyp/@{{1.2,1},{-1.3,-1},{-1.5,1},{1.6,-1}};
pts={f1,"(Subscript[F, 1])",f2,"(Subscript[F, 2])",o,"O",Orthocenter[tri],"H",
	tri[[1]],"A",tri[[2]],"B",tri[[3]],"C",tri[[4]],"D"};

tri={{-1,-.8},{-.6,.6},{.6,.8},{1.3,-.7}};

RectangularHyperbola[tri_List]:=Module[
{c=CircleCircleIntersections[NinePointCircles[tri]],d,i,j,ctr,
f},
d=Chop[Table[(c[[1,i]]-c[[2,j]]).(c[[1,i]]-c[[2,j]]),
{i,2},{j,2}]];
ctr=c[[1,Position[d,0][[1,1]]]]		
]

RectangularHyperbola[tri]

Properties {cosh(t),sinh(t)}

hyperbola[t_]:=a{Cosh[t],Sinh[t]}

Arc Length

v4.2.1

FullSimplify[ArcLength[hyperbola[t],t],a>0]

v5

ArcLength[hyperbola[t],t,Assumptions->t∈Reals&&a>0&&t≠0]

FullSimplify[%,t∈Reals]

Comparison

f[t_]:=-I EllipticE[I t,2]

g[t_]:=(1/(Sqrt[2]))((Sqrt[(E^(-2 t))+(E^(2 t))])+(((Gamma[(3/4)])^2)/(Sqrt[π]))-(1/(Sqrt[1+(E^(4 t))]))(2 (Sqrt[(E^(-2 t))+(E^(2 t))]) Hypergeometric2F1[-(1/4),(1/2),(3/4),-(E^(4 t))]))

FullSimplify[(Sqrt[(((E^(-2 t))+(E^(2 t)))/(1+(E^(4 t))))]),t>0]

FullSimplify[(Sqrt[(E^(-2 t))+(E^(2 t))])]

(Sqrt[2]) (Sqrt[Cosh[2 t]])+(((Gamma[(3/4)])^2)/(Sqrt[π]))

h[t_]:=(Sqrt[Cosh[2 t]])+(((Gamma[(3/4)])^2)/(Sqrt[2π]))-Sqrt[2](E^(-t)) Hypergeometric2F1[-(1/4),(1/2),(3/4),-(E^(4 t))]

Plot[Evaluate[f[t]-g[t]],{t,0,1},PlotStyle->Red]

FullSimplify[g[t],t>0]

FullSimplify[h[t],t>0]

i[t_]:=((1/4)+(I/4)) Beta[-(E^(4 t)),-(1/4),(1/2)]+(Sqrt[Cosh[2 t]])+(((Gamma[(3/4)])^2)/(Sqrt[2 π]))

Plot[Evaluate[Abs[{g[t],h[t],i[t]}-f[t]]],{t,0,1},PlotStyle->{Red,Green,Blue}]

{f[t],h[t],i[t]}//ColumnForm//TraditionalForm

v6

ArcLength[hyperbola[t],t,Assumptions->t∈Reals&&a>0]

Curvature

v5

FullSimplify[Curvature[hyperbola[t],t],a>0]

v6

FullSimplify[Curvature[hyperbola[t],t],a>0]

Tangential Angle

v5

FullSimplify[TangentialAngle[hyperbola[t],t]]

v6

FullSimplify[TangentialAngle[hyperbola[t],t,Assumptions->t∈Reals]]

Properties {sec(t),tan(t)}

hyperbola=a{Sec[t],Tan[t]}

ArcLengthFunction

(s=ArcLength[hyperbola,t,Assumptions->t>0&&a>0])//Timing

FullSimplify[(1/(-3+Cos[2 t]))a (Sqrt[1+((Sin[t])^2)]) ((Sqrt[6-2 Cos[2 t]]) (EllipticE[t,-1]-EllipticF[t,-1])+Sin[2 t]-4 Tan[t]),-π/2<t<π/2]

expr1=(1/(-3+Cos[2 t])) (Sqrt[1+((Sin[t])^2)]) ((Sqrt[6-2 Cos[2 t]]) (EllipticE[t,-1]-EllipticF[t,-1])+Sin[2 t]-4 Tan[t]);

