Instead of fitting one high-degree polynomial across an entire interval, piecewise polynomial interpolation uses low-degree polynomials on smaller subintervals.

A cubic spline uses a cubic polynomial on each interval between neighboring data points and joins the pieces so that the function, first derivative and usually second derivative are continuous at the interior knots.

This produces a smooth interpolant while keeping each polynomial local. Changing one data point influences nearby pieces much more directly than it would in one global high-degree polynomial.

Additional endpoint conditions, such as specified endpoint derivatives or zero second derivatives, determine the remaining degrees of freedom.

Splines provide a practical compromise between smoothness, locality and numerical behavior and are widely used for curves, tabulated data and numerical approximation.