Hyperbola Pedal Curve

Author

Eric W. Weisstein
June 27, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Equation

Cosh/Sinh

hyperbola={Cosh[t],Sinh[t]};

(pedal=PedalCurve[hyperbola,{x0,y0},t])//FullSimplify

hyperbola2={Sec[t],Tan[t]};

(pedal2=PedalCurve[hyperbola2,{x0,y0},t])//FullSimplify

Sec/Tan

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

FullSimplify[PedalCurve[hyperbola,{(Subscript[x, 0]),(Subscript[y, 0])},t]]//TraditionalForm

eqn=FullSimplify[PedalCurve[hyperbola,{x0,y0},t]]

eqn2=Together[TrigExpand[eqn]/.{Sin[t]->sin,Cos[t]->cos}]

eqns=(-(a^2)-2 (b^2)+(a^2) (cos^2)-(a^2) (sin^2)){x,y}-Numerator[eqn2]

gb=Factor/@GroebnerBasis[Append[eqns,sin^2+cos^2-1],{x,y},{sin,cos}]//FullSimplify

Length[gb]

AlgebraicDegree[gb[[1]],{x,y}]

Properties

par={((2 a ((b^2) Cos[t]+Sin[t] (b y0+a x0 Sin[t])))/((a^2)+2 (b^2)-(a^2) Cos[2 t])),((2 b (b y0+a (x0-a Cos[t]) Sin[t]))/((a^2)+2 (b^2)-(a^2) Cos[2 t]))};

ArcLength

(s=ArcLength[par,{t,0,2π},Assumptions->a>0&&b>0&&(x0|y0)∈Reals&&x0^2+y0^2<a^2])//Timing

(s=ArcLength[par,{t,0,2π},Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2==a^2])//Timing

(s=ArcLength[par,{t,0,2π},Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2>a^2])//Timing

ArcLengthFunction

ArcLength[par,t,Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2<a^2&&t>0]//Timing

ArcLength[par,t,Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2==a^2&&t>0]//Timing

ArcLength[par,t,Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2>a^2&&t>0]//Timing

Curvature

FullSimplify[Curvature[par,t],a>0&&b>0&&-Pi<t<Pi]//Timing

ic=Curvature[(-(a^2) (((x-x0))^2)+(b^2) (y^2)+((((x^2)-x x0+(y^2)))^2)-2 y ((b^2)+(x^2)-x x0+(y^2)) y0+((b^2)+(y^2)) (y0^2)) (((((a^2)+(b^2)-(x0^2)))^2)+2 ((a^2)+(b^2)+(x0^2)) (y0^2)+(y0^4)),{x,y}]

(ic2=Simplify[ic,a>0&&b>0])//Timing

ByteCount/@ic2

(ic3=FullSimplify[ic2[[1]],a>0&&b>0])//Timing

Slope

Slope[{(2 a (b^2 Cos[t]+Sin[t] (b y0+a x0 Sin[t])))/(a^2+2 b^2-a^2 Cos[2 t]),(2 b (b y0+a (x0-a Cos[t]) Sin[t]))/(a^2+2 b^2-a^2 Cos[2 t])},t]//FullSimplify

Slope[(-a^2 (x-x0)^2+b^2 y^2+(x^2-x x0+y^2)^2-2 y (b^2+x^2-x x0+y^2) y0+(b^2+y^2) y0^2) ((a^2+b^2-x0^2)^2+2 (a^2+b^2+x0^2) y0^2+y0^4)==0,{x,y}]//FullSimplify

SpecialAffineCurvature

sac=SpecialAffineCurvature[par,t]

(sac2=Simplify[sac,a>0&&b>0&&(x0|y0)∈Reals])//Timing

(sac3=FullSimplify[sac2,a>0&&b>0&&(x0|y0)∈Reals])//Timing

TangentialAngle

FullSimplify[TangentialAngle[par,t],a>0&&b>0]//Timing

FullSimplify[TangentialAngle[par,t],a>0&&b>0&&-π<t<π]//Timing

Plot

Focus Cosh/Sinh

FullSimplify[pedal/.{x0->Sqrt[2],y0->0}]

TrigExpand[Cosh[2t]]

Eliminate[{x==(1/(Sqrt[2]))+(-(1/(Sqrt[2]))+cosh[t]) /(((cosh[t])^2)+((sinh[t])^2)),y==(-1+(Sqrt[2]) cosh[t])/(((cosh[t])^2)+((sinh[t])^2)) sinh[t],
cosh[t]^2-sinh[t]^2==1},{sinh[t],cosh[t]}]

focus=Show[{
PedalCurvePlot[{-Cosh[t],Sinh[t]},{Sqrt[2],0},{t,-10,10},{-2,2,.3},DisplayFunction->Identity],PedalCurvePlot[hyperbola,{Sqrt[2],0},{t,-10,10},{-2,2,.3},DisplayFunction->Identity]},
DisplayFunction->$DisplayFunction,PlotRange->{{-2,2},{-1.2,1.2}}
]

Focus Sec/Tan

FullSimplify[pedal2/.{x0->Sqrt[2],y0->0}]

TrigExpand[Cos[2t]]

Eliminate[{x==-((2 (cos[t]+(Sqrt[2]) ((sin[t])^2)))/(-3+(((cos[t])^2)-((sin[t])^2)))),y==((2 (-(Sqrt[2])+cos[t]) sin[t])/(-3+(((cos[t])^2)-((sin[t])^2)))),
cos[t]^2+sin[t]^2==1},{sin[t],cos[t]}]

Center Cosh/Sinh

FullSimplify[pedal/.{x0->0,y0->0}]

FullSimplify[Eliminate[{x==cosh[t] /(((cosh[t])^2)+((sinh[t])^2)),y==-1/(((cosh[t])^2)+((sinh[t])^2))sinh[t],
cosh[t]^2-sinh[t]^2==1},{sinh[t],cosh[t]}]]

ctr=Show[{
PedalCurvePlot[Evaluate[hyperbola],{0,0},{t,-10,10},{-2,2,.4},DisplayFunction->Identity],PedalCurvePlot[Evaluate[-hyperbola],{0,0},{t,-10,10},{-2,2,.4},DisplayFunction->Identity]
},PlotRange->{{-1.5,1.5},{-1.2,1.2}},DisplayFunction->$DisplayFunction
]

Focus Sec/Tan

FullSimplify[pedal2/.{x0->0,y0->0}]

TrigExpand/@{Cos[2t],Sin[2t]}

FullSimplify[Eliminate[{x==-((2 cos[t])/(-3+(((cos[t])^2)-((sin[t])^2)))),y==((2sin[t]cos[t])/(-3+(((cos[t])^2)-((sin[t])^2)))),
cos[t]^2+sin[t]^2==1},{sin[t],cos[t]}]]

Both

Show[GraphicsArray[{focus,ctr}]]

Vector Properties

parm={(2 a (b^2 Cos[t]+Sin[t] (b y0+a x0 Sin[t])))/(a^2+2 b^2-a^2 Cos[2 t]),(2 b (b y0+a (x0-a Cos[t]) Sin[t]))/(a^2+2 b^2-a^2 Cos[2 t])};
assum=a>0&&b>0&&-π<t<π&&(x0|y0)∈Reals;
xform={Abs[x_]^(n_/;Mod[n,2]==0):>x^n};

VectorLength

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

NormalVector

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

TangentVector

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