Lituus

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Plot

PolarPlot[(θ^(-1/2)),{θ,.05,100},Ticks->None,PlotRange->{{-1,5},{-1,1}},PlotStyle->Red]

Properties

curve=a t^(-1/2){Cos[t],Sin[t]};

Arc Length

ArcLength[(θ^(-1/2)),θ]

FullSimplify[ArcLength[(θ^(-1/2)),{θ,(Subscript[θ, 0]),θ},Assumptions->θ>(Subscript[θ, 0])>0]]

Curvature

Parametric

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

Polar

FullSimplify[Curvature[a q^(-1/2),q],a>0&&q>0]

Inflection Points

FullSimplify[D[κ,t],t>0]

soln={ToRules[Reduce[(-3+40 (t^2)+16 (t^4))==0,t,Reals]]}

pts=(curve/.#)&/@soln//FullSimplify

N[%]

Block[{a=1},
PolarPlot[a (θ^(-1/2)),{θ,.05,100},Ticks->None,PlotRange->{{-1,5},{-1,1}},PlotStyle->Red,Epilog->{PointSize[.02],Point[Last[pts]]}]]

Slope

Slope[t^(-1/2) {Cos[t],Sin[t]},t]//FullSimplify

Slope[a q^(-1/2),q]//FullSimplify

SpecialAffineCurvature

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

Tangential Angle

TangentialAngle[curve,t]

Vector Properties

parm=a t^(-1/2) {Cos[t],Sin[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]]