# Spur gear (ISO 53)

Involute spur gear, cut from the standard basic rack, as a plain disc with a round bore.

## Geometry

Axis along Z, centred on the origin. Pitch diameter m z, tip diameter m (z + 2), root diameter m (z − 2.5), base diameter m z cos 20°, tooth thickness π m / 2 on the
pitch circle. Flanks are smooth splines on computed involute points, so they sit very close to the true involute, not exactly on it; the root curve is a polyline. Below them runs the trochoid that the basic rack's tip corner cuts, so small tooth counts are undercut as
a real cutter leaves them. No profile shift is applied, and the rack's tip radius isn't modelled.

`detail = "simple"` is a disc at the tip diameter, for big assemblies. A bore must leave at least one module of wall under the root circle.

## Source

- **Basic rack:** ISO 53:1998, standard basic rack profile: pressure angle 20°, addendum 1.0 m, dedendum 1.25 m.
- **Modules:** ISO 54:1996, preferred modules (1.25 and others from its series).
- **Involute geometry:** derived from these with the standard relations for an unshifted gear (radii above; flank angle
  from the involute function inv φ = tan φ − φ; the root curve is the path of the rack's tip corner as it rolls on the pitch circle).

Tooth counts, bores and face widths come from the step.parts catalogue metadata (MIT, `library/NOTICE.md`): they are the
gear's size, set by the designer. Tooth quality and bore fit aren't stated: neither standard above sets them for a gear.