FullSimplify[expr1]

From full hyperbola:

FullSimplify[Limit[-b EllipticE[t,-(a^2/b^2)]+b EllipticF[t,-(a^2/b^2)]+(Sqrt[a^2+2 b^2-a^2 Cos[2 t]] Tan[t])/Sqrt[2],b->a],a>0]

expr2=- EllipticE[t,-1]+ EllipticF[t,-1]+(( (Sqrt[3-Cos[2 t]]) Tan[t])/(Sqrt[2]));

FullSimplify[expr1-expr2]

ByteCount/@{expr1,expr2}

Curvature

Parametric

FullSimplify[Curvature[hyperbola,t],a>0&&-π<t<π]

Polar

FullSimplify[Curvature[a Sqrt[Sec[2 q]],q],a>0&&0<q<2π]

Refine[%,a>0]

Implicit

FullSimplify[Curvature[x^2-y^2==a^2,{x,y}]]

ByteCount/@%

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[hyperbola,t],a>0&&t∈Reals]

FullSimplify[-(Sec[t]^2/(a^4 Sec[t]^6)^(1/3)),a>0&&t∈Reals]

FullSimplify[Refine[%,a>0&&t∈Reals],t∈Reals]

expr=-((((Cos[t])^4) (((((Sec[t])^6)))^(2/3)))/(a^(4/3)));

FullSimplify[expr,t∈Reals]

FullSimplify[((Cos[t])^4) (((((Sec[t])^6)))^(2/3)),3π/2<t<2π]

FullSimplify[((Cos[t])^4) (((((Sec[t])^6)))^(2/3))-1,3π/2<t<2π]

Plot[-1+((Cos[t])^4) (((((Sec[t])^6)))^(2/3)),{t,0,2π},PlotStyle->{Red,Thick}]

Table[FullSimplify[expr,(n-1)π/2<t<n π/2],{n,4}]

Plot[((Cos[t])^4) (((((Sec[t])^6)))^(2/3)),{t,-10,10},PlotStyle->{Red,Thick}]

Differential Equation

xy=Numerator[Together[((((x-x0))^2)/(a^2))-((((y[x]-y0))^2)/(a^2))-1]]

eqs=Table[D[xy,{x,i}],{i,0,3}];

gb=GroebnerBasis[eqs,{x,y[x],y'[x],y''[x],y'''[x]},{x0,y0,a},MonomialOrder->EliminationOrder]

gb[[1]]//InputForm

Slope

Slope[{a Sec[t],a Tan[t]},t]//FullSimplify

Slope[a Sqrt[Sec[2 q]],q]//FullSimplify

Slope[x^2-y^2==(a^2),{x,y}]//FullSimplify

Tangential Angle

FullSimplify[TangentialAngle[hyperbola,t,Assumptions->t∈Reals]]

VectorLength

FullSimplify[VectorLength[hyperbola,t],a>0&&t∈Reals]

FullSimplify[(Sqrt[Sec[t]^2+Tan[t]^2]),t∈Reals]

Vector Properties

parm=a{Sec[t],Tan[t]};
assum=a>0&&h>0&&-Pi<t<Pi;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n,Sqrt[x_^(n_/;Mod[n,2]==0)]:>Abs[x]^(n/2)};

VectorLength

Assuming[assum,FullSimplify[Norm[parm]/.xform]]/.xform

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]/.xform]]/.xform

{-(1/(Sqrt[1+((Sin[t])^2)])),((Sign[Cos[t]] Tan[t])/(Sqrt[((Sec[t])^2)+((Tan[t])^2)]))}

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]/.xform]]/.xform

{((Sign[Cos[t]] Tan[t])/(Sqrt[((Sec[t])^2)+((Tan[t])^2)])),(1/(Sqrt[1+((Sin[t])^2)]))}