Linear Interpolation Calculator

Frequently Asked Questions

What is linear interpolation?

Estimating a value on the straight line between two known points: y = y₁ + (x − x₁)·(y₂ − y₁)/(x₂ − x₁). It assumes a constant rate of change between the points.

What is the difference from extrapolation?

Interpolation estimates within the known range; extrapolation projects beyond it and is far less reliable because the linear trend may not continue.

When is linear interpolation accurate?

When the underlying relationship is roughly linear over the interval. For curved data it introduces error - use closely spaced points or a non-linear model.

Where is it used?

Reading between table values (steam tables, tax brackets), graphics/animation tweening, sensor calibration, and filling gaps in data series.

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How This Calculator Works

Linear interpolation assumes the relationship between two known points (x₁, y₁) and (x₂, y₂) is a straight line and reads off the value anywhere along it. The slope is m = (y₂ − y₁) ÷ (x₂ − x₁), and the interpolation formula is y = y₁ + (x − x₁) × m. Rearranged, it solves for x given y: x = x₁ + (y − y₁) ÷ m. When the target lies between the two anchors it is interpolation; when it lies outside the bracket it is extrapolation - the same arithmetic, but the result is an estimate beyond the data and this tool flags it. Geometrically you are evaluating the unique line through two points, equivalently a weighted average: y = (1 − t)·y₁ + t·y₂ where t = (x − x₁)÷(x₂ − x₁) is the fractional position between them.

Worked Example: x = 22 between (10, 20) and (30, 60)

Slope m = (60 − 20) ÷ (30 − 10) = 40 ÷ 20 = 2. Interpolate at x = 22: y = 20 + (22 − 10) × 2 = 20 + 24 = 44. As a weighted average, t = (22 − 10) ÷ 20 = 0.6, so y = 0.4×20 + 0.6×60 = 8 + 36 = 44 - same answer. Solving the inverse: what x gives y = 50? x = 10 + (50 − 20) ÷ 2 = 10 + 15 = 25. Extrapolation example: x = 40 lies outside [10, 30], so y = 20 + (40 − 10) × 2 = 80 is an extrapolated estimate - valid only if the linear trend really continues past the data.

Where Interpolation Is Used

Engineering tables: steam tables, thermodynamic and material properties between tabulated entries.

Finance: bootstrapping yield curves and pricing instruments between quoted maturities.

Graphics & animation: tweening - computing in-between frames; texture and color blending (lerp).

Data science: resampling time series and filling gaps in evenly trending measurements.

Common Mistakes & Edge Cases

Extrapolating far beyond the data and trusting it - linearity rarely holds at the extremes; this tool warns when the target is outside the bracket.

Assuming the underlying relationship is linear when it is curved - interpolating a few points of an exponential or log curve introduces real error; use more points or a spline.

x₁ = x₂ gives a vertical line with undefined slope - rejected here.

Solving for x when the slope is zero (a horizontal line) has no unique solution - also rejected.

Swapping point order changes nothing for the value but flips the sign of intermediate differences; the formula is order-independent.

Related Concepts & Limits

Linear interpolation is the degree-1 case of polynomial interpolation; with more points, Lagrange or Newton interpolation, or piecewise cubic splines, capture curvature without the oscillation that high-degree single polynomials suffer (Runge's phenomenon). Bilinear and trilinear interpolation extend the same weighted-average idea to 2D images and 3D volumes. The reliability of any interpolation depends entirely on how well a line approximates the true function over the interval - closely spaced anchors on a smooth curve are accurate; widely spaced ones on a bending curve are not. Computations use double precision (≈ 15 significant digits). For non-linear data, treat this as a first-order estimate and validate against additional points.

Linear Interpolation Calculator

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