Circle Parallel Curves

Author

Eric W. Weisstein
June 6, 2014

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

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

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

Equation

FullSimplify[ParallelCurve[a{Cos[t],Sin[t]},t,k],a>0]

ParallelCurvesPlot[{Cos[t],Sin[t]},{-1,1,.2},{t,0,2Pi}]

Properties

Curvature

Parametric

FullSimplify[Curvature[(a+k) {Cos[t],Sin[t]},t],a>0&&k∈Reals]

Polar

FullSimplify[Curvature[a+k,q],a>0&&k∈Reals]

Implicit

FullSimplify[Curvature[x^2+y^2==(a+k)^2,{x,y}],a>0&&k∈Reals]

Slope

Parametric

Slope[(a+k) {Cos[t],Sin[t]},t]//FullSimplify

Polar

Slope[a+k,q]

Implicit

Slope[x^2+y^2==(a+k)^2,{x,y}]

Slope[x^2+y^2==(a+k)^2,{x,y}]

Vector Properties

parm=a{Cos[t]+t Sin[t],Sin[t]-t Cos[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