Involute Gear Design

I · The flank. The working flank is the involute of the base circle r_b = r cos α: parametrically x = r_b(cos u + u sin u), y = r_b(sin u − u cos u). Its angular position comes from the exact thickness relation ψ(r) = s/d + inv α − inv φ_r, cos φ_r = r_b/r, inv φ = tan φ − φ. Every plotted flank point satisfies the involute's defining property — its normal is tangent to the base circle — which is verified in the engine's test suite to a few parts in 10⁵ along with the curvature identity ρ_curv = √(r² − r_b²).

II · The root. The fillet is generated, not drawn: as the rack rolls on the pitch circle, its rounded tip corner (radius ρ*·m) sweeps a family of circles whose envelope is the trochoidal fillet. The meshing law puts each generated point on the line from the corner center through the pitch point, so the envelope is explicit. When the tooth count drops below z_min = 2(h_f* − x − ρ*(1 − sin α))/sin²α the corner sweeps past the interference point and undercuts — the same equations produce the neck automatically, trimmed against the involute at the true form diameter.

III · Over pins. A pin of diameter d_p centered in a tooth space touches both flanks where inv φ_M = s/d + inv α + d_p/d_b − π/z. The measurement is M = d_b/cos φ_M + d_p (even z) or M = d_b·cos(90°/z)/cos φ_M + d_p (odd z). Run backwards — measured M → φ_M → tooth thickness s → x = (s − πm/2)/(2m tan α) — it recovers the rack shift the machine actually cut, which is exactly how this sheet adjusts the profile. The span measurement W_k = m cos α·(π(k − ½) + z inv α) + 2xm sin α inverts the same way. The engine cross-checks the closed forms against a brute-force geometric pin-contact solve to ~10⁻⁷. Both measurements are also reality-checked: a pin must touch above the form diameter (not in the fillet), and the span anvils contact at d_Mk = √(d_b² + W_k²) (DIN 3960), which must land between the form and tip circles — outside that band the sheet flags the number as unmeasurable and suggests a usable k.

III·b — Thinning is a deeper cut. A gear cut thin for backlash is not a different curve: feeding the hob deeper by e reduces the thickness by 2e·tan α, drops the root by e, and shifts the trochoid — exactly the geometry of generating at x_gen = x − e/m. The sheet exploits that identity: Δs enters as x_gen = x + Δs/(2m tan α) for the flank, root, fillet and every measurement, while the tip circle keeps the design x (the blank was turned before hobbing). That is also why a print's OD and root Ø are honored verbatim when you type them: the OD is a lathe dimension, and the root tells you how deep the cutter really went (the sheet back-derives the effective cutter dedendum from it). Chordal numbers are the classic caliper pair: s̄ = d·sin(s/d) across the chord, checked at depth h̄_a = d_a/2 − (d/2)cos(s/d) set from the OD.

IV · Accuracy — certified three independent ways. All geometry is closed-form in double precision; curves are emitted as full-precision polylines refined until the chordal deviation is below your tolerance (default m/10,000 — about 12 µin on a DP 8 tooth). The engine's certification suite pins this down with hard absolute bounds of 1 µm and 0.0001 in: (1) known formulas — every diameter, thickness, the over-pin M and the span W re-evaluated from their textbook definitions agree to ~10⁻⁶ µm; (2) the string definition — an involute is a taut string unwound from the base cylinder, so for every emitted flank point the arc length wound must equal the straight length unwound (r_b·Δθ = √(r²−r_b²)) — verified point-by-point to ~10⁻¹¹ µm, with the string leaving the cylinder exactly tangentially; (3) computational audit — every emitted point is re-derived from an independent reimplementation of the exact curves (involute, rack-corner envelope, arcs) to ~10⁻¹¹ µm, the true mid-chord deviation of every segment is measured against the tolerance contract (sup below tol), and a tol → tol/4 → tol/16 refinement study shows point counts doubling (second-order sampling) with the enclosed area converging while the analytic outputs do not move at all. The solid is the same outline extruded and capped, vertex-welded and verified watertight (every edge shared by exactly two triangles; volume matches the cap-area × width identity to 10⁻⁶).