The paper presents the results of investigations in the field of kinematic geometry of the spatial curve of a line. The basis of the research is the method of the movable trihedron of the curve. The components of the trihedron motion along the spatial curve are considered and it is shown that its resultant instantaneous motion is screw motion. This result differs from the representation of the motion of a trihedron known in geometry as a rotation described by the Darboux vector. An analytical description of the set of axes of instantaneous helical motions of a trihedron in a moving and fixed system of assigning a spatial curve is given. The possibility of applying the obtained general results to the investigation of a plane curve is shown. The paper proposes a flat tooth gearing model based on the geometric interpretation of the motions of a trihedron of a plane curve and known in the geometric theory of plane mechanisms of the construction of Bobillier. The geometric scheme of this construction is expanded due to the introduction of evolutes simulating instantaneous motions of trihedron of the corresponding construction curves. As a result, a geometric model is obtained, which is more complete in comparison with the known models of flat gearing. It allows to perform both direct and inverse tasks of profiling the teeth of the wheels while simultaneously obtaining the curvature of the desired profiles in the absence of such. The proposed model can be used as the basis for the development of gears with a planar gearing scheme by the condition of achieving the necessary transmission performance due to the geometric shape of the teeth of the wheels.