Build accurate interpolated curves from ordered coordinate pairs. Compare segments, derivatives, and estimated query values. Download reports instantly for study, validation, and presentation needs.
Enter ordered x values, matching y values, and one or more query positions. Choose natural endpoints or supply clamped endpoint slopes for derivative-controlled interpolation.
This sample demonstrates a smooth interpolating curve through five known coordinates. Use it to test the calculator before entering your own dataset.
| Point | x | y | Suggested query |
|---|---|---|---|
| 1 | 0 | 1 | 0.5 |
| 2 | 1 | 3 | 1.5 |
| 3 | 2 | 2 | 2.5 |
| 4 | 3 | 5 | 3.5 |
| 5 | 4 | 4 | — |
Piecewise spline model: For each interval [xi, xi+1], the calculator builds Si(x) = ai + bi(x - xi) + ci(x - xi)² + di(x - xi)³.
Interval width: hi = xi+1 - xi. The x values must be strictly increasing so each segment width stays positive.
Interior system: αi = 3[(yi+1 - yi)/hi] - 3[(yi - yi-1)/hi-1] for interior nodes.
Natural boundary: c0 = 0 and cn-1 = 0, which forces zero second derivative at both ends.
Clamped boundary: The calculator uses user-entered endpoint slopes to constrain the first derivative at both ends and solve a modified tridiagonal system.
Recovered coefficients: bi = (yi+1 - yi)/hi - hi(ci+1 + 2ci)/3 and di = (ci+1 - ci)/(3hi).
1. Enter x coordinates in ascending order. Then enter matching y coordinates with the same number of values.
2. Add one query x value or several query values to evaluate the spline between known points.
3. Choose natural mode for free endpoints, or clamped mode when you know the first derivative at both ends.
4. Set the decimal precision that best matches your reporting or classroom requirements.
5. Press Calculate spline to place the results above the form, directly beneath the page header.
6. Review the query table, coefficient table, and interval equations. Download the output as CSV or PDF when needed.
It creates smooth third-degree polynomial segments between known data points. The segments join continuously, preserving value, slope, and curvature behavior better than many simpler interpolation methods.
Choose natural mode when endpoint curvature is unknown or assumed minimal. It sets the second derivative to zero at both ends and often works well for general educational and numerical tasks.
Use clamped mode when you know the slope at the first and last data points. This helps the spline follow measured endpoint behavior more accurately than unconstrained endpoint curvature.
Spline intervals depend on positive widths between consecutive x coordinates. Repeated or decreasing x values create undefined segments and can make the linear system impossible to solve correctly.
Yes. The calculator accepts multiple query x positions in one submission. It evaluates every point, shows the matching interval, and reports spline value plus first and second derivatives.
No. Query points outside the domain are flagged instead of extrapolated. That protects users from treating unsupported edge estimates as reliable interpolation results.
They define each interval polynomial. Coefficient a starts the segment value, b controls local slope, c shapes curvature, and d controls the cubic bending rate.
You can save query results, coefficient tables, and equations for assignments, reports, validation records, or sharing with collaborators who need a fixed copy.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.