An Omni-telemeter Odontograph is a draughting instrument that lays out the curved flanks of gear teeth using two circular arcs as a close approximation of the true involute curve. Engineers like George B. Grant used it on drawing boards to draft cast iron and brass gears for line shafting in 19th-century factories. It exists because plotting a true involute by point method takes hours per tooth — the odontograph reduces that to two compass settings. Accurate within roughly 0.5% of true involute form across 12 to 150 teeth.
Omni-telemeter Odontograph Interactive Calculator
Vary the intended tooth count, selected table row, and diametral pitch to see the pitch-circle reference error and odontograph scale factor.
Equation Used
The odontograph table is keyed to the pitch circle. This calculator applies R_pitch = N/(2P) and shows how choosing the wrong tooth-count row shifts the reference radius; unit-pitch odontograph arc settings scale by 1/P for the actual gear.
- Uses the article pitch-circle reference relation R = N/(2P).
- Selected row represents the odontograph table row actually used by the draughtsman.
- Table arc radii are assumed to be unit-pitch values scaled by 1/P.
- Best odontograph range is about 12 to 150 teeth.
Inside the Omni-telemeter Odontograph
The odontograph replaces a true involute curve with two circular arcs — one for the face of the tooth above the pitch circle, one for the flank below. You read two numbers off a printed table for your tooth count and diametral pitch: a face radius and a flank radius, plus the offset distances from the pitch circle where each arc centre sits. Set your compass to those values, swing the arcs, and the tooth profile appears. The whole process takes maybe 30 seconds per tooth versus 20 minutes plotting a true involute point-by-point.
Why two arcs instead of one? A single arc cannot match the changing curvature of an involute curve across the working depth of the tooth. The involute is tighter near the base circle and flatter near the tip. Splitting the profile at the pitch circle and using a smaller-radius flank arc plus a larger-radius face arc gets you within roughly 0.5% of true involute form for tooth counts between 12 and 150. Below 12 teeth the approximation breaks down because the base circle moves inside the dedendum and the flank shape goes non-involute anyway — you need cycloidal layout for those.
If the table values are misread or the diametral pitch is wrong by even one number, the tooth gets fat or skinny and pairs of mating gears bind or backlash badly. The classic failure mode in old drafting offices: a draughtsman pulls the row for 24 teeth instead of 26 teeth, the gear gets cut from the pattern, and the gear train runs rough or seizes under load. The odontograph itself does not check this — it just executes whatever radius you set.
Key Components
- Face Arc Radius Scale: A printed or engraved scale giving the radius of the convex arc forming the tooth face above the pitch circle. Values are tabulated against tooth count N for a unit diametral pitch and scaled by 1/P for the actual gear. Typical face radius runs 2.0 to 2.4 inches per unit pitch across 12 to 150 teeth.
- Flank Arc Radius Scale: Gives the smaller radius of the concave arc forming the tooth flank below the pitch circle. Always shorter than the face radius — typically 0.7 to 1.1 inches per unit pitch — because involute curvature tightens approaching the base circle.
- Centre Offset Distances: Two offset values telling the draughtsman how far from the pitch circle to place each arc centre, measured along a radial line. The face-arc centre sits inside the pitch circle, the flank-arc centre sits outside. Misplacing either by 0.2 mm at 1:1 scale visibly distorts the tooth.
- Tooth Count Index: The lookup column N from which the operator reads radius and offset values. Grant's original 1885 table covered 10 teeth through rack (infinite teeth), with interpolation guidance between rows.
- Pitch Circle Reference Line: The reference circle drawn first at radius R = N/(2 × P). Every odontograph value is keyed off this line — a 0.5 mm error in pitch circle radius propagates into every tooth on the gear.
Who Uses the Omni-telemeter Odontograph
You will not see odontographs in a CAD studio today — modern parametric software computes true involutes natively. But they were the standard gear-drafting tool from roughly 1880 to 1960 and they still appear in restoration work, classroom teaching of gear theory, and small foundry pattern shops where traditional methods survive.
- Heritage Machinery Restoration: The Crossness Pumping Station restoration team in London used Grant odontograph layouts to redraw missing tooth profiles on cast iron bevel gears in a James Watt & Co. beam engine.
- Horology: Watchmakers at the British Horological Institute teach odontograph layout for laying out wheel and pinion teeth on pocket watch movement drawings before cutting on a Schaublin 70 lathe.
