Fermat's Spiral

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Plot

Polar

Show[GraphicsArray[
Block[{$DisplayFunction=Identity},
{
PolarPlot[Sqrt[t],{t,0,6 Pi},Ticks->None,PlotStyle->Red],
PolarPlot[{Sqrt[t],-Sqrt[t]},{t,0,6 Pi},Ticks->None,PlotStyle->{Red,Blue}]
}
]]]

Parametric

ParametricPlot[Sqrt[t]{Cos[t],Sin[t]},{t,0,20},PlotStyle->Red,ImageSize->200]

Equation

Cartesian

r^2-a^2θ/.{r^2->x^2+y^2,θ->ArcTan[y/x]}

FullSimplify[%]

ContourPlot[(x^2)+(y^2)==ArcTan[x,y],{x,-2,2},{y,-2,2}]

ContourPlot[Tan[(x^2)+(y^2)]==(y/x),{x,-4,4},{y,-4,4},AspectRatio->Automatic]

Properties

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

Arc Length

Polar

ArcLength[a θ^(1/2),θ,Assumptions->a>0&&θ>0]

FullSimplify[FunctionExpand[%],θ>0]

Plot[((1/8)-(I/8))  Beta[-4 (θ^2),(1/4),(3/2)],{θ,0,2π},PlotStyle->Red]

Parametric

Simplify[ArcLength[curve,t]]

Curvature

Paramatric

Simplify[Curvature[curve,t]]//Factor

Polar

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

Slope

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

Slope[a Sqrt[q],q]//FullSimplify

Slope[Tan[(x^2+y^2)/a^2]==y/x,{x,y}]//FullSimplify

ByteCount/@%

SpecialAffineCurvature

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

Tangential Angle

TangentialAngle[curve,t]

Vector Properties

parm=a Sqrt[t] {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]]