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Spline Interpolation

Spline interpolation creates smooth curves by connecting data points with piecewise polynomials. It combines the best aspects of linear and polynomial interpolation.

The Concept​

Instead of using a single high-degree polynomial, splines use multiple low-degree polynomials (typically cubic) between consecutive points. These pieces are carefully matched to ensure smoothness.

Types of Splines​

Cubic Splines​

Most common type, using cubic polynomials between each pair of points.

Properties:

  • Passes through all data points
  • Continuous first and second derivatives
  • Minimizes curvature

Natural Cubic Splines​

Cubic splines with zero second derivative at endpoints.

Hermite Splines​

Require both values and derivatives at data points.

B-Splines​

Offer even more local control, don't necessarily pass through points.

Advantages​

✅ Smooth: Continuous derivatives
✅ Stable: No Runge's phenomenon
✅ Local control: Changes affect only nearby segments
✅ Accurate: Good approximation of smooth functions
✅ Flexible: Different boundary conditions available

Disadvantages​

❌ More complex: Harder to understand than linear
❌ Computational cost: More expensive than linear
❌ Requires contiguous data: Best with ordered data points

When to Use Spline Interpolation​

Splines are ideal when:

  • You need smooth curves
  • Data represents a smooth underlying function
  • You have many data points
  • Local control is important
  • Derivatives matter (e.g., velocity from position)

Example in Excel​

Using our Excel add-in:

=INTERP.SPLINE(x_values, y_values, x_new)

Sample Data​

XY
00
11
24
39
416

To interpolate at X = 2.5:

=INTERP.SPLINE(A2:A6, B2:B6, 2.5)
// Result: ~6.25 (smooth curve)

Mathematical Details​

For cubic splines between points ii and i+1i+1, each segment is:

Si(x)=ai+bi(x−xi)+ci(x−xi)2+di(x−xi)3S_i(x) = a_i + b_i(x-x_i) + c_i(x-x_i)^2 + d_i(x-x_i)^3

The coefficients are chosen to ensure:

  1. Si(xi)=yiS_i(x_i) = y_i (passes through points)
  2. Si(xi+1)=yi+1S_i(x_{i+1}) = y_{i+1} (continuity)
  3. Si′(xi+1)=Si+1′(xi+1)S'_i(x_{i+1}) = S'_{i+1}(x_{i+1}) (smooth first derivative)
  4. Si′′(xi+1)=Si+1′′(xi+1)S''_i(x_{i+1}) = S''_{i+1}(x_{i+1}) (smooth second derivative)

Visualization​

Data Points

Divide into Segments

Fit Cubic Polynomial per Segment

Match Derivatives at Joints

Smooth Continuous Curve

Comparison with Other Methods​

FeatureLinearPolynomialSpline
Smoothness❌✅✅
Stability✅❌✅
Speed✅✅✅✅
Accuracy⚠️✅✅✅
Best Choice

For most applications requiring smooth interpolation, cubic splines are the gold standard.

Practical Applications​

Engineering​

  • CAD/CAM curve design
  • Path planning for robotics
  • Signal interpolation

Finance​

  • Yield curve construction
  • Volatility surface modeling

Graphics​

  • Animation curves
  • Font rendering
  • Image scaling