Sinusoidal Spiral

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Plot

<<Utilities`Typesetting`

Off[ParametricPlot::"pptr"]

Show[GraphicsArray[Partition[Block[{$DisplayFunction=Identity},
PolarPlot[((Cos[#[[1]] θ])^(1/#[[1]])),Evaluate[{θ,Sequence@@#[[2]]}],
PlotStyle->Red,PlotRange->#[[3]],
PlotPoints->50,AspectRatio->Automatic,Ticks->None,PlotLabel->TraditionalForm[n==InlineFraction[#[[1]]]]]&/@{
{-2,{-π,π},{1.5{-1,1},{-1,1}}},
{-1,{0,2π},{{-.2,1.1},{-1,1}}},
{-1/2,3{-1,1},Automatic},
{-1/3,3.5{-1,1},{{-12,2},3{-1,1}}},
{1/3,{0,3π},Automatic},
{1/2,{0,2π},Automatic},
{1,{0,π},Automatic},
{2,{0,2π},Automatic}
}
],4]],GraphicsSpacing->{0,.1}]

Properties

sinuspiral=a Cos[n t]^(1/n);

Arc Length

v5

ArcLength[sinuspiral,t]//Timing

FullSimplify[(Sqrt[(a^2) ((Cos[n t])^(2/n))+(a^2) ((Cos[n t])^(-2+(2/n))) ((Sin[n t])^2)]),a>0]

Integrate[a (Sqrt[((Cos[n t])^(-2+(2/n)))]),{t,0,t}]//Timing

FullSimplify[((a (Sqrt[π]) (Sqrt[(n^2) (t^2)]) Gamma[1+(1/(2 n))])/(n t Gamma[((1+n)/(2 n))]))-(1/2) a (Sqrt[((Cos[n t])^(-2+(2/n)))]) ((Csc[n t])^2) Hypergeometric2F1[(1/(2 n)),(1/2),(1/2) (2+(1/n)),((Cos[n t])^2)] (Sqrt[((Sin[n t])^2)]) Sin[2 n t],(n|t)∈Reals]

v6

Out-of-the-box arc length

Arc Length

ArcLength[sinuspiral,t,Assumptions->n∈Reals&&a>0&&t>0]//Timing

FullSimplify[%8,t>0&&n>0&&2n t<π]

FullSimplify[%8,t>0&&n<0&&π+2n t>0]

With[{a=1,n=2.3},Plot[Subtract@@{-((a ((Sqrt[π]) Gamma[1+(1/(2 n))]+((Beta[((Cos[n t])^2),(1/(2 n)),(1/2)] Gamma[(1/2) (3+(1/n))])/(1+n))))/(Gamma[((1+n)/(2 n))])),((a (-Abs[Sin[n t]] Beta[((Cos[n t])^2),(1/(2 n)),(1/2)] Csc[n t]+(((Sqrt[π]) Gamma[(1/(2 n))])/(Gamma[((1+n)/(2 n))]))))/(2 n))}

Using simplified form of integral with no assumptions

Integrate[a (Sqrt[((Cos[n t])^(-2+(2/n)))]),{t,0,t},Assumptions->n∈Reals&&a>0&&t>0]//Timing

Curvature

v5

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

v6

FullSimplify[Curvature[sinuspiral,t],a>0&&(n|t)∈Reals]

Tangential Angle

v5

TangentialAngle[sinuspiral,t]

v6

TangentialAngle[sinuspiral,t]//FullSimplify