Gear with teeth whose profile is an involute of a circle
Two involute gears, the left driving the right: Blue arrows show the contact forces between them (1) downward force applied by the left gear and (2) upward resistance by the right gear. The force line (or line of action) runs along the long leg of dashed blue line which is a tangent common to both base circles. The involutes here are traced out in converse fashion: points of contact move along the stationary force-vector "string" as if it was being unwound from the left rotating base circle, and wound onto the right rotating base circle. In this situation, there is no force, and so no contact needed, along the opposite [lower left to upper right] common tangent (not shown). In other words, if the teeth were slightly narrower while everything else remained the same there would be a gap above each tooth on the left gear, because downward force is being applied by it.Construction of an involute curve from the surface of a circle; this can be seen as the path traced by the end of a string being unwound from a disc. Involute gear teeth are not precisely this shape, due to material allowances like fillets et cetera.
The involute gear profile is the most commonly used system for gearing today, with cycloid gearing still used for some specialties such as clocks. In an involute gear, the profiles of the teeth are involutes of a circle. The involute of a circle is the spiraling curve traced by the end of an imaginary taut string unwinding itself from that stationary circle called the base circle, or (equivalently) a triangle wave projected on the circumference of a circle.
