Helical Internal Ring Gear Design

I · Two planes, one ring tooth. A helical ring tooth is a spur ring tooth in the plane perpendicular to itself (the normal plane, where the cutter, the caliper and the ball live) wound onto a helix of angle β at the pitch cylinder. The plane of rotation (transverse) sees everything stretched by the wrap: m_t = m_n/cos β, tan α_t = tan α_n/cos β, d = z·m_n/cos β, while radial heights keep NORMAL proportions (addendum m_n(h_a*+x_n) inward, dedendum m_n(h_f*-x_n) outward). The shift and thinning map as x_t = x_ncos β and Δs_t = Δs_n/cos β. On the base cylinder the helix flattens to tan β_b = tan β·cos α_t, the angle that runs the metrology below. The lead is L = πd/tan β; one axial pitch is p_x = π·m_n/sin β. One internal-specific bonus falls straight out: the tips of a ring tooth must stay OUTSIDE the base circle for an involute to exist there, z ≥ 2(h_a*+x)_t/(1-cos α_t) in the transverse plane, and since α_t grows with β while the coefficients shrink by cos β, the classic ≈34-tooth floor of 20° full-depth spur rings falls fast with helix.

II · The flank, inside out. In the transverse plane the working flank is the same involute of the base circle r_b = r cos α_t, used on its CONCAVE side. Thickness opens OUTWARD: ψ(r) = s/d - inv α_t + inv φ_r, cos φ_r = r_b/r, so the tooth is thinnest at the tips (the minor Ø, the pre-machined bore) and thickest at the root (the major Ø). The identical closed forms (and the identical certification) as the spur ring sheet, run at (m_t, α_t, x_t).

III · The root, cut by a real cutter. The fillet is the envelope of the shaper cutter's tip-corner circle as the cutter (an external gear of z_c teeth, HELICAL, same hand and helix as the ring) rolls INSIDE the ring on the true generation circles, all of it in the transverse plane where shaping kinematics live. Conjugacy fixes everything: the ring's thickness sets the working pressure angle through inv α_w = inv α_t + 2(x - x_c)_t tan α_t/(z - z_c), the base-circle identity a·cos α_w = r_b - r_b,c sets the feed depth, and the cutter's tip reach is whatever produces the specified root at that depth. The generated point obeys the law of gearing, its normal passes through the rolling pitch point, so the envelope is explicit, the root tangency is closed-form, and the fillet-involute hand-off happens at the roll instant where the contact crosses the cutter's own flank-to-corner blend. Change the cutter tooth count and the fillet changes with it.

IV · Between balls, not pins. Two balls in opposite spaces, an inside mic between them. A cylindrical pin in a helical space lies along the helix and touches off-plane; a ball contacts both flanks in one transverse section. On the base-tangent plane the concave flank unrolls to a straight line inclined at β_b, and the ball clears it by half the ball measured in that plane, which inserts the base helix into the internal closure in two places: inv φ_M = inv α_t + π/z - s_t/d - d_ball/(d_b·cos β_b), center circle r_M = r_b/cos φ_M, M = 2r_M - d_ball (even z; odd z gets the cos 90°/z correction), contact at tan-length r_btan φ_M + (d_ball/2)cos β_b. Note the internal signs: a BIGGER ball sits DEEPER and reads SMALLER. The engine does not take the closure on faith: the test suite builds the exact internal involute helicoid point-by-point and finds the true 3-D tangency of the ball by brute force, the closed form matches to microns on even and odd counts, shifted and thinned. Run backwards, a measured M recovers x_n (or Δs_n), which is exactly how this sheet adjusts the profile. Everything reduces to the spur ring's between-pins equations at β = 0.

V · The twisted annulus, certified. The 3-D solid is the certified transverse section swept along the helix: each slice is the exact section rotated by (twist/width)·z, no re-approximation, with slice count set by your chord tolerance, and the mesh verified watertight (every directed edge used exactly twice) with volume matching the cap-area × width identity (Cavalieri, a twist does not change volume) and every rim vertex exactly on the rim cylinder (the blank does not twist). The STEP flanks are tensor-product B-spline surfaces built the same way: the u-direction carries the certified 2-D profile spline, the v-direction interpolates each control point's own helix track with exact helix end-tangents (see the method note on the helical sheet for what that is worth), so every constant-v section of the surface is the bottom section rotated and lifted, a property the test suite re-audits by parsing the emitted file and evaluating the surfaces against R(θv)·S(u,0) + Wv·ẑ. Tip lands live on one true minor CYLINDRICAL_SURFACE, the rim is two straight cylinder halves, and the edge graph is checked closed on the emitted bytes. All geometry is closed-form double precision underneath, inheriting the gear engine's three-way certification.