Helical Gear Design

Exact external helical-gear geometry from the normal module or normal diametral pitch and the helix angle. Enter a measurement over balls (helicals measure over balls, not pins) and the sheet back-solves the true profile shift and redraws the as-built tooth — then export a true twisted STEP AP203 solid, the transverse DXF, or an STL.

Updated: 7/4/2026

The live solid — drag to orbit. Every number, the exact profile and the exports follow below.

1 Type what you know — teeth, helix + one size is enough 2 Check it on balls — or type what the shop measured 3 Read every diameter, the lead & both planes 4 Inspect the exact transverse profile 5 Export twisted STEP / DXF / STL

1 · The gear — size, helix, form & blank

Metric — module English — DP

Teeth z

Normal diametral pitch P_nd

1/in

Helix angle β (deg)

Hand

Pressure angle α_n

Profile shift x_n

auto: no undercut

Thinning Δs_n

Bore Ø

Min wall

Face width

Chord tolerance

Reading a print with half the numbers? Set the helix angle (it is on every helical print — or measure the lead L and use tan β = πd/L), then type any two independent facts below (or one plus the tooth count above) — the rest fill in gray as live results. Type over any gray value to redefine the gear; clear a field to hand it back. A typed outside Ø or root Ø is honored exactly in every view and export — the blank is turned (and the cutter fed) to the print, not to the textbook formula.

Diameters

Outside Ø

Pitch Ø

Root Ø

Base Ø

Pitches & tooth

Transv. circular pitch

Transv. base pitch

Normal thickness

Chordal thickness

Chordal addendum

Addendum

Dedendum

Whole depth

A standard full-depth 20°-normal, 15° right-hand helical loads immediately, and everything re-solves the moment you stop typing. Count the teeth, read β off the print and mic the OD — that is enough. Entering the pitch or module directly always works too; typing it hands the helper fields back to output duty.

auto: optimal rack — full depth, max full-round corner default

Addendum h_a*

Dedendum h_f*

Tip radius ρ*

Auto rack is ON: standard full-depth proportions (h_a* = 1.00, h_f* = 1.25) with the largest full-round corner the rack tip carries at this normal pressure angle — ρ* = 0.471, the full-fillet hob and the strongest standard root. Untick to spec your own cutter.

These describe the generating rack (hob) in its NORMAL section — the same hob cuts spurs and any helix, which is the whole point of the normal system. The root fillet in every view is the true generated trochoid of this corner in the transverse plane — not a pasted arc — so undercut, when it happens, is the real cut shape.

2 · Measure over balls — and set the profile from what the shop measured

Reading drives

profile shift x_n thinning Δs_n

Ball Ø

Measured M

→ shift x_n

→ thinning Δs_n

Measured span W

→ shift x_n

→ thinning Δs_n

Ball diameter (auto — contacts at pitch Ø)

0.21378 in

Measurement over two balls

3.39864 in

Ball contacts flank at Ø

3.10583 in

Span W over 4 teeth

1.33794 in

Face needed for the span

0.32540 in

Measure helicals over balls — a cylindrical pin lies along the tooth space helix and reads differently. Even tooth count — balls sit diametrically opposite.

The classic shop loop: hob the gear, mic over two balls (or span k teeth square to the flank), and adjust. This sheet runs the same loop backwards, LIVE: type what the mic read and it drives the whole model as you type — the reading stays in its field, Reading drives chooses whether it owns the profile shift x_n (the classic decode of an unknown gear, the default) or the thinning Δs_n with the print's shift trusted (tip and root stay put; only the tooth gets thinner, exactly like feeding the hob deeper). The owned field is locked and follows the reading; clear the reading and your own numbers come back untouched. The two decode fields beside each reading always show both interpretations — click one to copy it if you want to keep it in the design. When both M and W are typed, the one you edited last drives. A blank ball field bakes the suggested ball in the moment M starts driving, so the assumed ball cannot drift with the shift. Even tooth counts measure straight across; odd counts get the cos(90°/z) correction automatically. The ball equations carry the base helix angle exactly (a ball reads differently from a spur pin on the very same tooth thickness), and the span readout is checked against the involute band and the face width — the base-tangent plane crosses the face at β_b, so a span that needs more face than the gear has is flagged as unmeasurable.

