Linear Interpolation Calculator (Lerp)
Plug in two known points P₁(x₁, y₁) and P₂(x₂, y₂) plus a target x to instantly compute y from the linear formula y = y₁ + (x − x₁) · (y₂ − y₁) / (x₂ − x₁), along with the line's slope, y-intercept and equation. When x falls inside [x₁, x₂] the result is flagged as interpolation; outside that interval it is flagged as extrapolation with a reminder that accuracy degrades the further you push beyond the data. Typical uses include reading between rows of a table (steam tables, tax brackets, rate sheets), sensor calibration, animation/graphics tweening (lerp), and estimating a third sample from two measured ones.
Enter finite numbers and make sure x₁ ≠ x₂ (the two points cannot share an x-coordinate).
Estimated y
50
Slope m
10
y-intercept b
0
Line equation
y = 10x
Displayed values are rounded to 6 decimals; the URL preserves your original inputs.
Formula
y = y₁ + (x − x₁) · (y₂ − y₁) / (x₂ − x₁); equivalently y = m·x + b with m = (y₂ − y₁)/(x₂ − x₁) and b = y₁ − m·x₁
- · Linear interpolation assumes the underlying relationship between the two known points is a straight line. When the true function is noticeably curved (exponentials, logarithms, trigonometric functions, …) this introduces error — usually acceptable over small intervals, which is what makes table look-ups practical. (Reference: Burden & Faires, Numerical Analysis, 9th ed., §3.1.)
- · If x lies inside [min(x₁, x₂), max(x₁, x₂)] the result is interpolation; outside that interval it is extrapolation. Extrapolation error generally grows with the distance from the data because the linear assumption may no longer hold.
- · When x₁ = x₂ the two points share an x-coordinate and there is no unique y for the target — the tool flags the input as invalid rather than returning NaN or ±∞.
- · The output is independent of the order in which you enter the two points: swapping (x₁, y₁) with (x₂, y₂) gives the same y, slope and intercept.
- · In computer graphics, games and animation, "lerp(a, b, t)" is the special case x₁ = 0, x₂ = 1, x = t, giving y = a + t·(b − a). t ∈ [0, 1] is interpolation; outside that range is extrapolation.
- · Engineering table look-ups (steam tables, ASHRAE charts, between tax-bracket rows), multi-point sensor calibration, and estimating a value between two measurements all rely on this formula as their first-order approximation.
- · Displayed values are rounded to 6 decimal places for readability; the URL preserves your original inputs so a shared link reproduces the exact same calculation.
Frequently asked
What is the difference between linear interpolation and extrapolation?
Both use the exact same formula y = y₁ + (x − x₁) · (y₂ − y₁) / (x₂ − x₁); only the location of x differs. When x lies between the two known points (min(x₁,x₂) ≤ x ≤ max(x₁,x₂)) you are interpolating, which is generally reliable. When x lies outside that interval you are extrapolating — the straight-line assumption may no longer hold and error typically grows with distance from the data. The tool labels which case your input falls into.
When is linear interpolation good enough, and when do I need something fancier?
Linear interpolation is good enough whenever the x-gap between the two known points is small and the underlying function is smooth — engineering tables, sensor calibration and reading between rate-table rows are typical cases. When the gap is large, the underlying function is clearly curved (e.g. exponential decay or a sine wave), or you need to differentiate / integrate the interpolated result, switch to polynomial interpolation (Lagrange, Newton divided differences), cubic-spline interpolation, or a proper nonlinear fit — error drops dramatically.
How does the "lerp" function in graphics / games relate to this?
It is the same formula in different clothing. lerp(a, b, t) is the special case x₁ = 0, y₁ = a, x₂ = 1, y₂ = b, x = t, which collapses to y = a + t · (b − a). t = 0 returns a, t = 1 returns b, 0 < t < 1 is interpolation, and t < 0 or t > 1 is extrapolation (sometimes used deliberately for overshoot or bounce). If your inputs are already in 0/1 form, you can plug them directly into this tool with identical results.
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