Bilinear Interpolation Calculator
Enter the four corner values of a rectangular grid and the x, y point to query. The calculator applies bilinear interpolation in two stages — first along x for each row, then along y — and shows every intermediate step.
Bilinear interpolation using the four corner values
Normalise x position within the grid (tₓ)
Interpolate along x at y = y₁ (bottom edge) → R₁
Interpolate along x at y = y₂ (top edge) → R₂
Normalise y position, then interpolate between R₁ and R₂
- 1
Normalise x position (tₓ)
tₓ = (2 − 0) ÷ (4 − 0) = 0.5 - 2
Bottom-edge interpolation R₁ (at y₁)
100 × (1 − 0.5) + 200 × 0.5 = 150 - 3
Top-edge interpolation R₂ (at y₂)
400 × (1 − 0.5) + 300 × 0.5 = 350 - 4
Normalise y (tᵧ), then interpolate between R₁ and R₂
tᵧ = 0.5; 150 × (1 − 0.5) + 350 × 0.5 = 250The result equals a weighted sum of all four corner values.
How does this calculator work?
Bilinear interpolation: normalise x position as tₓ = (x−x₁)/(x₂−x₁), interpolate along x at each edge to get R₁ and R₂, then interpolate along y: f = R₁(1−tᵧ) + R₂tᵧ. Result is a weighted sum of all four corner values. Useful in image scaling, GIS rasters and gridded data.
Formula
How this is calculated
Bilinear interpolation estimates a value at an arbitrary point inside a 2D rectangular grid by performing three linear interpolations. The grid is defined by its four corners: (x₁,y₁), (x₂,y₁), (x₁,y₂) and (x₂,y₂), each with a known function value. Step 1 interpolates linearly along x at the bottom edge (y = y₁) using the bottom-left and bottom-right corner values to get an intermediate result R₁. Step 2 does the same at the top edge (y = y₂) to get R₂. Step 3 interpolates vertically between R₁ and R₂ using the y position of the query point.
The algorithm is equivalent to a weighted sum of all four corners, where each weight equals the area of the opposite sub-rectangle divided by the total grid area. This guarantees that the interpolated surface is exactly linear along any horizontal or vertical line through the grid, and that the result matches each corner value exactly when the query point sits on a corner.
Bilinear interpolation is widely used in computer graphics (texture filtering), geographic information systems (resampling raster maps), finite-element analysis and meteorology (interpolating gridded weather data). A key limitation is that the interpolated surface is not globally smooth — the partial derivatives are discontinuous across grid lines, making bicubic or higher-order methods preferable when smoothness across the entire surface matters.
Frequently asked questions
Linear interpolation works in one dimension — it estimates a value between two known points on a line. Bilinear interpolation extends this to two dimensions by applying linear interpolation twice: first in the x direction (once for each horizontal edge), then in the y direction between the two intermediate results.
Yes — the calculator still computes a result when the query point lies outside the grid, but the result is an extrapolation, not interpolation. Extrapolation assumes the linear trend continues beyond the data, which may be unreliable far from the grid. The calculator warns you when this occurs.
No. Bilinear uses only four corner points and produces a surface that is linear along any axis-aligned line. Bicubic interpolation uses a 4×4 neighbourhood of 16 points and fits cubic polynomials, producing a smoother surface. Biquadratic uses 9 points and quadratic polynomials. Bilinear is the simplest and most efficient of the three.
Also known as
TG we-Calculate Editorial Team. (2026). Bilinear Interpolation Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/bilinear-interpolation-calculator
TG we-Calculate Editorial Team. "Bilinear Interpolation Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/bilinear-interpolation-calculator.
TG we-Calculate Editorial Team, "Bilinear Interpolation Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/bilinear-interpolation-calculator
@misc{wecalculate_bilinear_interpolation_calculator, title = {Bilinear Interpolation Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/bilinear-interpolation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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