3 · Every dimension that matters — both planes

Normal diametral pitch P_nd	8.0000 1/in
Teeth z · normal pressure angle α_n	24 · 20.00°
Helix angle β · hand at the pitch cylinder	15.000° · right hand
Base helix angle β_b tan β_b = tan β · cos α_t	14.076°
Transverse DP · pressure angle α_t the plane the profile below lives in	7.7274 1/in · 20.647°
Profile shift x_n	0.0000
Pitch diameter d z·m_n/cos β	3.10583 in
Base diameter d_b	2.90635 in
Tip (outside) diameter d_a	3.35583 in
Root diameter d_f	2.79333 in
Form diameter (true SAP / TIF) fillet → involute hand-off	2.93727 in
Lead L one full wrap of the helix	36.41455 in
Axial pitch p_x	1.51727 in
Normal base pitch p_bn base-pitch checker reading	0.36902 in
Transverse circular / base pitch	0.40655 in / 0.38044 in
Virtual (equivalent) teeth z_v z / (cos²β_b · cos β)	26.41
Overlap ratio ε_β b · sin β / (π·m_n)	1.153
Tooth thickness s_n @ pitch normal plane · transverse s_t = 0.20328 in	0.19635 in
Chordal thickness s̄ caliper in the normal plane (virtual gear)	0.19623 in
Chordal addendum h̄_a caliper depth, set from the OD	0.12792 in
Tip thickness s_a (transverse)	0.09455 in
Span over 4 teeth W base tangent, normal plane	1.33794 in
Roll angles form / pitch / tip transverse involute-checker settings	8.38° / 21.59° / 33.07°

4 · The exact transverse profile

Scroll to zoom, drag to pan. Transverse section (the plane ⊥ the axis) — 24 exact teeth of the mt = 0.12941 in transverse involute, true gen — static preview; the interactive view loads with JavaScript.

Scroll to zoom, drag to pan. Transverse section (the plane ⊥ the axis) — 24 exact teeth of the m_t = 0.12941 in transverse involute, true generated trochoid root fillets, sampled at 0.01 thou·10⁻³ chordal tolerance — 10,032 points. The helix twists this section along the axis; the table above carries the normal-plane numbers.

5 · Take it with you — twisted STEP, DXF & STL

STEP AP203 is the real thing — an analytic boundary representation of the twisted solid, not facets: planar caps (the top one rotated by the twist), an exact cylindrical tip land, a straight bore, and flank surfaces written as tensor-product B-spline helicoids — built so every constant-height section is exactly the certified transverse profile rotated along the helix, with true helix end-tangents (the test suite audits the sweep property against the file itself). Units are declared in the file (inch or millimetre per your toggle), so it imports at true scale and machines from solid. The DXF is one closed polyline of the transverse section (plus the bore circle) at the chord tolerance above — a flat file cannot carry the helix, so use it for the section, not the part. The STL is the watertight twisted faceted solid — STL carries no units, so tell your slicer it is in inches (English mode) or millimeters (metric mode).

Parasolid (.x_t)? Parasolid is Siemens’ closed kernel format — there is no faithful third-party writer, so this sheet will not pretend to make one. The standard route: import the STEP into any Parasolid-based CAD (SolidWorks, NX, Solid Edge) and Save As → .x_t; the helicoid arrives as smooth analytic geometry, and the kernel does the conversion natively.

6 · The twisted solid — bore, face width, every tooth

Watertight twisted solid (right-hand helix, lead 36.41 in): 406,464 triangles, 203,232 welded vertices · volume 12.348 in³ · drag to orbit. — static preview; the interactive view loads with JavaScript.

Watertight twisted solid (right-hand helix, lead 36.41 in): 406,464 triangles, 203,232 welded vertices · volume 12.348 in³ · drag to orbit. The STL button exports exactly this mesh.

I · Two planes, one tooth. A helical tooth is a spur tooth in the plane perpendicular to itself (the normal plane — where the hob, 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)) — the same hob cuts every helix. 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.

II · The section is the spur machinery, exactly. In the transverse plane the flank is the true involute of d_b = d·cos α_t and the root fillet is the generated trochoid of the hob corner — the identical closed forms (and the identical certification) as the spur sheet, run at (m_t, α_t, x_t). One classic consequence falls straight out: the undercut limit z_min = 2(h_f,t* − x_t − ρ_t*(1 − sin α_t))/sin²α_t improves with helix (α_t grows, the coefficients shrink by cos β) — a 12-tooth pinion that undercuts as a spur cuts clean at β = 30°. The virtual spur gear seen in the normal plane has z_v = z/(cos²β_b·cos β) teeth of module m_n — the sheet uses it for the chordal caliper pair, and it is why helicals also run like bigger gears than they count.

