Geometrical Method for Increasing Precision of Machine Building Parts

Calculation of complex curves with high accuracy is challenging. The curve is approximated by segments of straight lines and carried out smoothing. A new method of analytic geometry to calculate the trajectories of mechatronic systems and CAD/CAM is offered. The theoretical basis of the method is symmetries in the Euclidean plane. The method can accurately calculate the trajectory for the centrally symmetric conic sections and, in some cases, arbitrary differentiable planar curves. The method gives an accurate analytical solution without using radicals. This allows you to find the formulaic dependencies for further calculations. The existence of Lissajous figures allows assuming the presence of a universal method of calculation of complex trajectories based on planar differentiable curves. Some examples of the design of the kinematic mechanisms are presents additionally.