Spline interpolation constructs a spline that passes through prescribed data points. Given distinct nodes 
and values 
, a cubic spline interpolant 
satisfies 
, is a cubic polynomial
on every interval ![[x_i,x_(i+1)]](/images/equations/SplineInterpolation/Inline5.svg)
,
and has continuous first and second derivatives.
Two additional endpoint conditions determine a unique cubic spline. A natural cubic spline uses 
, while a clamped cubic spline specifies
the two endpoint slopes. Spline interpolation generally avoids the large oscillations
that can occur with a single high-degree interpolating polynomial.
