plotting - Problem with ParametricPlot

I'm plotting gear curves, and I observe that for some parameter values ParametricPlot[] does not plot all the teeth:

gearCurve[a_, b_, n_] := ParametricPlot[
  {
    (a + 1/b Tanh[b Sin[n t]]) Cos[t], 
    (a + 1/b Tanh[b Sin[n t]]) Sin[t]
  }, 
  {t, 0, 2 Pi}, 
  Axes -> False];

  gearCurve[10, 5, 38]

produces the following image:

problematic gear

What is going on?

Answer

Yaroslav Bulatov posted a great answer addressing this problem on StackOverflow.

Among several good illustrations the answer includes this undocumented option:

Method -> {Refinement -> {ControlValue -> (*radians*) }}

Alexey Popkov also gave an excellent analysis of the Plot algorithm.

His answer includes the apparently equivalent but cleaner (in degrees rather than radians as above):

Method-> {MaxBend -> (*degrees*) }

Example:

gearCurve[a_, b_, n_] :=
  ParametricPlot[
    {(a + 1/b Tanh[b Sin[n t]]) Cos[t],
     (a + 1/b Tanh[b Sin[n t]]) Sin[t]},
    {t, 0, 2 Pi},

    Axes -> False,
    Method-> {MaxBend -> 1}
  ];

gearCurve[10, 5, 38]

Mathematica graphics

Comments

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