Many tasks from the natural and engineering sciences require precision solutions with complex curves. The main obstacle is the lack of the necessary mathematical apparatus. The analysis of symmetries on the Euclidean plane by Dieudonne and the figure by Weyl allowed us to formulate a new method for obtaining the parameters of linear transformation alternative to classical. It can be used for an ellipse, hyperbola, as well as complex flat curves. The method is analyzed for trajectories having symmetries. A theorem to obtain the parameters of the transformed curve in the general case is formulated. Theoretical calculations and the results of experimental studies using the method of geometric modeling are given. The method is very new, so it may not work for some curves. There is the possibility of obtaining it, since the research uses the simplest apparatus.