Interpolate a value inside a 2D rectangular grid from four corner values. See the step-by-step breakdown, corner weights, partial derivatives, and a contribution table.
Bilinear interpolation is a technique for estimating the value of a function at an arbitrary point within a rectangular grid, given the function's values at the four surrounding grid corners. It works by performing linear interpolation first in one direction (say, along x) and then in the perpendicular direction (along y), producing a smooth surface that passes exactly through the known corner values. The method is widely used in image processing (resizing and rotating images), geographic information systems (interpolating elevation or temperature data), computer graphics (texture mapping), and any scientific application where data is sampled on a regular 2D grid.
This calculator accepts the coordinates of the four corners (x₁, y₁) to (x₂, y₂), the function values at each corner Q₁₁, Q₂₁, Q₁₂, Q₂₂, and the target point (x, y). It instantly returns the interpolated value along with normalized coordinates, a verification via an alternative interpolation path, the value range, and approximate partial derivatives. A visual bar chart shows how much weight each corner contributes, and a detailed step-by-step table walks you through every calculation stage. Eight presets cover common use cases — from simple unit-square interpolation to temperature grids and terrain elevation lookups — so you can explore the method without manual setup.
Bilinear Interpolation Calculator helps you solve bilinear interpolation problems quickly while keeping each step transparent. Instead of redoing long algebra by hand, you can enter x₁, y₁, x₂ once and immediately inspect Interpolated Value, Verification (alt path), Normalized Coords (t, u) to validate your work.
This is useful for homework checks, classroom examples, and practical what-if analysis. You keep the conceptual understanding while reducing arithmetic mistakes in multi-step calculations.
f(x,y) ≈ Q₁₁(1−t)(1−u) + Q₂₁·t(1−u) + Q₁₂(1−t)u + Q₂₂·t·u, where t = (x−x₁)/(x₂−x₁) and u = (y−y₁)/(y₂−y₁).
Result: Interpolated Value shown by the calculator
Using the preset "Simple unit square", the calculator evaluates the bilinear interpolation setup, applies the selected algebra rules, and reports Interpolated Value with supporting checks so you can verify each transformation.
This calculator takes x₁, y₁, x₂, y₂ and applies the relevant bilinear interpolation relationships from your chosen method. It returns both final and intermediate values so you can audit the process instead of treating it as a black box.
Start with the primary output, then use Interpolated Value, Verification (alt path), Normalized Coords (t, u), Value Range to confirm signs, magnitude, and internal consistency. If anything looks off, change one input and compare the updated outputs to isolate the issue quickly.
A strong workflow is manual solve first, calculator verify second. Repeating that loop improves speed and accuracy because you learn to spot common setup errors before they cost points on multi-step algebra problems.
It is used to estimate values on a 2D grid: resizing digital images, interpolating weather data between sensor stations, texture mapping in 3D graphics, and elevation look-ups in topographic maps. Use this as a practical reminder before finalizing the result.
Linear interpolation works in 1D (between two points). Bilinear interpolation extends this to 2D by performing two nested linear interpolations — one in each axis direction.
Mathematically the formula still applies outside the rectangle, but extrapolation is unreliable because bilinear interpolation assumes the surface is well-represented by the four corners. Keep this note short and outcome-focused for reuse.
Bilinear interpolation requires a true rectangle with non-zero width and height. If one dimension collapses, the problem reduces to standard 1D interpolation.
No. Whether you interpolate first in x then y, or first in y then x, the result is identical. The "Verification (alt path)" output confirms this.
It produces a continuous surface (C⁰) but the first derivatives are discontinuous across grid cell boundaries. For smoother results, bicubic interpolation provides C¹ continuity.