Symmetries of discrete curves and point clouds via trigonometric interpolation

We formulate a simple algorithm for computing global exact symmetries of closed discrete curves in the plane. The method is based on a suitable trigonometric interpolation of vertices of the given polyline and consequent computation of the symmetry group of the obtained trigonometric curve. The algorithm exploits the fact that the introduced unique assignment of the trigonometric curve to each closed discrete curve commutes with isometries. For understandable reasons, an essential part of the paper is devoted to determining rotational and axial symmetries of trigonometric curves. We also show that the formulated approach can be easily applied on unorganized clouds of points. A functionality of the designed detection method is presented on several examples.