Linear Interpolation Calculator

Fundamental Interpolation Formula

Given two known points (x_1, y_1) and (x_2, y_2), the interpolated value y at any point x between x_1 and x_2 can be calculated using the linear interpolation formula. This formula essentially uses the point-slope form of a line, where we find the slope between the two known points and apply it to find the y-coordinate at the desired x-coordinate.

y = y_1 + (x - x_1)(y_2 - y_1)/(x_2 - x_1)

Geometric Interpretation

Linear interpolation can be understood geometrically as finding a point on the straight line segment connecting two known points. The ratio (x - x_1)/(x_2 - x_1) represents how far along the line segment from the first point to the second point we need to travel. This ratio, often denoted as t where 0 ≤ t ≤ 1, creates a weighted average: y = (1-t)y_1 + ty_2, showing that interpolation is essentially a blend between the two known values.

t = (x - x_1)/(x_2 - x_1), \quad y = (1-t)y_1 + ty_2

Mathematical Properties

Linear interpolation possesses several important mathematical properties. It is exact at the endpoints (when x = x_1, we get y = y_1, and when x = x_2, we get y = y_2). The method is also continuous, meaning there are no jumps in the interpolated function. Additionally, linear interpolation preserves monotonicity—if the data points are increasing or decreasing, the interpolated values will maintain that trend.

y(x_1) = y_1, \quad y(x_2) = y_2, \quad \textContinuous on [x_1, x_2]

Error Analysis and Accuracy

The accuracy of linear interpolation depends on how closely the actual function behaves like a straight line between the data points. For functions with small second derivatives, the interpolation error is bounded by |E| ≤ (h^2)/(8)\max|f''(\xi)|, where h = x_2 - x_1 is the spacing between points. This means the error decreases quadratically as we reduce the spacing between points. Linear interpolation works best for smooth, slowly varying functions.

|E| ≤ ((x_2 - x_1)^2)/(8)\max_x_1 ≤ \xi ≤ x_2|f''(\xi)|

Interpolation vs Extrapolation

When x lies between x_1 and x_2, we perform interpolation, which is generally reliable. However, when x falls outside this range, we are performing extrapolation, which assumes the linear relationship continues beyond our data. Extrapolation is inherently less reliable because we have no data to validate that the trend continues. The further we extrapolate, the less confident we can be in our estimates.

\textInterpolation: x_1 ≤ x ≤ x_2, \quad \textExtrapolation: x < x_1 \text or x > x_2

Piecewise Linear Interpolation

When we have more than two data points, we can apply linear interpolation piecewise between each consecutive pair of points. This creates a continuous piecewise-linear function that passes through all data points. While simple to implement, this approach results in discontinuous derivatives at the data points (the slope changes abruptly), which may not be desirable for some applications.

y(x) = y_i + (x - x_i)\fracy_i+1 - y_ix_i+1 - x_i, \quad x \in [x_i, x_i+1]

Computational Efficiency

Linear interpolation is computationally inexpensive, requiring only basic arithmetic operations: two subtractions, one division, and one multiplication per interpolation. This makes it ideal for real-time applications, embedded systems, and situations where millions of interpolations must be performed. The simplicity and speed of linear interpolation often outweigh its limitations in accuracy for many practical applications.

\textComplexity: O(1) \text per evaluation

Applications in Science and Engineering

Linear interpolation finds extensive use in computer graphics (rendering pixels between vertices), numerical analysis (solving differential equations), signal processing (resampling audio and images), financial modeling (calculating interest rates between periods), geographic information systems (elevation modeling), and data analysis (filling missing values in datasets). Its simplicity and reliability make it a go-to method for quick estimations.

\textUsed in: graphics, numerics, signals, finance, GIS, data science

Limitations and Alternatives

Linear interpolation assumes a constant rate of change between points, which may not reflect reality for curved functions. It cannot capture local maxima or minima between data points. For more accurate results with curved data, alternatives include quadratic interpolation (using three points), cubic spline interpolation (ensuring smooth derivatives), polynomial interpolation (fitting a single polynomial through all points), or rational function interpolation. The choice depends on data smoothness, computational resources, and required accuracy.

\textAlternatives: Quadratic, Cubic Spline, Polynomial, Rational

Numerical Stability

Linear interpolation is numerically stable and well-conditioned. Unlike higher-order polynomial interpolation, it does not suffer from Runge's phenomenon (oscillations near boundaries). The method handles large ranges of values well and does not amplify rounding errors significantly. However, care must be taken when x_2 - x_1 is very small, as this can lead to division by numbers close to zero.

\textCondition number: \kappa = (|x_2 - x_1| + |y_2 - y_1|)/(|x_2 - x_1|)

Worked Example

Inputs

• x1: 0
• y1: 0
• x2: 10
• y2: 20
• x: 5

Result: 10

Explanation

For points (0,0) and (10,20), interpolating at x = 5: y = 0 + (5-0)(20-0)/(10-0) = 5 × 20/10 = 10.

Second Scenario

Inputs

• x1: 0
• y1: 0
• x2: 12
• y2: 20
• x: 5

Result: 10

Explanation

This scenario uses different inputs (x1 = 0, y1 = 0, x2 = 12, y2 = 20, x = 5) to show how changing one variable affects the linear interpolation result. Run the calculator above with these values to get the exact updated output with step-by-step work.