What is bilinear interpolation?

Bilinear interpolation extends linear interpolation to two dimensions. It first interpolates along one axis (x) at two rows, then interpolates the two resulting values along the second axis (y), producing a smooth estimate across a rectangular cell.

What are the four corner points?

The four corners of the rectangular grid cell are labeled Q11 (x1,y1), Q21 (x2,y1), Q12 (x1,y2), and Q22 (x2,y2), where x1 < x2 and y1 < y2. Each corner has a known value that you supply.

Does the query point have to be inside the cell?

No — the formula works for any (x, y) coordinate, but if the point lies outside the rectangle the result is a bilinear extrapolation, which carries more uncertainty. The calculator will flag this with a warning.

How accurate is bilinear interpolation?

Bilinear interpolation is exact at the four corners and provides a smooth, continuous estimate in between. It assumes the function is approximately linear along each axis within the cell, which is valid when the grid is fine enough relative to the rate of change.

What is the difference between bilinear and nearest-neighbor interpolation?

Nearest-neighbor simply copies the value of the closest corner, producing blocky results. Bilinear interpolation blends all four corners weighted by distance, giving a smoother and generally more accurate estimate.

Can x1 equal x2 or y1 equal y2?

No — both axis ranges must be non-zero. If x1 = x2 or y1 = y2 the grid cell is degenerate (zero width or height) and the formula is undefined. The calculator will display an error in that case.

Bilinear Interpolation Calculator

Enter four corner values of a rectangular grid cell and a query point (x, y) to calculate the bilinearly interpolated value.

Grid Cell Definition

X-Axis Range

x₁ (left)

x₂ (right)

Y-Axis Range

y₁ (bottom)

y₂ (top)

Corner Values

Q₁₁ at (x₁, y₁)

Q₂₁ at (x₂, y₁)

Q₁₂ at (x₁, y₂)

Q₂₂ at (x₂, y₂)

Query Point

x (query)

y (query)

Result

Fill in all fields above and click Calculate.

Grid Cell Diagram

Bilinear Interpolation Formula

t = (x − x1) / (x2 − x1)

u = (y − y1) / (y2 − y1)

R1 = (1−t) × Q11 + t × Q21

R2 = (1−t) × Q12 + t × Q22

P = (1−u) × R1 + u × R2

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Summary

Bilinear interpolation estimates an unknown value at an interior point (x, y) of a rectangular grid cell by performing linear interpolation first along one axis and then along the other. It requires the four corner coordinates and their known values: Q11 (x1,y1), Q12 (x1,y2), Q21 (x2,y1), and Q22 (x2,y2). This tool computes the result entirely in your browser with a step-by-step breakdown.

How it works

Enter the x-axis range: the left edge x1 and right edge x2 of the grid cell.
Enter the y-axis range: the bottom edge y1 and top edge y2 of the grid cell.
Enter the four corner values: Q11, Q12, Q21, Q22 at (x1,y1), (x1,y2), (x2,y1), (x2,y2).
Enter the query point coordinates (x, y) where you want to estimate the value.
Click Calculate — the tool interpolates first along x, then along y, and shows all steps.

Use cases

Estimate temperature or pressure at an interior grid point in meteorology.
Resample image pixel colors during texture mapping or image scaling.
Interpolate elevation values in a digital terrain model.
Find intermediate stress or strain values in a finite-element grid.
Estimate a lookup-table output between four neighboring entries.
Perform 2D function approximation in numerical analysis coursework.
Derive intermediate values from tabulated engineering data with two parameters.
Reconstruct missing sensor readings on a 2D measurement grid.

Frequently Asked Questions

Last updated: 2026-05-29 · Reviewed by Nham Vu