What is cubic spline interpolation?

Cubic spline interpolation is a way to find new points between some points you already know. It gives you a smooth curve that goes through all your original points.

Imagine you drew some dots on a piece of paper. How could you connect the dots with a smooth curving line? If you tried to use a ruler, you would get a jagged line with sharp corners. But you could carefully sketch a smooth curve that bends just right to hit each dot. That’s kind of like what cubic spline interpolation does – it math to finds the smoothest curve to connect known points.

Why is it called “cubic” spline interpolation? The smooth curve it makes is defined using cubic polynomials. A cubic polynomial is a math equation that has variables that go up to the power of 3 (the “cube” of a number). For example:

y = a·x³ + b·x² + c·x + d

Between each pair of known points, cubic spline interpolation finds the cubic polynomial curve that fits just right to connect them smoothly. Then all those curve segments get joined together into one long smooth spline.

How does cubic spline interpolation work?

Let’s say you have a set of n points (x₁, y₁), (x₂, y₂), … (xn, yn). The x-coordinates are the input values (like time) and the y-coordinates are your data points (like stock prices).

Cubic spline interpolation constructs a piecewise cubic function that passes through all n points. “Piecewise” means it’s made up of separate pieces or segments – in this case, a different cubic curve between each pair of adjacent points. For n points, you end up with n-1 cubic segments joined together.

To find the right cubic polynomials for the segments, there are a few conditions:

1. Each segment must start at one data point and end at the next.
2. The segments have to join up smoothly – the curve can’t have any gaps or sharp corners.
3. The slope of the curve has to be continuous all the way through – it can’t suddenly change direction.
4. The rate of change of the slope (the “curvature”) also has to match up between segments.

There are different methods to set up and solve the system of equations to find all the cubic polynomials. A common technique is the tridiagonal matrix algorithm.

Once you have the piecewise function with all the cubic formulas plugged in, you can use it to calculate a y-value for any x-value. The interpolation function fills in all the gaps between your original data points with nice smooth estimates.

When is cubic spline interpolation used?

Cubic splines are often used when you need to estimate values between some data points you collected. This comes up in:

• Computer graphics for making smooth curves and surfaces
• Animation for moving objects along paths between key points
• Robotics to plan smooth motion trajectories
• CAD and 3D modeling to design curvy shapes
• Signal processing to reconstruct signals from samples
• Economics and finance to model yield curves and forward rates
• Science and engineering to fit data and approximate functions

In general, cubic spline interpolation is good when you want a smooth curve fit and continuity at the data points is important. It gives a more natural looking result than simple straight lines.

Pros and cons of cubic spline interpolation

Some advantages of cubic splines are:

• Smoothness: Cubic splines produce very smooth interpolated curves with continuous slopes and curvatures. There are no sharp corners or abrupt changes in direction.
• Accuracy: Cubic polynomials can fit complex curves pretty well. The interpolation errors are usually small if the data points aren’t too far apart.
• Simplicity: The theory behind cubic splines is well-established and not too hard to understand compared to some other methods. There are easy algorithms to set up and solve for the polynomial coefficients.
• Efficiency: The piecewise nature of splines means each segment is a pretty simple low-order polynomial. Evaluating the interpolated points is fast. You can also control the tightness of the fit by adding or removing knots.

Some disadvantages are:

• Overshooting: Splines can sometimes wiggle and oscillate, especially if the data has big vertical jumps. The curve might go way above or below the data range, which isn’t always desirable.
• Sensitivity: The interpolated curve can change a lot if just one data point changes. Cubic splines don’t handle outliers well – one bad apple can spoil the curve.
• Boundary behavior: The curve beyond the first and last data points is determined by the end conditions chosen. It may look strange if not constrained well.
• Underfitting: While piecewise cubics are flexible, they may miss fine details and nuances in very complicated data sets. Higher-order splines or other models might be needed.

So cubic spline interpolation is a good all-around curve fitting method but it’s not perfect for all situations. It’s a popular starting point that works well on fairly smooth data.

Alternatives to cubic spline interpolation

While cubic splines are very common, there are other ways to interpolate curves too:

• Linear interpolation: Connects the dots with straight line segments. It’s simple but not very smooth.
• Polynomial interpolation: Fits one high-degree polynomial through all the data points. It can wiggle wildly.
• Bezier curves: Cubic Bezier curves are another way to make smooth interpolating paths in computer graphics. The curve doesn’t necessarily pass through the control points.
• Trigonometric interpolation: Uses sines and cosines to build a curve with repeating patters and periodicity.
• Gaussian process regression: Gives a smooth mean curve with error bands to show uncertainty. Handles multiple variables and “noisy” data.
• Smoothing spline: A variation of cubic splines that trades off between smoothness and faithfulness to the data. It can adjust to avoid overfitting.
• Radial basis functions: Build up the curve using sums of circular bell-like shapes around each data point. They can handle unstructured or scattered data.

The best choice depends on the type of data and situation. More powerful interpolation methods are better at capturing complex shapes but they have more knobs and dials to tune.