To reduce some confusion on teeth and trapezoids:
Re-reading the multiple papers by Polowniak/Sobolak, there are different versions of trapezoids. In their mathematical model of a worm gear, there is a small circumferential clearance between two trapezoids, one for the wheel, and one for the worm. Only one trapezoid is shown in the attached .3dm file, which could be for the tooth notch in the wheel, or for the worm, or for a cutting hob, which resembles a worm, for cutting the wheel. So the 2 dimensional trapezoid shown in the attached .3dm file is in the plane x=0, and could represent the 2 dimensional worm wheel notch between 2 teeth of the wheel, or a tooth of a hob cutting tool which could machine the wheel. Or the trapezoid could be a worm tooth profile. In machining a worm, a trapezoidal cutting blade can be used.

Worm wheel teeth are 3 dimensional. The upper and lower edge of the worm tooth cause the wheel teeth to be tapered...
The papers have more matrix math for them...

- Brian

Bug: If the globoid net is closed, ( gear teeth = worm turns, ) there is a duplicate line rung at the end, which needs to be deleted before network will work.

Beta3 does not add the last line segment, if the two globoid curves are closed. (gearTeeth = wormTurns = 1)
- Brian

Free gear book pdf.
https://www.academia.edu/36781112/Gear_Geometry_and_Applied_Theory_pdf?email_work_card=view-paper