Gear Undercut Minimum Teeth and Backlash
Two checks spur-gear-geometry says it omits, both closed-form. Nmin = 2k/sin^2(phi) reproduces the familiar minimums exactly - 32 teeth at 14.5 degrees, 18 at 20, 12 at 25 - which is why the industry moved to higher pressure angles for small pinions; a 14-tooth 20-degree pinion is undercut and weakened right where bending stress peaks. And backlash from opening the center distance is 2 dC tan(phi), not one-for-one: 0.010 in of center distance makes 0.0073 in of backlash.
Formula and source
Undercut minimum Nmin = 2k / sin^2(pressure angle), k = addendum coefficient (1.0 full depth, 0.8 stub); backlash from center-distance change B = 2 x dC x tan(pressure angle).
Involute gear geometry as compiled in Machinery's Handbook and the AGMA standard proportions, by name - closed-form, no table reproduced.
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