Galilean Spiral

Author

Eric W. Weisstein
April 12, 2021

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

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

©2021 Wolfram Research, Inc. except for portions noted otherwise

Sources

https://mathcurve.com/courbes2d.gb/galilee/galilee.shtml

https://encyclopediaofmath.org/wiki/Galilean_spiral

Parametric equations

GraphicsGrid[Partition[With[{a=-1},Table[PolarPlot[a+b (q^2),{q,0,20},Ticks->None,PlotLabel->Row[{Style["b",Italic]," == ",b}],LabelStyle->Directive[Black,FontFamily->"Times"],PlotStyle->Red],{b,{-1,-.5,0,.125,.5,1}}]],UpTo[3]],Dividers->All]

GraphicsGrid[Partition[With[{a=-1},Table[PolarPlot[a+b (q^2),{q,-10,10},Ticks->None,PlotLabel->Row[{Style["b",Italic]," == ",b}],LabelStyle->Directive[Black,FontFamily->"Times"],PlotStyle->Red],{b,{-1,-.5,0,.125,.5,1}}]],UpTo[3]],Dividers->All]

GraphicsGrid[Partition[With[{a=0},Table[PolarPlot[a+b (q^2),{q,-10,10},Ticks->None,PlotLabel->Row[{Style["b",Italic]," == ",b}],LabelStyle->Directive[Black,FontFamily->"Times"],PlotStyle->Red],{b,{-1,-.5,0,.125,.5,1}}]],UpTo[3]],Dividers->All]

Properties

<<MathWorld`Curves`

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

ArcLength

Assuming[a>0&&b∈Reals,ArcLength[curve,{t,0,t0}]]

Assuming[a>0&&b∈Reals,MyArcLength[b (q^2)-a,{q,0,θ}]]

Curvature

FullSimplify[Curvature[curve,t],a>0&&b∈Reals]

ImplicitEquation

x^2+y^2==(b ArcTan[y/x]^2-a)^2/.{x->r Cos[θ],y->r Sin[θ]}//FullSimplify

With[{a=1,b=1/8},PolarPlot[b q^2-a,{q,-10,10}]]

With[{a=1,b=1/8},ContourPlot[x^2+y^2==(((b ArcTan[y,x]^2-a))^2),{x,-10,10},{y,-10,10},Axes->True]]

With[{a=-1,b=1/8},ContourPlot[x^2+y^2==(((b ArcTan[y,x]^2-a))^2),{x,-10,10},{y,-10,10},Axes->True]]

NormalVector

FullSimplify[NormalVector[curve,t],a>0&&b∈Reals&&t>0]

ParametricEquations

With[{a=1,b=1/8},ParametricPlot[(b t^2-a) {Cos[t],Sin[t]},{t,-10,10}]]

PolarCurvature

FullSimplify[Curvature[(b (q^2)-a),q]]

PolarEquation

With[{a=1,b=1/8},PolarPlot[b q^2-a,{q,-10,10}]]

PolarSlope

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

PolarTangentialAngle

FullSimplify[TangentialAngle[(b (q^2)-a),q]]

Slope

FullSimplify[Slope[curve,t],a>0&&b∈Reals&&t>0]

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[curve,t],a>0&&b∈Reals&&t>0]

TangentialAngle

TangentialAngle[curve,t]

TangentVector

FullSimplify[TangentVector[curve,t],a>0&&b∈Reals&&t>0]

VectorLength

FullSimplify[VectorLength[curve,t],a>0&&b∈Reals&&t>0]