Symmetries of discrete curves and point clouds via trigonometric interpolation

TL;DR

The paper introduces a straightforward algorithm that uses trigonometric interpolation to accurately identify all symmetries of closed discrete curves and point clouds in the plane.

Contribution

It presents a novel, efficient method for detecting symmetries of discrete curves and point clouds using trigonometric interpolation, applicable to organized and unorganized data.

Findings

The algorithm accurately detects rotational and axial symmetries.

It can be applied to both organized curves and unorganized point clouds.

Demonstrated effectiveness on multiple example datasets.

Abstract

We formulate a simple algorithm for computing global exact symmetries of closed discrete curves in 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 assigning 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 nonorganized clouds of points. A functionality of the designed detection method is presented on several examples.