How this calculation works
Solve the tridiagonal system for node second derivatives, then build one cubic per adjacent interval. The pieces match supplied values and join with matching first and second derivatives. Natural means zero endpoint second derivatives, not zero endpoint slope.
Sᵢ(x)=aᵢ+bᵢq+cᵢq²+dᵢq³, q=x−xᵢ; S″ at both endpoints = 0A worked example
For (0,0), (1,1), (2,0), the first interval is S(x)=1.5x−0.5x³. At x=0.5, the interpolated value is 0.6875, slope 1.125 and second derivative −1.5.
Good to know
Natural boundary conditions only, 2–50 well-spaced ordered nodes. No extrapolation, smoothing, statistical fitting or overshoot prevention. The unknown original function need not share the spline’s endpoint curvature. Coefficients and SVG are floating-point approximations.
Reference: SciPy: cubic splines and natural boundary conditions (opens in a new tab) ↗
Natural refers to endpoint curvature
A cubic spline uses one cubic on each interval between consecutive x values. Its pieces meet with matching values and first two derivatives. Two additional endpoint conditions are needed to determine the complete spline.
The natural choice sets the two endpoint second derivatives to zero. It does not force either endpoint slope to zero. A straight line is the natural spline through two nodes; more nodes can introduce curvature and overshoot.
The linear interpolation calculator constructs one straight reference segment. A natural cubic spline calculator uses all ordered nodes and a stated endpoint-curvature condition.
Read each cubic in its local coordinate
The coefficient table uses q=x−xᵢ for each interval. A row’s a,b,c,d therefore represent a+bq+cq²+dq³, not a polynomial in unshifted global x unless the interval starts at zero.
For nodes (0,0), (1,1), (2,0), the first interval is 1.5x−0.5x³. At 0.5 it gives 0.6875, slope 1.125 and second derivative −1.5. The adjoining interval joins smoothly at x=1.
Interpolation does not remove noise
The spline passes through every supplied node. If those nodes contain measurement noise, interpolation preserves that noise rather than estimating a smooth statistical trend. Natural endpoint curvature is a modeling choice and need not match an unknown underlying function.
The linear interpolation calculator describes a straight segment between two references. This page uses all ordered nodes and a smooth piecewise-cubic construction. It refuses extrapolation because carrying the outer cubic beyond the supplied range would add a different unsupported assumption.
A few questions, answered
Why must x coordinates increase?
Each x must identify one distinct node and a positive interval width. Repeated or reversed x values do not fit this interpolation setup.
Does a natural spline preserve monotonicity?
No. A natural cubic spline can overshoot or change direction between nodes, even when the supplied data are monotone.
Is this a best-fit regression curve?
No. It interpolates the nodes exactly up to floating-point roundoff; it does not estimate regression parameters or uncertainty.