Cubic spline: piecewise cubic S(x) = aᵢ + bᵢ(x−xᵢ) + cᵢ(x−xᵢ)² + dᵢ(x−xᵢ)³ on each interval. Conditions: interpolation (S(xᵢ)=yᵢ), C¹ (S' continuous), C² (S'' continuous). Natural: S''(x₀)=S''(xₙ)=0.
How to Use
Enter data:
- Points: (x₀,y₀), ...
- Evaluate: x value
- Output: S(x) + coefficients
vs Polynomial
Polynomial interpolation through n points uses degree n−1 → Runge oscillations. Splines use degree 3 everywhere → no oscillations, guaranteed smooth. Industry standard for curves.
Boundary Types
- Natural: S''=0 at endpoints
- Clamped: S'=given at endpoints
- Not-a-knot: S''' continuous at 2nd/penultimate points
Step-by-Step Instructions
- 1Enter data points.
- 2View spline coefficients.
- 3Enter x to evaluate.
- 4Get smooth interpolation.
- 5Compare segments.