Linear InterpolationEdit
Linear interpolation is a straightforward method for estimating unknown values that lie between measured data points. By connecting neighboring points with straight line segments, it yields a simple, fast, and predictable means of filling gaps in data. This makes it a staple in engineering practice, computer graphics, and data processing where speed and robustness are valued over perfect fidelity to an underlying model. In its most common form, linear interpolation uses two known points to interpolate values on the interval between them, and it can be extended piecewise across a whole data set.
In practice, linear interpolation serves as a building block rather than a final destination. It is widely used to resample measurements, animate motion between keyframes, and provide a quick estimate when a more sophisticated model would be unnecessary or too costly. Because the method is local and linear, it exhibits excellent stability and interpretability, which is why it remains a standard tool in Numerical analysis and in many applied fields. For additional context, see how linear interpolation relates to broader ideas in interpolation and how it contrasts with higher-order approaches found in Lagrange interpolation or Spline theory.
Overview
Concept and scope: linear interpolation constructs a continuous, piecewise-linear function that passes through all known data points. It is defined on an interval [x0, x1] by a straight line that joins the two known values y0 and y1 at x0 and x1, respectively. For points within that interval, the estimate is y = y0 + (y1 - y0) · (x - x0) / (x1 - x0). When many data points are present, the same idea is applied on each successive pair, yielding a path composed of straight segments. This approach is often preferred when the primary concern is fast, reproducible results and when the underlying phenomenon is not expected to behave nonlinearly between samples.
Locality and robustness: the estimate in any subinterval depends only on the endpoints of that subinterval. This locality makes linear interpolation resistant to global data anomalies and easy to audit, which is important in safety-critical or compliance-conscious settings where reproducibility matters. For context, see Interpolation and comparisons with global polynomial strategies discussed in Polynomial interpolation.
Relationship to real-world tasks: in computer graphics, linear interpolation underpins color and position blending between keyframes; in sensor networks and geographic information systems, it provides quick estimates between known measurements; in digital signal processing, it serves as a fast way to resample or align samples across different rates.
Mathematical formulation
Let (x0, y0) and (x1, y1) be two known data points with x0 < x1. For any x in [x0, x1], the interpolated value is y = y0 + (y1 - y0) · (x - x0) / (x1 - x0).
When the data set contains many points { (xi, yi) }, linear interpolation is applied piecewise on each consecutive interval [xi, xi+1]. The resulting function is continuous on the domain and linear on each subinterval, but its derivative is not continuous at the knots xi (the points where the pieces meet). This makes the method computationally light while delivering predictable behavior.
A common practical note is the use of linear extrapolation beyond the end points. If x < x0 or x > xN, applying the end-segment line is possible, but care is advised because extrapolation can amplify errors if the data do not reflect the true trend outside the observed range. See further on error behavior in the next section.
For a broader view of how the same principle appears in higher dimensions, see Bilinear interpolation for 2D cases and its extensions to 3D as Trilinear interpolation.
Error analysis
Linear interpolation provides a straightforward error bound when the target function is smooth. If f is twice continuously differentiable on the interval and h is the maximum subinterval width (the largest distance between successive abscissas), then for x in [a, b] the interpolation error satisfies |f(x) - ℓ(x)| ≤ (M/8) h^2, where M ≥ max |f''(ξ)| over the interval, and ℓ is the linear interpolant that passes through the known points. The key takeaway is that the error scales with the square of the subinterval size and with the curvature of the underlying function. In practice, reducing h (i.e., using more sample points) yields a noticeably tighter approximation, but at the cost of additional storage and computation elsewhere in the system.
When the function being approximated has sharp turns or high curvature, linear interpolation may require very small subintervals to keep errors within tolerance. In such cases, practitioners may switch to higher-order methods (such as spline or polynomial interpolation) or adopt adaptive schemes that refine the sampling where needed. See discussions of Spline and Lagrange interpolation for contrasts in how different methods trade off smoothness, stability, and accuracy.
Variants and extensions
Piecewise linear interpolation: The baseline approach described above, applied to each consecutive pair of data points. This is the most common form and is valued for simplicity and speed. See also Piecewise linear interpolation in related literature.
Multidimensional extensions:
- Bilinear interpolation: Extends the one-dimensional idea to two dimensions by performing linear interpolation first in one direction and then in the other. This is widely used in image resampling and texture mapping; see Bilinear interpolation for details.
- Trilinear interpolation and beyond: In higher dimensions, repeated application of one-dimensional linear interpolation governs the generally accepted approach for simple grid-based data.
Higher-order alternatives:
- Polynomial interpolation (e.g., Lagrange interpolation or Newton interpolation): Fits a single polynomial to all data points. While potentially more accurate for well-behaved data, these methods can exhibit Runge’s phenomenon (oscillations at the edges) and compactly computed, stable implementations often require careful handling.
- Spline interpolation (e.g., Spline theory): Uses low-degree polynomials stitched together with continuity constraints to produce smooth, global or local fits. Cubic splines in particular are popular for providing smooth first and second derivatives, which is advantageous in animation and engineering simulations.
Computational considerations:
- Evaluation speed: Linear interpolation requires only a couple of arithmetic operations per segment, making it ideal in real-time systems and environments with limited processing power.
- Memory and precomputation: Efficient implementations can store just the nodes and reuse simple formulas for each interval, which simplifies auditing and verification in regulated workflows.
Applications
Real-time graphics and animation: Interpolating positions and colors between keyframes is a core use. The method’s determinism and low overhead make it attractive for rendering pipelines and hardware-accelerated processes. See Computer graphics and Animation discussions in related articles.
Geographic information systems and data resampling: When converting map data or sensor grids from one sampling pattern to another, linear interpolation offers a robust default that avoids overfitting and excessive computation. Related topics include Geographic information systems and Spatial data practices.
Engineering and physics simulations: In time-stepping schemes or along a spatial grid, linear interpolation provides stable estimates between known points without introducing artificial oscillations. See also Numerical analysis foundations and comparisons with higher-order methods.
Signal processing and data alignment: Used to align signals sampled at different rates or to reconstruct intermediate samples when storage or bandwidth is constrained.
Computation and implementation
Practical implementation favors keeping the method straightforward. The essential steps are: - Identify the interval [x0, x1] containing the target x. - Compute the slope (y1 - y0) / (x1 - x0). - Apply the linear formula to obtain y.
Numerical stability is typically not a major concern for linear interpolation, but attention should be given to floating-point precision when x is very close to x0 or x1, or when data spans many orders of magnitude. See Floating-point and Numerical stability for deeper discussions, and note how linear interpolation interacts with error propagation in a larger computational pipeline.
In data pipelines that implement higher-level methods, linear interpolation often appears as a subroutine inside a broader approach, such as a multistep resampling strategy or a first-pass estimator before a more refined model is applied. Users may encounter it alongside Two-point interpolation concepts and in contexts that involve Data interpolation.