Bilinear / Double Interpolation Calculator
Perform double linear interpolation on 2D regular grid matrices. Ideal for thermodynamic steam tables, engineering reference charts, image resampling, and digital elevation models.
How the Bilinear Interpolation Calculator (Double Interpolation) Works
In physical sciences and engineering, data is frequently tabulated as a function of two independent variables—such as pressure and temperature, or latitude and longitude. When your desired condition falls between four bounding table cells, our bilinear interpolation calculator (also known as a double interpolation calculator) provides the standard method to estimate intermediate values with complete mathematical certainty.
The technique executes three successive 1D linear interpolations: first, two horizontal interpolations across the bounding rows at $y_1$ and $y_2$ to find intermediate values $R_1$ and $R_2$; followed by a final vertical interpolation between $R_1$ and $R_2$ at the target $y$. This double interpolation process maintains continuity across regular 2D grids and lookup tables.
Bilinear Interpolation Formula
- f(x, y)=Interpolated 2D value: The calculated output value at the target coordinate (x, y).
- x₁, x₂=Grid X boundaries: The bounding column coordinates along the horizontal axis.
- y₁, y₂=Grid Y boundaries: The bounding row coordinates along the vertical axis.
- Q₁₁, Q₂₁=Row 1 corner values: Function values f(x₁, y₁) and f(x₂, y₁) along the bottom row.
- Q₁₂, Q₂₂=Row 2 corner values: Function values f(x₁, y₂) and f(x₂, y₂) along the top row.
Note: When normalized coordinates u = (x - x₁)/(x₂ - x₁) and v = (y - y₁)/(y₂ - y₁) are used, the equation simplifies to f(u, v) = (1-u)(1-v)Q₁₁ + u(1-v)Q₂₁ + (1-u)vQ₁₂ + uvQ₂₂.
Step-by-Step Worked Examples
Walk through two complete real-world calculation problems or click any example to load its numbers directly into the calculator.
Thermodynamic Steam Table Enthalpy Lookup
An engineer needs the specific enthalpy h (kJ/kg) of superheated steam at pressure P = 14 bar and temperature T = 260°C. The steam table lists values at 10 bar and 20 bar across 200°C and 300°C.
- Compute spans: Δx = 20 - 10 = 10 bar, Δy = 300 - 200 = 100 °C
- Interpolate in X along Row 1 (200°C): R₁ = ((20-14)/10)×2828 + ((14-10)/10)×2797 = 0.6×2828 + 0.4×2797 = 2815.6 kJ/kg
- Interpolate in X along Row 2 (300°C): R₂ = ((20-14)/10)×3051 + ((14-10)/10)×3023 = 0.6×3051 + 0.4×3023 = 3039.8 kJ/kg
- Interpolate in Y between R₁ and R₂: ((300-260)/100)×2815.6 + ((260-200)/100)×3039.8 = 0.4×2815.6 + 0.6×3039.8 = 2950.12 kJ/kg
Digital Elevation Model (DEM) Terrain Height
A GIS surveyor wants to determine the ground elevation at GPS coordinate (x = 45 m, y = 70 m) located between four survey grid points spaced 100 meters apart.
- Normalized weights: u = (45 - 0) / 100 = 0.45, v = (70 - 0) / 100 = 0.70
- X-interpolation at y = 0: R₁ = (1 - 0.45)×120 + 0.45×140 = 0.55×120 + 0.45×140 = 129.0 m
- X-interpolation at y = 100: R₂ = (1 - 0.45)×130 + 0.45×180 = 0.55×130 + 0.45×180 = 152.5 m
- Y-interpolation: f(45, 70) = (1 - 0.70)×129.0 + 0.70×152.5 = 0.30×129.0 + 0.70×152.5 = 38.7 + 106.75 = 145.45 m
Bilinear Interpolation FAQs
What is bilinear interpolation?
Bilinear interpolation (also known as double interpolation) extends 1D linear interpolation to two-dimensional regular grids. It computes an interpolated value at (x, y) by performing linear interpolation first in one direction (usually along the x-axis for both bounding rows) and then linearly interpolating the results in the second direction (along the y-axis).
Does the order of interpolation matter (X first or Y first)?
No. Bilinear interpolation is mathematically symmetric. Whether you interpolate along the X direction first and then along Y, or interpolate along Y first and then along X, the resulting value is identical because multiplication and addition of the weighting factors are commutative.
Why is it called bilinear if the result is not a flat plane?
It is called 'bilinear' because it is strictly linear when considered along any line parallel to either the X or Y coordinate axes. However, because the formula contains cross-product terms (x · y), the interpolated surface is actually a hyperbolic paraboloid rather than a flat plane.
Can bilinear interpolation be used on irregular or rotated grids?
Standard bilinear interpolation assumes a rectilinear grid where coordinate axes are orthogonal and parallel to the bounding box. For irregular triangles or scattered coordinate clouds, triangular interpolation (TIN) or barycentric coordinates are typically employed.
Where is bilinear interpolation commonly applied?
Bilinear interpolation is ubiquitous in engineering and computing: reading 2D lookup tables (thermodynamic steam tables, aerodynamic lift coefficients), digital image resampling and anti-aliasing, geographic information systems (GIS DEM elevations), and weather forecast grid downscaling.