Hyperbola Evolute

Author

Eric W. Weisstein
March 16, 2008

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

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

©2008 Wolfram Research, Inc. except for portions noted otherwise

Equation

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

evolute=Evolute[hyperbola,t]//FullSimplify

Eliminate[
{
x==((((a^2)+(b^2)) ((Cosh[t])^3))/a),
y==-((((a^2)+(b^2)) ((Sinh[t])^3))/b),
Cosh[t]^2-Sinh[t]^2==1
},
{Cosh[t],Sinh[t]}]//Timing

Eliminate[
{
x==((((a^2)+(b^2)) ((cosh[t])^3))/a),
y==-((((a^2)+(b^2)) ((sinh[t])^3))/b),
cosh[t]^2-sinh[t]^2==1
},
{cosh[t],sinh[t]}]//Timing

GroebnerBasis[{x-((((a^2)+(b^2)) (cosht^3))/a),y+((((a^2)+(b^2)) (sinht^3))/b),cosht^2-sinht^2-1},{sinht,cosht},MonomialOrder->EliminationOrder]//Factor//Simplify

FullSimplify[(a^(2/3))((Abs[((((a^2)+(b^2)) ((Cosh[t])^3))/a)])^(2/3))-(b^(2/3))((Abs[-((((a^2)+(b^2)) ((Sinh[t])^3))/b)])^(2/3)),a>0&&b>0&&t∈Reals]

Figure

EvolutePlot[hyperbola/.{a->1,b->1},{t,-2,2}]