Spline Interpolation

Spline interpolation uses piecewise polynomials to achieve smooth approximation without the oscillations of high-degree polynomial interpolation, providing the foundation for modern computer graphics and CAD systems.

DefinitionCubic Spline

Given data points $(x_0, y_0), \ldots, (x_n, y_n)$ with $x_0 < x_1 < \cdots < x_n$ , a cubic spline $S(x)$ satisfies:

$S(x)$ is a cubic polynomial $S_i(x)$ on each interval $[x_i, x_{i+1}]$
$S(x_i) = y_i$ for all $i$ (interpolation condition)
$S_i(x_{i+1}) = S_{i+1}(x_{i+1})$ (continuity)
$S_i'(x_{i+1}) = S_{i+1}'(x_{i+1})$ (smooth first derivative)
$S_i''(x_{i+1}) = S_{i+1}''(x_{i+1})$ (smooth second derivative)

Additional boundary conditions are needed to make the system determined. Common choices include natural spline ( $S''(x_0) = S''(x_n) = 0$ ), clamped spline (specified $S'(x_0)$ , $S'(x_n)$ ), or periodic spline.

Cubic splines balance computational efficiency with smoothness. Linear splines ( $C^0$ continuous) are simple but have discontinuous derivatives. Quadratic splines require asymmetric boundary conditions. Cubic splines provide $C^2$ continuity and are the lowest-degree polynomials achieving this naturally.

ExampleNatural Cubic Spline Construction

For three points $(0,0)$ , $(1,1)$ , $(2,0)$ , construct the natural cubic spline.

On $[0,1]$ : $S_0(x) = a_0 + b_0 x + c_0 x^2 + d_0 x^3$ On $[1,2]$ : $S_1(x) = a_1 + b_1 x + c_1 x^2 + d_1 x^3$

The interpolation and continuity conditions give 6 equations in 8 unknowns. Natural boundary conditions $S_0''(0) = 0$ and $S_1''(2) = 0$ provide 2 more, yielding a tridiagonal linear system:

$S_0(x) = x + \frac{3}{4}x^2 - \frac{3}{4}x^3$ $S_1(x) = -7 + 9x - \frac{15}{4}x^2 + \frac{3}{4}x^3$

This spline is $C^2$ continuous at $x=1$ and interpolates all three points.

Remark

Optimality of Cubic Splines: Among all $C^2$ functions interpolating given data, the natural cubic spline minimizes: $\int_{x_0}^{x_n} [f''(x)]^2 \, dx$

This variational property models a thin elastic beam passing through the data points, explaining why splines appear "natural" and smooth. The minimization leads to a tridiagonal system that can be solved efficiently in $O(n)$ operations.

Practical spline implementations use parametric representations for curves and surfaces. B-splines provide local control (changing one control point affects only nearby curve segments) and form the basis of NURBS (Non-Uniform Rational B-Splines) used in CAD/CAM systems. Tensor product splines extend to multidimensional approximation for image processing and scientific visualization.

Splines avoid the Runge phenomenon that plagues high-degree polynomial interpolation. While polynomial error grows as $O(h^{n+1})$ with degree $n$ and spacing $h$ , cubic spline error is $O(h^4)$ , independent of the number of points. This makes splines practical for large datasets where high-degree polynomials would be numerically unstable.