Spline Interpolation Calculator

Cubic spline interpolation is a method of constructing a smooth curve through a set of known data points. It builds piecewise cubic polynomials between consecutive points, ensuring that the curve and its first and second derivatives are continuous at every knot (data point). This avoids the oscillation problems seen with high-degree polynomial interpolation (Runge's phenomenon).

The Natural boundary condition sets the second derivative to zero at both endpoints, giving a curve that bends least at the edges. Not-a-Knot forces the third derivative to be equal across the first and last interior knots, producing a smoother transition. The Quadratic (clamped/quadratic end) condition approximates the slope at the endpoints using a quadratic fit, which can be more accurate when the data trend is known.

Cubic spline algorithms build the coefficient matrix based on the sequential spacing (h) between consecutive x values. If x values are not sorted in ascending order, the spacing calculations become invalid and produce incorrect or undefined results. Always sort your data from smallest to largest x before entering it.

You need a minimum of 3 data points for cubic spline interpolation. With only 2 points, a linear interpolation is all that is possible. More points generally produce a more accurate and representative spline, but the method works well even for small datasets of 4–10 points.

Each spline segment is expressed as S(t) = a + b·t + c·t² + d·t³, where t = x − xᵢ is the local offset from the left endpoint of the segment. The coefficient 'a' is the y-value at the left knot, 'b' is the slope, 'c' relates to the second derivative (curvature), and 'd' governs how the curvature changes across the segment.

This calculator is designed for interpolation — estimating values within the range of your data. Extrapolating beyond the first or last x value using spline polynomials can produce wildly inaccurate results because the cubic polynomials are only calibrated to fit the interior data. For extrapolation, consider using a different method or extending your dataset.

Polynomial interpolation fits a single polynomial of degree n−1 through all n data points. For large datasets this polynomial can oscillate severely (Runge's phenomenon). Cubic spline interpolation instead uses low-degree (cubic) polynomials on each sub-interval, which remain smooth and well-behaved even with many data points.

Different boundary conditions impose different constraints on the curvature at the endpoints, which propagates through the entire system of equations used to find the spline coefficients. In practice, the differences are usually small in the interior of the data range but can be more noticeable near the endpoints. The Natural condition is the most commonly used default.