Plane Curves

Implementation

Needs["Graphics`Colors`"]

Needs["CURVES2D`"]

Needs["CSPROGS`"]

Epitrochoid

The epitrochoid is one of the many interesting
plane curves that can be studied by means of
Mathematica.

?epitrochoid

ParametricPlot[epitrochoid[18,2,6][t]//Evaluate,
{t,0,2Pi},AspectRatio->Automatic];

Curvature of an Epitrochoid

The function kappa2 will compute the signed curvature
of a plane curve.  When the formula for the signed
curvature is especially complicated, it can be
plotted instead.

Plot[kappa2[epitrochoid[18,2,6]][t]//Evaluate,
{t,0,2Pi}];

Pedal Curve of an Epitrochoid

Mathematica can be used to plot new curves from
given curves.  Here is the plot of the pedal curve
of an epitrochoid.

ParametricPlot[pedal[epitrochoid[18,2,6]][t]//Evaluate,
{t,0,2Pi},AspectRatio->Automatic];

Determining a curve from its curvature

The fundamental theorem of plane curves states that
up to a rigid motion a plane curve is determined
by its signed curvature.  The function plotintrinsic
draws a curve whose signed curvature is a given function.

?plotintrinsic

plotintrinsic[#Sin[#]&,
{0,0,0},MaxSteps->700,{-20,20},
PlotPoints->100];