Generating Cycloidal Gears for 3D Printing

This article describes an algorithm for producing, for any desired resolution and any desired numbers of wheel and pinion teeth, polygonal approximations to the shapes of a pair of cycloidal gears that mesh correctly. The larger (with the larger number of teeth) of the two gears is called the wheel; the smaller of the two gears is called the pinion. An Octave implementation of the algorithm, mostly written in 2014, is included. The Octave implementation contains a (crude, but evidently adequate, at least for reasonable numbers of wheel and pinion teeth) solution of the problem of finding (iteratively, since I am not aware of any analytical solution) the generating wheel angle corresponding to the tips of the tooth addenda. (We'll call this the “tooth tip problem”). The only previous solution to this problem of which I am aware is somewhat more complicated. However, this Octave implementation does not contain a good solution to the problem (not discussed in the literature at all, as far as I am aware) of automatically determining the generating wheel angles required to produce a polygon which approximates the curved parts of the teeth (the addenda) to a resolution specified by the user (in order to match the resolution of the 3D printer that will be used to physically produce the gears). I was unsure as to how to solve this latter problem (we'll call this the “generating wheel step problem”) until about six months ago, when I realised that there is (if I am not mistaken) a simple solution involving a priority queue. A sketch of this proposed solution is included here. However, since implementing a priority queue in Octave would be a bit cumbersome, my intention is to port the implementation to Java (which seems to have support for priority queues in its standard class libraries), and then attempt to implement the algorithm described here for automatically determining the generating wheel angles.