A new gear set consisting of a planar noncircular gear and a specific face gear called undulating face gear is presented, which are able to transmit varying angular velocity. Based on the kinematics of planar noncircular pinion and rack, the formation mechanism of pitch curves of the undulating face gear is given, with the continuous transmission condition of the gear set discussed. Tooth surfaces of the undulating face gear are enveloped by the teeth of a shaper cutter when its pitch circle rolls on the pitch curve of the undulating face gear without sliding. According to the spatial meshing theory, the mathematical model of the face gear tooth surface is built. The phenomenon of tooth undercutting and tip-cutting of the face gear is investigated through three-dimensional machine design software to obtain its minimum inner radius and maximum outer radius. Furthermore, a computer program is developed to calculate the data of the face gear tooth surface and a motion simulation of the gear set without undercutting and tip-cutting is made, which is able to illustrate validity and correctness of the design theory. (C) 2014 Elsevier Ltd. All rights reserved.