n0=  n1=  n2=  n3= .
h=  p=  phi=o   .

Second order roulette or gear-in-hoop-in-wheel parametric equation as described in blog post. Click on update to redraw. Scroll down for more controls and examples. Good values for hoop offset h is 0.4 - 0.6, while the pen-hole relative distance p is usually 0.7 - 0.9. Url with current parameters is available below for ctrl-c.
hole=mm  show circles  show symmetry  show f0 

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Inversion of frequencies F=(T1/T,T3/T); but T<T1<T3:

 T1=  T3=  T= .
. . . .

Examples (click):