Hobbing is a common manufacturing method when producing helical, involute gears. In order to give the manufactured gear a controlled surface smoothness, a method to, very accurately, determine the achieved surface geometry is needed. In this report, the cutting surfaces of the tool, of which the cutting edges are the boundaries, are assumed to be plane in arbitrary directions. They are mathematically described using parametric and analytically differentiable functions. These functions give the possibility to determine the geometry of the three-dimensional surface of the manufactured gear, without any additional numeric approximations. By comparing this surface with the smooth surface of an ideal gear, the roughness of the surface can be determined. An example is given in which the surface topology and the characteristic surface roughness parameters are determined.