This chapter delves into the principles and methods for reconstructing curves and surfaces from discrete data using variational techniques. We begin with an introduction to classical interpolation methods, focusing on polynomial Lagrange interpolation and its comparison with cubic spline interpolation. These approaches are evaluated for their smoothness and accuracy, particularly in the case of clean, structured data. The chapter then addresses the more challenging scenario of noisy data in one-dimensional cases, where simple interpolation fails to provide reliable reconstructions. To handle noise, we introduce variational regularization techniques, which stabilize the interpolation by balancing fidelity to the data and smoothness of the curve.