The Drawing
x — · y —
Flags
The Numbers
Every diameter on the drawing, and whether the tooth is whole.
The same wheels, as clockmakers engraved them. See The Complication
A gear is a curve with a reason.
x — · y —
Every diameter on the drawing, and whether the tooth is whole.
The same wheels, as clockmakers engraved them. See The Complication
Four sentences on why the involute keeps a constant speed.
Wrap a string around a circle, tie a pencil to its end and unwind it taut: the pencil draws an involute, and the string is always the curve's normal. Two such curves pressed together touch where their normals coincide, so the common normal at every contact point is the one line tangent to both base circles. That line cuts the line of centres at a fixed point, the pitch point, and the gears turn as if two cylinders rolled there without slipping: the speed ratio is the ratio of base circle radii and nothing else. Move the centres apart and the pressure angle opens, the base circles stay what they were, and the ratio does not change.
A rack cutter's tip reaches below the base circle when z < 2(ha* − x)/sin²α: 17.1 teeth at 20°, so the handbooks say 17. Below that count the rack tip, swinging round on its fillet, scoops out the foot of the involute it has just cut. The flank loses its lowest part and the tooth gets a neck. Positive profile shift lifts the cutter and buys teeth back: at x = 0.3 a 12-tooth pinion clears, and the page says so the moment you type it.
The standards, the textbook and the tests the page passes.
Every gear here is generated the way a rack cutter generates it. The basic rack is ISO 53 profile A: addendum 1.0 m, dedendum 1.25 m, root fillet 0.38 m, at the pressure angle you choose. The flank is the exact involute of the base circle db = m z cos α, drawn as a parametric curve from the fillet junction to the tip, never as arcs. The root fillet is the envelope of the rack tip's fillet circle as the rack rolls along the pitch line: a trochoid offset by the fillet radius, computed from the rolling motion and sampled to the chordal tolerance you set. When the tooth count drops under the undercut threshold the page finds where the trochoid cuts back through the involute and stops the involute there.
inv α = tan α − α inv αw = inv α + 2 (x₁ + x₂) tan α / (z₁ + z₂) a = m (z₁ + z₂) / 2 · cos α / cos αw ε = (√(ra₁² − rb₁²) + √(ra₂² − rb₂²) − a sin αw) / (π m cos α)
Pairs follow ISO 21771: the operating pressure angle from the involute function by Newton's method, the centre distance from it, the path of contact between the tip circles along the line of action, and the contact ratio as that path over the base pitch. Internal pairs add the involute-interference test z₁/z₂ ≥ 1 − tan αa₂ / tan αw (Maitra §2.26). Trains are exact: ratios are fractions of integers, reduced, and the planetary cases come from the Willis equation (ωs − ωc)/(ωr − ωc) = −zr/zs. The animation integrates those speeds and nothing else; every planet both spins and orbits at the rate the fraction says.
The test suite holds 78 tests with 182 assertions in six files. Each one names a number from a standard, a handbook or an independent computation, and the tolerance was never loosened to pass.
| Receipt | Expected | This page | Verdict |
|---|---|---|---|
| inv(20°) | 0.014904 | 0.0149044 | pass, 1e-6 |
| m 2, z 20, 20°: db · da · df | 37.588 · 44 · 35 | 37.588 · 44 · 35 | pass |
| undercut limit at 14.5° · 20° · 25° | 31.9 · 17.1 · 11.2 → 32 · 17 · 12 | 31.903 · 17.097 · 11.199 | pass |
| 20/40 shifted x = 0.3 each: αw, a | Newton = bisection | 22.72°, 61.125 mm, Δ < 1e-10 | pass |
| contact ratio, 20/40 standard | 1.5 to 1.7 | 1.635 | pass, pinned |
| planetary 20/20/60 | 4 : 1, N 4 yes, N 3 no, −3 | 4 : 1, 80 mod 4 = 0, 80 mod 3 = 2, −3 | pass |
| involute equation on sampled flank | r = rb / cos φ | < 1e-9 mm | pass |
| rack swept through its motion, z 20 and z 12 | never cuts the finished tooth | deepest 7.2e-6 mm, touches at 1e-8 | pass |
| kerf 0.1 on the polygon | tip + 0.1 exactly | +0.1 within the chord tolerance; 4.7e-4 over at 0.01 | exact to tolerance, pinned |
| DXF read back by its own reader | AC1009, one closed polyline per outline | 4 polylines, 2 layers, closed | pass |
The one pinned miss: the kerf offset of the sampled polygon overshoots the true offset by up to the chord sagitta at a tip corner (4.7e-4 mm at a 0.01 mm tolerance, 5e-5 at 0.001). Tighten the tolerance and it shrinks with it.
ISO 53:1998 for the basic rack; ISO 21771:2007 for the geometry and the pair; Litvin and Fuentes, Gear Geometry and Applied Theory (2004), §6.9–6.10 for rack generation, fillet and undercutting; Maitra, Handbook of Gear Design (1994), chapter 2 and chapter 9; DIN 6885-1 for the keyways. The file formats: SVG 1.1 with 1 user unit = 1 mm, DXF R12 (AC1009) with closed POLYLINE entities and one layer per part, $INSUNITS set to millimetres.
Add the page to your home screen and it opens offline. The address bar carries the whole train: copy the link and the drawing travels with it.