Constructing chaotic systems tailored for each particular real-world application has been a long-term research desideratum. We report a solution for this problem based on the concept of relative motion. We investigate the periodic motion on a closed contour of a coordinate frame in which a chaotic system evolves. By combining these two motions (periodic on a close contour and chaotic) new customized shape trajectories are acquired. We demonstrate that these trajectories obtained in the stationary frame are also chaotic and, moreover, conserve the Lyapunov exponents of the initial chaotic system. Based on this finding we developed an innovative method to construct new chaotic systems with customized shapes, thus fulfilling the requirements of any particular application of chaos.