The involute of a circle, the tooth it becomes, and why two gears mesh with a constant angular-speed ratio. Drag the sliders; every quantity is recomputed live (rounded if necessary).
Involute of a circle. Unwind a taut string from the base circle: its free end traces the involute. For t > 0, the string is tangent to the base circle and normal to the involute, with length and curvature radius rbt. The unwrapped angle records complete turns as well as the final direction.
Tooth profile. Each brass flank is an involute segment from max(rb, rd) to the addendum circle. The dotted strings are tangent to the base circle and normal to the flank. When the root circle is below the base circle, red dashed radial segments serve as placeholders, not cutter-generated root fillets. The tip follows the addendum circle; generated roots and undercut are shown in Tab 4.
Two gears meshing. For interference-free involute contact, the common normal is the line of action through the pitch point P, tangent to both base circles. The red segment is limited by the two drawn involute flanks. Its length divided by the base pitch πm cos α gives the contact ratio, the average number of tooth pairs in contact. The pitch circles roll without slipping; the tooth surfaces generally slide except at P. Interfering settings are shown only as a geometric construction, not a working gear pair.
Rack cutter and undercut. The gear is a numerical approximation obtained by subtracting sampled rack positions from a blank disc. The straight flanks generate involutes, while the rounded tips generate root fillets (envelope curves). Green shows the exposed fillet envelope; red dashed curves trace the flank–fillet junctions and are not themselves the root boundary. Undercut occurs when the straight-flank end reaches deeper than T. Large positive profile shifts can instead produce pointed teeth below the blank addendum circle.