interpolate

Interpolation is a method for estimating unknown values that fall within the range of a set of known data points. Given a set of inputs and their corresponding values, interpolation constructs a function that passes through the known points and provides estimates at in-between inputs. It is used to fill gaps, resample data, or construct smoother representations. Interpolation is distinct from extrapolation, which estimates values outside the known range.

Common interpolation methods include linear interpolation, which connects adjacent points with straight lines; polynomial interpolation, which fits a single polynomial to all data points (using approaches such as Lagrange or Newton forms); and spline interpolation, which uses piecewise polynomials (often cubic) with continuity constraints to ensure smoothness. Nearest-neighbor interpolation assigns each unknown point the value of the closest known point. More advanced techniques include radial basis functions and kriging, which are used in multivariate or geostatistical contexts.

In multidimensional settings, interpolation generalizes to surfaces and higher-dimensional fields. Different methods trade off simplicity, smoothness, and accuracy: linear and spline methods are common for regular grids, while radial basis and kriging are used for irregular data.

Applications span data analysis, numerical methods, computer graphics, image and audio resampling, and geographic information systems. Important considerations include the behavior of the underlying function, the spacing of known points, and the potential for artifacts such as Runge’s phenomenon with high-degree polynomial interpolants. Interpolation aims to provide plausible estimates consistent with the known data within the observed domain.