This paper studies planar curve fitting from sampled points under two related viewpoints. In
the first part, open and closed curves are represented by parametric functions and approximated
with cubic spline interpolation, B-spline interpolation, polynomial least-squares fitting, and B
spline least-squares fitting. The comparison is carried out under different parameterization rules,
different numbers of sample nodes, and several noise levels. In the second part, closed curves are
modeled as periodic signals in the complex plane and reconstructed by truncated Fourier series.
This formulation explains why a closed contour can be viewed as a superposition of circular
motions and also provides a natural way to analyze the effect of the harmonic order. Quantitative
evaluation is based mainly on Chamfer distance, together with visual inspection of local details
and reconstruction stages. The experiments show that spline-based methods are more reliable
on curves with local bends, that parameterization strongly affects interpolation quality on non
uniformly curved shapes, and that Fourier reconstruction achieves a clear accuracy gain as more
harmonics are introduced, especially for contours with repeated oscillatory structure.
