Fitting a Closed Curve to Unordered Planar Points

Fitting smooth closed curves to unordered planar point clouds arises in various applications, including shape reconstruction, image processing, geometric modelling, and computational geometry. We propose a numerical framework for this problem under an approximate star-shaped assumption. The method combines a deterministic centre-selection strategy with a trigonometric least-squares approximation of the radial function. By minimising a radial ambiguity functional, an interior centre is identified to enable a well-defined polar representation without point ordering or neighbourhood reconstruction. A truncated Fourier series is then fitted to obtain a smooth periodic parametric curve. In contrast to approaches that require point ordering, neighbourhood reconstruction, or nonlinear optimisation, the proposed framework reduces the two-dimensional fitting problem to a one-dimensional least-squares approximation in the radial coordinate, resulting in a smooth analytic representation obtained through a linear and computationally efficient procedure. Accuracy is evaluated using normalised RMSE and maximum point-to-curve deviation. Experiments on a 250-point unstructured dataset and on invariant curves arising from dynamical systems demonstrate accurate approximation with moderate Fourier order. The method is effective for planar point clouds that admit an approximate star-shaped structure.