Hyperbolic Spiral

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Plot

PolarPlot[1/θ,{θ,0,100},PlotStyle->Red,Ticks->None]

Equations

Parametric

ParametricPlot[{Cos[t],Sin[t]}/t,{t,.01,500}]

Cartesian

Eliminate[Thread[{x,y}=={Cos[t],Sin[t]}/t],t]

Eliminate[Thread[{x,y}^2=={1-Sin[t]^2,Sin[t]^2}/t],t]

ContourPlot[(y/x)==Tan[(1/(Sqrt[(x^2)+(y^2)]))],{x,-.02,.02},{y,-.02,.02},PlotPoints->10,MaxRecursion->4]

Properties

curve=a{Cos[t],Sin[t]}/t;

Arc Length

ArcLength[curve,t]

FullSimplify[Curvature[curve,t],t>0&&a>0]

Curvature

Parametric

FullSimplify[Curvature[a{Cos[t],Sin[t]}/t,t],a>0&&t>0]

Polar

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

Slope

Slope[a {Cos[t],Sin[t]}/t,t]//FullSimplify

Slope[a/q,q]//FullSimplify

FullSimplify[Slope[y/x==Tan[1/Sqrt[x^2+y^2]],{x,y}],(x|y)∈Reals]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&t>0]

Tangential Angle

TangentialAngle[curve,t]

Vector Properties

parm=a {Cos[t],Sin[t]}/t;
assum=a>0&&t>0;
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]]