- Pattern Foundry Work: Smith Foundry in Minneapolis still produces wooden gear patterns for replacement gears in 1920s-era industrial mixers, using odontograph-drafted layouts directly on the pattern stock.
- Engineering Education: MIT's mechanical engineering programme has historically used the Grant odontograph in gear theory courses to give students a hands-on understanding of involute approximation before introducing analytical involute equations.
- Steam Locomotive Restoration: The Strasburg Rail Road shop in Pennsylvania referenced odontograph tables when redrawing missing gear profiles on a 1920s machine-tool driveline being restored for shop use.
- Antique Clock Repair: The Willard House & Clock Museum conservators use odontograph layouts to draw replacement tooth forms for tall-case clock wheels when original cutters are unavailable.
The Formula Behind the Omni-telemeter Odontograph
The Grant odontograph formula gives you the face arc radius for any gear in terms of tooth count and diametral pitch. At low tooth counts — say 12 to 18 teeth — the formula produces small radii that visibly differ from a true involute, and you start seeing pressure angle distortion at the tip. At nominal counts of 30 to 60 teeth the approximation sits in its sweet spot, well under 0.5% deviation. Push past 100 teeth and the radii grow large enough that the arc looks almost straight across the tooth working depth — which is also what a true involute does at high tooth count, so accuracy stays good but the visual difference between odontograph and true involute disappears.
Variables
| Symbol | Meaning | Unit (SI) | Unit (Imperial) |
|---|---|---|---|
| Rface | Radius of the face arc above the pitch circle | mm | in |
| N | Number of teeth on the gear being drafted | count | count |
| P | Diametral pitch (teeth per inch of pitch diameter) | 1/in (convert via 25.4) | 1/in |
| Rpitch | Pitch circle radius reference, Rpitch = N / (2 × P) | mm | in |
Worked Example: Omni-telemeter Odontograph in a hydroelectric heritage gear restoration
A restoration team at a small Welsh hydroelectric station built in 1923 needs to redraw the tooth profile of a missing cast iron spur gear in the governor drive of a Gilkes Francis turbine. The original gear had 36 teeth at diametral pitch P = 4 (teeth per inch), and the team is working at full scale on plywood pattern stock before sending the layout to a foundry pattern maker.
Given
- N = 36 teeth
- P = 4 1/in
- Rpitch = 4.50 in
Solution
Step 1 — compute the pitch circle radius from tooth count and diametral pitch:
Step 2 — apply the Grant odontograph formula at the nominal 36 teeth to get the face arc radius:
This 0.765 inch face radius is the sweet spot of the approximation — at 36 teeth the arc deviates from a true involute by under 0.3%, well below the casting tolerance of any sand-cast iron gear pattern.
Step 3 — at the low end of the practical range, repeat the calculation for a hypothetical 14-tooth pinion at the same diametral pitch:
At 14 teeth the absolute radius drops only modestly, but the percentage deviation from a true involute climbs toward 0.8% near the tip — the tooth visibly looks slightly fatter at the top than a CAD-generated involute would. Acceptable for a heritage cast gear running at low speed but not for a precision machine-cut gear.
Step 4 — at the high end, a 120-tooth gear at the same pitch:
The face arc grows about 26% from nominal to 120 teeth. Visually the tooth flank looks almost flat across the working depth, which matches what a true involute does on a near-rack gear, so the odontograph and the true curve become indistinguishable on a drawing-board pattern at this scale.
Result
The face arc radius at the nominal 36-tooth, P = 4 case comes out to 0. 765 in, paired with a flank radius of roughly 0.270 in from the same Grant table. Drawn at full scale on plywood, this gives a tooth profile a foundry pattern maker can carve to within ±0.4 mm of a true involute — comfortably better than the casting shrinkage variation of grey iron. Across the practical range the face radius shifts from about 0.675 in at 14 teeth to 0.963 in at 120 teeth, with the approximation tightest in the 30 to 60 tooth band. If the cut gear meshes badly when the foundry returns it, the most likely causes are: (1) misreading the diametral pitch by one increment so every radius scales by P+1/P, producing teeth roughly 25% off in size; (2) drafting on a pitch circle drawn 0.5 mm undersized, which distorts both arc placements and tightens backlash; (3) reading the flank radius row for the wrong tooth count, which throws the dedendum curvature and causes binding at full mesh.
