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Interpolation is the mathematical technique of estimating unknown values that fall between known data points. It is one of the oldest and most practical tools in applied mathematics, used whenever you have discrete measurements and need to estimate values at intermediate positions. Linear interpolation — the simplest and most widely used form — assumes the data changes at a constant rate between two known points and draws a straight line between them to estimate intermediate values.
The Interpolation Calculator performs linear interpolation given two known points $$(x_0, y_0)$$ and $$(x_1, y_1)$$ and a target x-value. The formula $$y = y_0 + \frac{(x_{\text{target}} - x_0)(y_1 - y_0)}{x_1 - x_0}$$ computes the estimated y-value by finding what fraction of the way the target lies between the known x-values and applying that same fraction to the y-values.
Linear interpolation appears in an extraordinary range of practical applications. In engineering, it is used to read values from tables (steam tables, material property tables) between tabulated entries. In computer graphics, it blends colors, positions, and textures — the "lerp" function is ubiquitous in game engines and rendering pipelines. In signal processing, it reconstructs analog signals from digital samples. In geographic information systems (GIS), it estimates elevation or temperature between measurement stations.
In finance, linear interpolation constructs yield curves from observed bond yields at specific maturities, estimates implied volatilities between strike prices in options pricing, and fills gaps in time-series data. In manufacturing, quality control uses interpolation to estimate measurements between calibration points. In meteorology, weather forecasts interpolate between observation stations to create continuous temperature and pressure maps.
The key assumption of linear interpolation is that the true function is approximately linear between the known points. This assumption is more accurate when the points are close together (small Δx) or when the underlying function is smooth. For highly curved functions, higher-order methods like polynomial interpolation (Lagrange, Newton), spline interpolation (cubic splines), or trigonometric interpolation (Fourier) provide better accuracy, but linear interpolation remains the fastest, simplest, and most robust method.
The calculator also displays the slope (rate of change between the known points), the fraction along the interval (how far the target lies between the known x-values, where 0 = at x₀ and 1 = at x₁), and the deltas (total change in x and y). A fraction outside [0, 1] means you are extrapolating beyond the known data — a practice that carries increasing risk of error the further you go beyond the known range.
Enter two known data points and the target x-value. The calculator computes:
Linear Interpolation Formula:
$$y_{\text{target}} = y_0 + (x_{\text{target}} - x_0) \cdot \frac{y_1 - y_0}{x_1 - x_0}$$
This can be rewritten as:
$$y_{\text{target}} = y_0 + t \cdot \Delta y$$
where $$t = \frac{x_{\text{target}} - x_0}{\Delta x}$$ is the fraction along the interval, $$\Delta x = x_1 - x_0$$, and $$\Delta y = y_1 - y_0$$.
Slope: $$m = \frac{\Delta y}{\Delta x} = \frac{y_1 - y_0}{x_1 - x_0}$$
If $$0 \leq t \leq 1$$, the target lies between the known points (interpolation). If $$t < 0$$ or $$t > 1$$, the target lies outside (extrapolation).
The interpolated y is the estimated value at your target x, assuming a linear relationship between the two known points. The slope shows the rate of change — how much y changes per unit of x. The fraction along interval shows the relative position: 0.0 means the target equals x₀, 1.0 means it equals x₁, 0.5 means it is exactly halfway. Values outside [0, 1] indicate extrapolation, which is less reliable than interpolation. The deltas (Δx, Δy) show the total span between known points — smaller spans generally yield more accurate interpolation for curved functions.
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Given temperatures 10°C at hour 1 and 30°C at hour 5, the estimated temperature at hour 3 is 20°C. The target is 50% of the way through the interval, so the temperature is 50% of the way from 10 to 30.
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With data at x=0 (y=100) and x=10 (y=200), predicting at x=15 gives y=250. The fraction of 1.5 confirms extrapolation — the target is 50% beyond the known range, increasing uncertainty.
Linear interpolation estimates a value between two known data points by assuming a constant rate of change (straight line) between them. The formula $$y = y_0 + (x - x_0) \cdot \frac{y_1 - y_0}{x_1 - x_0}$$ finds the y-value on the line connecting $$(x_0, y_0)$$ and $$(x_1, y_1)$$ at the target x. It is the simplest and most common interpolation method.
Interpolation estimates values between known data points (fraction 0-1). Extrapolation estimates values beyond known data (fraction <0 or >1). Interpolation is generally reliable because the data brackets the estimate; extrapolation is riskier because it assumes the trend continues unchanged beyond the observed range.
Linear interpolation is most accurate when: (1) the underlying function is approximately linear between the points; (2) the known points are close together; (3) the function is smooth (no abrupt changes). For highly curved functions, errors increase. The maximum interpolation error is bounded by $$\frac{1}{8}(\Delta x)^2 \cdot \max|f''|$$, where f'' is the second derivative.
lerp(a, b, t) = a + t × (b - a) is the standard linear interpolation function used ubiquitously in game engines, animation, and rendering. It blends between value a (at t=0) and value b (at t=1). Colors, positions, transparency, and other visual properties are interpolated this way to create smooth transitions.
Engineering steam tables list thermodynamic properties at discrete temperatures and pressures. To find properties at intermediate values, engineers use linear interpolation between adjacent table entries. For example, if steam enthalpy is 2676 kJ/kg at 100°C and 2716 kJ/kg at 110°C, the enthalpy at 105°C is approximately 2696 kJ/kg.
Beyond linear interpolation: Polynomial interpolation (Lagrange, Newton) fits curves through 3+ points. Spline interpolation uses piecewise polynomials for smoothness. Cubic Hermite also matches slopes. Barycentric interpolation is numerically stable. Higher-order methods are more accurate for curved data but can oscillate (Runge's phenomenon) with many points.
This calculator performs linear interpolation between two points. For multiple data points, you would use piecewise linear interpolation (connecting consecutive pairs with straight lines), polynomial interpolation (a single curve through all points), or spline interpolation (smooth piecewise polynomials). Each method has trade-offs between simplicity, accuracy, and smoothness.
The fraction t indicates the relative position of your target within the interval. t=0 means you are at x₀; t=1 means you are at x₁; t=0.5 means exactly halfway. If t is negative, your target is before x₀; if t>1, it is beyond x₁. Values of t far from [0,1] indicate significant extrapolation and reduced reliability.
Financial applications include: (1) Yield curves — interpolating interest rates between known maturities; (2) Implied volatility — estimating option volatilities between strike prices; (3) Time series — filling missing daily returns from weekly data; (4) Present value — interpolating discount factors. Bloomberg and other financial terminals use various interpolation methods for these tasks.
Bilinear interpolation extends linear interpolation to two dimensions. Given four corner values on a grid, it first interpolates linearly in the x-direction (top and bottom edges), then interpolates between those results in the y-direction. It is standard in image scaling, texture mapping in 3D graphics, and spatial data analysis (temperature maps, terrain elevation).
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The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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