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The most important section is on involute curves. It's the curve formed by unwinding a string against the circle:


It creates a constant angular velocity ratio at all points where the gears mesh (the law of gears).

In layman's terms, the tip of the tooth gets thinner so that the angular velocity there is reduced at that larger radius. Otherwise the gears advance/retreat as they rotate, which creates vibration.

I think there might be a whole host of curves that work for this, the other main one being a cycloid, which I'm not really familiar with:


I first learned about involute curves from a cousin that works as a machinist. Mr. Wizard also blew my young mind with noncircular wheels:


Edit: stumbled onto this technique to make involute gears in CAD:


If someone has a simpler method, I'd love to see it.

Well, I think the parametric formula for the involute of a circle is (1-it) exp(it). If you pop open Python with Numpy you can say

    t = np.linspace(0, 1); (1 - 1j*t) * np.exp(t * 1j)
And that gives you almost a radian of the involute, unless I've screwed something up. You can evaluate that at the desired number of points, clipped to the desired range of radii, export the coordinates to CSV if necessary, and import them into your CAD program as a smooth polyline. For example, with FreeCAD, you can directly script it in Python and https://forum.freecadweb.org/viewtopic.php?t=27866 Draft.makeBSpline will apparently do the job. Blender should be similar.

To me this sounds simpler.

I wrote a CAD package in Go that has a function for it.


  func InvoluteGear(
   numberTeeth int, // number of gear teeth
   gearModule float64, // pitch circle diameter / number of gear teeth
   pressureAngle float64, // gear pressure angle (radians)
   backlash float64, // backlash expressed as per-tooth distance at pitch circumference
   clearance float64, // additional root clearance
   ringWidth float64, // width of ring wall (from root circle)
   facets int, // number of facets for involute flank

Cycloid gears are used in roots-style superchargers. I learned about the shape back in college when I thought I was going to build one from scratch.


Geartrax software, it will draw perfect involute splines.

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