  A pair of noncircular gears can be used to generate a strictly increasing
continuous function $\psi(\varphi)$ whose derivative $\psi'(\varphi) =
\mathrm{d}\psi(\varphi)/\mathrm{d}\varphi > 0$ is $2\pi/n$-periodic, where
$\varphi$ and $\gamma = \psi(\varphi)$ are the angles of the opposite rotation
directions of the drive gear and the driven gear, respectively, and $n \in
\mathbb{N} \setminus \{0\}$. In this paper, we determine the geometry of both
gears for given transmission function $\psi(\varphi)$ when manufacturing with a
rack-cutter having fillets. All occurring functions are consistently derived as
functions of the drive angle $\varphi$ and the function $\psi$. Throughout the
paper, methods of complex algebra, including an external product, are used. An
effective algorithm for the calculation of the tooth geometries - in general
every tooth has its own shape - is presented which limits the required
numerical integrations to a minimum. Simple criteria are developed for checking
each tooth flank for undercut. The base curves of both gears are derived, and
it is shown that the tooth flanks are indeed the involutes of the corresponding
base curve. All formulas for both gears are ready to use.

Metric Geometry;Differential Geometry