(* ::Package:: *)

(* ::Text:: *)
(*The evolutes here arise by flowing each point along the unit normal at the same constant speed.  In the beginning, (2,3) Lissajous curve is smooth, but 6 singuarlities emerge.  By the end, only two remain as the (2,3) curve gets deformed into a large circle with doubled edges.  It is the "odometer" of the (2,3) curve under the Gauss map.  The original evolute uses the normal vector (second derivative) of the arc-length parameterized curve.  See also http://en.wikipedia.org/wiki/File:Evolute1.gif*)


(* ::Input:: *)
(*V[t_] := {Sin[2*t],Sin[3*t]};*)
(*A = ParametricPlot[v[t],{t,0,2 Pi}];*)
(*NN= 100;*)
(*Evolute[v_,k_] :=  ParametricPlot[v[t] - k {-v'[t][[2]], v'[t][[1]]}/Norm[v'[t]],{t,0,2 Pi}, PlotRange->5]*)
(*Animate[Evolute[V,k],{k,0,4,0.01}, AnimationRunning -> False, ControlPlacement-> Bottom]*)


(* ::ItemParagraph:: *)
(*It is possible to draw arrows directly in Mathematica.  This project arose from trying to draw the unit normal field of a planar curve.*)


(* ::Input:: *)
(*v[t_]:= {Cos[t],Sin[t]}*)
(*AA = ParametricPlot[v[t],{t,0,2 Pi}];*)
(*NN= 30;*)
(*BB = Graphics[Table[ Arrow[{{Cos[2 Pi k /NN ], Sin[2 Pi k /NN ]},{-Sin[2 Pi k /NN], Cos[2 Pi k/ NN]}}],{k,0,NN}]];*)
(*CC = Graphics[Table[ Arrow[{v[2 Pi k /NN ],v[2 Pi k /NN ] + 0.8v'[2 Pi k /NN]}],{k,0,NN}]];*)
(*DD = Graphics[Table[ Arrow[{v[2 Pi k /NN ],2*v[2 Pi k /NN ] }],{k,0,NN}]];*)
(*{Show[AA,BB, PlotRange-> 1],Show[AA,CC, PlotRange-> 1.5],Show[AA,DD,PlotRange-> 2]}*)