III · Balls, not pins. A cylindrical pin in a helical space lies along the helix and touches off-plane; the standard practice is a ball, which contacts both flanks in one transverse section. On the base-tangent plane the flank unrolls to a straight line inclined at β_b, and the ball center clears each flank by half the ball measured in that plane — which inserts the base helix into the spur closure in two places: inv φ_M = s_t/d + inv α_t + d_ball/(d_b·cos β_b) − π/z, center at r_M = r_b/cos φ_M, and M = 2r_M + d_ball (even z; odd z gets the cos(90°/z) correction). The engine does not take that on faith: the test suite builds the exact involute helicoid surface 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) exactly as on the spur sheet.

IV · The span lives in the normal plane. A disc-anvil micrometer spans k teeth square to the flanks: W_k = m_ncos α_n·(π(k − ½) + z·inv α_t) + 2x_nm_nsin α_n — normal quantities outside, the transverse involute function inside. Two reality checks gate it: the contact diameter d_Mk = √(d_b² + (W_k/cos β_b)²) must land between form and tip circles, and — special to helicals — the base-tangent plane crosses the face at β_b, so the measurement needs face width b ≥ W_k·sin β_b. A narrow face fails that floor and the sheet says so (use the balls instead).

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 normal thickness by 2e·tan α_n, drops the root by e, and shifts the trochoid — exactly the geometry of generating at x_n,gen = x_n + Δs_n/(2m_ntan α_n) for the flank, root, fillet and every measurement, while the tip circle keeps the design x_n (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).

V · The solid and the STEP helicoid — 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). 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, 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 CYLINDRICAL_SURFACE; the boundary columns of each flank double as the wall edge curves, so the shell is watertight by construction. All geometry is closed-form double precision underneath, inheriting the spur engine's three-way certification (textbook formulas, the taut-string involute property, and an independent re-derivation of every emitted point).

Method note — exact-tangent helicoid fitting, and what it is worth

STEP has no native "involute helicoid" surface, so every helical-gear exporter — this one included — must represent the flank as a NURBS surface. The standard construction interpolates a handful of cross-sections along the twist with cubic splines. It has a quiet failure mode that this sheet measures, fixes, and proves fixed on every build — the walk-through and the numbers follow.

1 · The construction. Each flank is a tensor-product B-spline: the u-direction carries the certified transverse profile spline (the same one the 2-D view certifies against the true involute and trochoid); the v-direction interpolates each profile control point's own helix track through N_v+1 sections, all tracks sharing one parameterization so the tensor product is legal and the boundary columns double as the wall edge curves (watertight by construction). Because rotation is linear, the surface then agrees with the rotated section exactly at every interpolated station — whatever error remains lives between stations, in the v-splines.

2 · The shortcut and its cost. A clamped cubic interpolant needs end derivatives, and the common shortcut is the chord slope between the first two (last two) points. For a circular helix that direction is off by half an angular step Δθ/2, and the biased clamp drags the first and last spans inward — a sag that measures ≈ R·Θ·Δθ/48 (R = radius, Θ = total twist), only second order in the section count. At the section counts a tolerance-driven exporter actually picks (4–5 across the twist), that is 6.8 to 194 µm on realistic gears — 21× to 342× past this sheet's own certification tolerance. In shop terms: a quarter-thou to eight thou of false lead, concentrated near the two faces — the size of a deliberate lead crown, or a whole lead-tolerance band, baked silently into the CAD. A CAM or CMM programmer consuming that file would chase a phantom lead error the hob never cut.

3 · The fix is free. The v-curves are not arbitrary data — they are exact helices, and a helix's tangent is closed-form: dP/dv = Θ·(−y, x, 0) + W·ẑ. Clamping the interpolant with the true end tangents restores the full fourth-order accuracy of cubic interpolation, and the residual lands on the textbook floor R·Δθ⁴/384 — the measured deviations below match that formula to two digits. No extra sections, no larger file: the same 7-column control net that missed by microns now misses by tenths of a nanometre.

4 · The proof loop. The exporter does not ask to be trusted: the test suite re-parses the emitted STEP text, rebuilds the surfaces with an independent de Boor evaluator, and audits the sweep law S(u,v) = R(Θv)·S(u,0) + Wv·ẑ across the parameter grid — so the helicoid property is verified on the very bytes a CAD system will read, every build.

