I like that “mechanism of simple construction“.
That’s so as not to frighten lawyers.
The mathematics of gears and their generation, espcially in such realms as helical bevels, are formidable indeed. Those who developed the geometry and designed the machines to follow those mathematical laws, deserve all of our respect.
Another geometrical task a shaper can perform, with the appropriate accessory, is cutting concave radii. Not, as one might imagine, by rotating the cutter head*, but by the locus of a sub-table carrying the work-piece, controlled by two slides, one horizontal, the other vertical, fixed rigidly with respect to the machine, not the traversing table.
*(Though I have finish-machined a 7.25” g. locomotive’s smokebox saddle on a big old shaper whose clapper-box was mounted on a worm-driven quadrant. It would be feasible to design and build a proper accessory cutter head for such duty.)