The Involute Curve — Definition and Fundamental Property
The involute of a circle is the curve traced by a point on a taut string as it is unwound from the circle’s surface. For a helical gear, this circle is the base circle — and the base circle radius d_b/2 is the single most geometrically important dimension of the gear because it determines the entire shape of the tooth flank. Two properties of the involute make it ideal for helical gear tooth forms:
- Constant pressure angle: At every point on the involute, the angle between the common tangent to the involute and the tangent to the base circle at the contact point equals the transverse pressure angle α_t. This is constant regardless of where on the involute the contact occurs — the key property that makes the involute gear transmit a constant angular velocity ratio even if the centre distance varies slightly.
- Self-consistency of meshing pairs: Two involutes generated from the same base circle (a gear and its pinion with equal or different tooth counts) will mesh correctly with a constant velocity ratio. No other curve has this property — it is the geometric reason why the involute became the universal helical gear tooth form in the 19th century and has never been superseded.
Key Circle Diameters — What They Mean and How to Calculate Them
A complete helical gear tooth form involves five concentric reference circles, each playing a different role in the gear geometry and inspection. For a helical gear with normal module Mn, tooth count z, normal pressure angle α_n = 20°, and helix angle β:
| Circle Name | Symbol | Diameter Formula (standard gear, x=0) | Role |
|---|---|---|---|
| Pitch circle | d | d = Mn × z / cos β | Reference circle where the gear is defined. Pitch-line velocity v_t = π × d × n / 60,000. Determines centre distance with mating gear: a = (d₁ + d₂) / 2. |
| Base circle | d_b | d_b = d × cos α_t = Mn × z × cos α_n / (cos β × cos α_t × cos β) … simplified: d_b = d × cos α_t | The circle from which the involute is generated. All tooth contact occurs on the involute — which begins at d_b. No involute exists below d_b. |
| Tip (addendum) circle | d_a | d_a = d + 2 × Mn (standard addendum h_a = 1.0 × Mn) | The outer diameter of the gear body. Contact ends at the tip circle. The tip is the most heavily stressed point of the tooth root of the mating gear during the approach phase. |
| Root (dedendum) circle | d_f | d_f = d − 2.5 × Mn (standard dedendum h_f = 1.25 × Mn) | The root circle at the tooth root. Not a contact surface — the root fillet begins here. The case depth ECD must exceed a minimum below d_f to prevent case crushing. |
| Form circle | d_F | d_F = √(d_b² + (d_a_mating × sinα_t)²) … approximate: d_F ≈ d_b + 2 × (design margin) | The smallest diameter at which the gear analyser starts the profile measurement. Below d_F the tooth fillet begins; above d_F the profile must follow the theoretical involute. The active profile extends from d_F to d_a. |
Example: M5, z=24, β=20°, α_n=20°
α_t = arctan(tan20°/cos20°) = arctan(0.3640/0.9397) = 21.17°
d = 5 × 24 / cos20° = 127.8 mm
d_b = 127.8 × cos21.17° = 127.8 × 0.9320 = 119.1 mm
d_a = 127.8 + 2×5 = 137.8 mm
d_f = 127.8 − 2.5×5 = 115.3 mm
Note: d_f (115.3 mm) < d_b (119.1 mm) — the root circle is INSIDE the base circle.
This means the tooth fillet region (from d_f to d_F) lies below the base circle and
cannot be an involute — it is a trochoidal fillet generated by the tool tip geometry.
The active involute profile begins at d_F (above d_b) and extends to d_a.
Close-up of a helical gear tooth flank: the active involute profile (the zone where mesh contact occurs with the mating gear) extends from the form circle d_F to the tip circle d_a. The root fillet below d_F is generated by the gear cutting tool’s tip radius and cannot be on the involute; this is the highest-stress zone of the tooth but not a contact surface
The Active Profile — What the Gear Analyser Actually Measures
The gear analyser measures the actual tooth flank profile along a straight line of roll in the transverse plane — starting at the form circle diameter d_F (the start of the useful involute) and ending at the tip circle diameter d_a. This measurement line is called the evaluation range L_αF. The profile deviations measured within this range describe how closely the actual tooth flank follows the theoretical involute:
Profile Deviation Parameters (DIN 3962 / ISO 1328-1)
The total deviation band [µm] within which the actual helical gear profile lies across L_αF. F_α is the primary DIN profile accuracy parameter: DIN Class 4 has F_α ≤ 7 µm for M5; DIN Class 7 has F_α ≤ 22 µm. F_α determines the transmission error amplitude at mesh frequency — directly affecting noise, vibration, and K_V.