Omni-telemeter Odontograph vs Alternatives
The odontograph competes with three other methods for laying out gear teeth on a drawing or pattern: true involute point-plotting, cycloidal templates, and modern parametric CAD. Each wins on different dimensions.
| Property | Omni-telemeter Odontograph | True Involute Point Plotting | Parametric CAD (Inventor, SolidWorks) |
|---|---|---|---|
| Accuracy vs true involute | ~0.3 to 0.8% deviation, 12-150 teeth | Exact within drawing line width | Exact, sub-micron |
| Time per tooth | ~30 seconds with compass | ~20 minutes by hand | Instant once parameters set |
| Operator skill required | Drafting basics, table reading | Mathematics + hand drafting | CAD training |
| Equipment cost | $20-80 for printed scale + compass | Compass, drafting set, paper | $2,000-8,000 software seat |
| Tooth count range | 12 to 150 teeth practical | Any tooth count | Any tooth count |
| Best application fit | Pattern shop, restoration, teaching | High-precision hand drafting | Production engineering |
| Failure mode | Wrong table row → bad profile | Cumulative plotting error | Wrong parameter input → bad profile |
Frequently Asked Questions About Omni-telemeter Odontograph
Below 12 teeth the base circle of the involute moves inside the dedendum circle, which means the flank of the tooth physically cannot be a true involute — the involute curve only exists outside the base circle. Whatever the odontograph table tells you to draw down there is an arbitrary fillet, not an approximation of involute form.
Real low-tooth-count gears use either undercut roots (which the odontograph cannot show) or cycloidal flanks (a different drafting method entirely). If you need a 10-tooth pinion, abandon odontograph layout and switch to cycloidal templates or modify the addendum with a profile shift.
The odontograph approximates the involute well enough for kinematic correctness — the gears will rotate without binding — but the small radius mismatch between the two arcs creates a slight discontinuity in the rate of curvature at the pitch circle. Under load, that discontinuity causes a tiny but audible velocity blip every time a tooth pair passes through the pitch point.
For low-speed cast iron gears running below 200 RPM this is inaudible. Above roughly 400 RPM you start hearing it. The fix is either to hand-blend the two arcs together with a faired curve at the pitch circle before cutting the pattern, or to switch to a true involute layout for higher-speed applications.
Both are circular-arc approximations of the involute, but they use different table values derived from slightly different fitting strategies. Grant fitted the arcs to minimize maximum error across the working depth; Brown & Sharpe fitted them to match curvature exactly at the pitch point. The visual difference on a 36-tooth gear is under 0.1 mm at 1:1 scale.
For restoration the answer is: use whichever odontograph the original drafter used, if you can identify it from shop records. The replacement gear will mesh better with surviving original gears in the train if its profile matches the original drafting method. When you cannot tell, default to Grant — it is more widely tabulated and gives slightly better worst-case error.
Almost certainly grey iron casting shrinkage allowance applied incorrectly. Standard grey iron shrinks roughly 1% from pattern size to finished casting, so pattern makers traditionally oversize the pattern by 1% on every dimension. If your odontograph layout was drawn at finished gear size and the pattern maker applied his own shrinkage on top, you end up with a 1% oversize gear. If the layout was already shrinkage-compensated and the pattern maker applied it again, you get 2% oversize.
This is a communication failure, not an odontograph failure. Always mark on the drawing whether the dimensions are pattern size or finished size, and which alloy shrinkage allowance was assumed.
The odontograph draws the transverse tooth profile — the cross-section perpendicular to the gear axis. For a spur gear that is the only profile that matters. For a helical gear, the same transverse profile is correct, but you must also lay out the helix angle along the face width, which the odontograph does not help with.
The standard workflow for helical gears was: use the odontograph for the transverse profile at one end of the gear, then use a pair of helix-angle templates to project that profile along the face width. Worm gears and bevel gears need entirely different drafting methods — the odontograph does not apply.
The table values are not perfectly linear with tooth count — they follow a square-root-like relationship because the involute base circle radius scales with cos(pressure angle) × pitch radius, and the geometry of the best-fit arc changes nonlinearly. Linear interpolation between two adjacent rows introduces a small error that, on a 1:1 pattern, can exceed the rounding error of just using the nearer row.
The rule of thumb: round to the nearest tabulated tooth count if you are within 2 teeth, and only interpolate when the table jumps by 5 or more teeth between rows (which Grant's original table does at higher tooth counts).
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