Interpolation

TL;DR

Estimates unknown values that fall between known data points.
Widely used in fields such as computer graphics and signal processing to reconstruct or approximate data.
Multiple methods exist that trade off simplicity, accuracy, and computational cost.

Definition

Interpolation is a mathematical method used to estimate a value within the range of a set of known data points.

Explanation

Interpolation computes an estimated value at a point that is not present in the original data by using the known data points around it. Different interpolation methods fit different kinds of functions to the known points — for example, straight lines, smooth curves, or polynomial functions — and are chosen based on the data and desired accuracy.

Examples

Temperature over a day

Given data points that represent the temperature at different times of the day, interpolation can estimate the temperature at a time not included in the original data set.

Position of an object over time

Given data points that represent the position of an object at different times, interpolation can estimate the object’s position at a time not included in the original data set.

Use cases

Computer graphics
Signal processing
Scientific modeling
Data analysis
Other mathematical and scientific fields

Notes or pitfalls

Linear interpolation connects two known points with a straight line; it is simple and fast but can be inaccurate when the relationship between points is non-linear.
Cubic spline interpolation uses a smooth curve through the data points; it is generally more accurate than linear interpolation but can be more computationally expensive.
Polynomial interpolation fits a polynomial function to the data points; it can be highly accurate but may be difficult to implement and slow to compute for large data sets.

Linear interpolation
Cubic spline interpolation
Polynomial interpolation