The systematic linear inclination of the helical gear mean profile from the involute [µm]. Positive f_Hα means the tooth is thicker at the tip — the pressure angle is effectively larger than specified. f_Hα governs the entry/exit impact during mesh — it is the target of tip relief modification (Art46). An f_Hα value within tolerance but close to the limit indicates a pressure angle error in the grinding wheel dressing.
The waviness of the helical gear actual profile around the mean line [µm] — the high-frequency component after the f_Hα slope is removed. f_f is the component that most directly excites noise at harmonic frequencies of the mesh frequency. It reveals grinding wheel vibration, spindle run-out, and thermal distortion during grinding. f_f on a helical gear cannot be reduced by profile shift or tip relief — only by better grinding control.
Why the Form Circle d_F Matters — Undercutting and Measurement Range
The form circle d_F marks the transition between the theoretical involute profile (above d_F, towards the tip) and the trochoidal root fillet (below d_F, towards the root). Two important consequences:
Consequence 1 — Undercutting Detection
If the active contact begins below the form circle d_F (i.e. the mating gear’s tip contacts the subject gear below where the involute starts), the contact occurs on the non-involute trochoidal fillet. This is the undercutting condition — the mating gear tip “undercuts” the fillet instead of running smoothly on the involute. Undercutting causes: an irregular velocity ratio at the affected part of the mesh cycle; weakening of the tooth root (material removed from the fillet zone); and, in severe cases, interference that prevents the gears from meshing altogether. Positive profile shift (Art61) moves d_F upward to prevent undercutting in low-tooth-count helical gear pinions.
Consequence 2 — Gear Analyser Measurement Start
The gear analyser must use the correct d_F for each helical gear — this is the starting point of the profile measurement. If d_F is set too small (below the actual fillet boundary), the analyser will try to measure the non-involute fillet region as if it were an involute and report false large deviations at the root end of the profile chart. Korea Ever-Power calculates d_F for every helical gear order and programs it into the gear analyser before measurement, confirming that the measurement range L_αF covers only the true involute zone.
Form circle diameter (approximate, for standard gear with x=0 and standard tip circle on mating gear):
d_F ≈ max(d_b, √(d_b² + [(d_a_mating/2)² – a² × sin²α_t]))
where: d_a_mating = tip circle diameter of the mating gear [mm]
a = centre distance [mm]
α_t = transverse pressure angle [degrees]
For a gear meshing with an equal gear (z₁ = z₂ = 24, M5, β=20°, a=127.8mm):
d_F ≈ √(119.1² + [(137.8/2)² − 127.8² × sin²21.17°])
d_F ≈ √(14184.8 + [4768.4 − 2136.5])
d_F ≈ √16816.7 ≈ 129.7 mm ← Measurement starts at d_F = 129.7 mm (above d_b = 119.1 mm)
Normal vs Transverse Plane — Why the Analyser Measures in the Transverse Plane
A helical gear drawing specifies α_n (the normal pressure angle — perpendicular to the tooth lead) because this is the angle of the cutting tool. However, the involute tooth form exists in the transverse plane (perpendicular to the gear axis). The gear analyser measures the profile deviation in the transverse plane — using the transverse pressure angle α_t (not α_n) as the basis for the theoretical involute. This distinction matters for interpreting the analyser chart: the theoretical involute in the chart is calculated from α_t, not α_n. If a gear engineer calculates the expected roll angle range for the measurement using α_n instead of α_t, the calculated d_F will be incorrect and the analyser chart will show false profile form deviations at the measurement boundaries.
