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]

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]

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

Among several good illustrations the answer includes this undocumented option:

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

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 -> {Refinement -> {ControlValue -> 1 Degree}}
  ];

gearCurve[10, 5, 38]

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]