Measured results — fitted flank-edge track vs the true helix (writer's own section count at the default tolerance ≈ 0.2–0.6 µm; identical fitting system, only the end condition differs):

Gear	Twist Θ	Sections	Chord-clamped	Exact-tangent	Improvement
z24 P_nd8 β15° · 1.00″ face	0.173 rad	4	6.8 µm	0.00039 µm	17,000×
z24 P_nd8 β15° · 1.75″ face	0.302 rad	4	20.5 µm	0.0036 µm	5,700×
z25 P_nd10 β28° · 1.00″ face	0.376 rad	4	28.3 µm	0.0077 µm	3,700×
z60 m_n2 β20° · 50 mm face	0.285 rad	4	28.2 µm	0.0044 µm	6,400×
z24 P_nd8 β15° · 3.00″ face	0.518 rad	4	60.2 µm	0.031 µm	1,900×
z12 m_n3 β30° · 40 mm face	1.111 rad	5	100 µm	0.15 µm	650×
z36 m_n4 β45° · 60 mm face	0.589 rad	4	194 µm	0.13 µm	1,500×

The chord column violates the certification tolerance in every row; the exact-tangent column is under it in every row. The same audit run on the actual emitted files for the 1″-face gear reads 6.31 µm with the chord clamp and 0.00039 µm with the exact tangents — and the study's replicated fit agrees with the shipped file to four digits (0.003611 µm both ways on the 1.75″ demo), so the numbers above are the numbers in the file.

Or spend sections instead: the chord clamp converges only as 1/N_v², so on the demo gear it needs 32 sections to scrape under tolerance and ≈ 300 to match what the exact tangent delivers with 4 — a 5× (certified) to 43× (equal-accuracy) difference in surface data for the same geometry. The exact tangent is the better deal in both directions: more accurate and smaller.

None of this is exotic mathematics — clamped splines with prescribed end derivatives are classical (de Boor). The point is quantitative: at the section counts real exporters use, the customary chord shortcut writes a lead-crown-sized artifact into helical gear CAD, the exact helix tangent removes it at zero cost because the tangent is closed-form, and an exporter can prove the property on its own output. Check any helical STEP you receive against its own sweep law — it is a five-line test, and this page's files pass it by three orders of magnitude.

Notes on using the results

Thrust is the price of quiet. The helix turns part of the tooth force axial: F_a = F_t·tan β at the pitch cylinder — a 20° helix pushes ~36% of the tangential load into the bearings. Locate the shaft with a bearing that takes thrust, or cancel it with a double-helical (herringbone) pair.
Hands must oppose on parallel shafts. An external helical pair on parallel axes meshes right-hand against left-hand at the same β. Same-hand external gears only mesh on crossed axes (a different, point-contact animal). The Hand selector drives the 3-D solid and the STEP so the part you export is the part you meant.
Face width earns the overlap. The smooth-running benefit comes from the overlap ratio ε_β = b·sin β/(π·m_n): at ε_β ≥ 1 at least one full tooth is always entering as another leaves. The sheet reads it out and flags a face too narrow to reach 1 — widen the face, steepen the helix, or accept spur-like noise.
Backlash & tolerances. A real mesh runs with backlash: gears are cut a few hundredths of a module thinner than theoretical. The Thinning Δs_n field carries that allowance separately from the design shift — tip and root stay at the print values while the tooth, the over-ball M, the span and the chordal caliper numbers all follow. Measure the actual part and choose whether the reading sets x_n (unknown gear) or Δs_n (known print, unknown allowance).
Mating checks. This sheet is single-gear geometry. Center distance, working pressure angle, total contact ratio (ε_α + ε_β) and root clearance of a pair are the natural next sheet.

References & further reading

tec-science — Profile shift — the x = 1 − z·sin²α/2 no-undercut relation this sheet applies in its cutter-true form.
Machinery's Handbook — Gearing section: helical gear formulas (normal/transverse), measurement over balls, span measurement.
KHK — Gear Technical Reference — helical gear calculations, normal vs transverse systems, over-ball tables.
Wikipedia — Gear (helical) — helix hand, thrust, parallel vs crossed axes.
Wikipedia — Involute gear — the involute function and its properties.
Litvin & Fuentes, Gear Geometry and Applied Theory — screw involute surfaces (involute helicoids), generation and envelopes.

Disclaimer

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