Korea Ever-Power — Profile Measurement Report with Every Helical Gear
Korea Ever-Power gear analyser profile chart for a DIN Class 5 precision ground helical gear — the chart shows the actual profile deviation (black line) within the evaluation range L_αF from form circle d_F to tip d_a. The slope f_Hα (the fitted mean line inclination) and the form deviation f_f (waviness around the mean) are calculated automatically. In this case: F_α = 6.2 µm, f_Hα = 3.1 µm, f_f = 4.8 µm — all within the DIN Class 5 tolerance for M5
Korea Ever-Power provides the full gear analyser profile chart (F_α, f_Hα, f_f — actual deviation plot) for every precision helical cut gear order of DIN Class 5 and above. The form circle d_F used in the measurement is documented on the certificate — confirming that the measurement range covers only the true involute zone. For helical gear orders with tip relief applied, the tip relief magnitude C_α and start angle are both confirmed on the profile chart — the chart shows the intentional positive deviation at the tip zone that constitutes the tip relief, and the linear region below that confirms the unmodified involute portion. As a direct helical gear manufacturer, Korea Ever-Power’s gear analyser uses a calibrated stylus traceable to national length standards — giving results traceable to ISO 1328-1 requirements. Browse the helical gear product range.
Frequently Asked Questions
A large f_Hα on a helical gear analyser chart indicates that the actual tooth flanks are systematically inclined relative to the theoretical involute — the tooth is effectively cut or ground at a slightly different pressure angle than specified. The most common cause: the grinding wheel dressing angle was set incorrectly (by a fraction of a degree), so every tooth was ground with a slightly wrong profile slope. Other causes: the grinding machine’s “involute kinematic” setting (the parameter that determines how the grinding wheel moves relative to the gear to generate the involute) was calibrated with a wrong base circle radius — which happens if the transverse pressure angle α_t was entered as the normal pressure angle α_n (a common error for helical gears). Korea Ever-Power verifies the α_t input (not α_n) for all grinding machine setups and includes f_Hα in the pre-shipment check.
Yes — F_α is the primary predictor of transmission error in a helical gear of the transmission error (TE) amplitude at mesh frequency. Approximately: TE ≈ F_α × (stiffness correction) / pairs in contact for the helical gear. For ε_γ = 2.0 (two tooth pairs sharing the load), the TE amplitude is approximately 0.35–0.5 × F_α. For a helical gear with F_α = 6 µm at DIN Class 5: TE ≈ 2–3 µm — the printing press specification (Art59) requires TE ≤ 3 µm, confirming DIN Class 5 is the minimum adequate. With F_α = 22 µm at DIN Class 7: TE ≈ 8–11 µm — three to four times above the printing press specification, confirming that hobbed DIN Class 7 is inadequate for precision printing applications.
The evaluation range L_αF in the gear analyser is the range over which the F_α, f_Hα, and f_f values are calculated — starting at the form circle d_F and ending 0.45–0.5 × Mn below the tip d_a (a small margin is excluded at the tip because the tip chamfer or radius creates a measurement artefact). The usable involute range is slightly narrower still — it excludes the tip and root zones where the profile deviation may be intentionally modified by tip relief or root fillet. For a helical gear with parabolic tip relief: the analyser chart shows the full evaluation range including the tip relief zone; F_α is calculated over the full range including the tip relief deviation, but f_Hα and f_f are calculated over the reference range (excluding the tip relief region) to show the quality of the unmodified involute separately from the intentional tip modification.
Not directly — d_b is a mathematical construction. It is verified in a helical gear indirectly through span measurement W_k (which measures the base tangent length — a quantity derived directly from d_b) or through the gear analyser profile measurement (which fits the theoretical involute generated from d_b to the actual profile). If W_k matches the calculated nominal within DIN 3967 tolerance, the helical gear base circle is confirmed correct. A W_k outside the expected range on a helical gear indicates an incorrect base circle — wrong module, tooth count, pressure angle, or profile shift. Korea Ever-Power cross-checks W_k against the gear analyser base circle determination for every helical gear at DIN Class 4–6.
Full Profile Chart with Every Helical Gear Order (DIN Class 5+)
Korea Ever-Power provides the gear analyser profile chart (Fα, fHα, ff — actual deviation plot plus form circle d_F and evaluation range L_αF) for every DIN Class 5 and above order. Tip relief is shown on the chart and confirmed against the specified C_α before shipment.
Fα · fHα · ff profile chart · d_F documented · α_t correctly applied · Tip relief confirmed · ISO 1328-1 traceable · Standard DIN 5+
Editor: Cxm
