About

Bilinear interpolation estimates a value f(x, y) at an arbitrary point within a rectangular grid defined by four known corner values. It performs three linear interpolations: two along one axis to produce intermediate values R₁ and R₂, then one along the perpendicular axis to yield the final result. The method assumes the function varies linearly between sample points, which introduces error proportional to the second derivative of the true surface. Misapplying it to non-rectilinear grids or extrapolating outside the cell produces incorrect results without warning. This calculator enforces strict domain checks: the interpolation point must lie within or on the boundary of the rectangle [x₁, x₂] × [y₁, y₂].

Applications span image resampling (where corner values are pixel intensities), GIS terrain modeling (elevation grids), computational fluid dynamics (pressure or velocity fields), and climate data interpolation (temperature at unsampled coordinates). The tool assumes a rectilinear grid with x₁ ≠ x₂ and y₁ ≠ y₂. For irregular point distributions, consider inverse distance weighting or kriging instead.

Formulas

Given four corner values on a rectilinear grid at coordinates (x₁, y₁), (x₂, y₁), (x₁, y₂), and (x₂, y₂), the interpolated value at point (x, y) is computed in two stages.

First, compute normalized distances:

t = x − x₁x₂ − x₁ , u = y − y₁y₂ − y₁

Interpolate along the x-axis at each y level:

R₁ = (1 − t) ⋅ Q₁₁ + t ⋅ Q₂₁

R₂ = (1 − t) ⋅ Q₁₂ + t ⋅ Q₂₂

Then interpolate along the y-axis:

P = (1 − u) ⋅ R₁ + u ⋅ R₂

Equivalently, the full expanded form:

f(x, y) = Q₁₁(x₂ − x)(y₂ − y) + Q₂₁(x − x₁)(y₂ − y) + Q₁₂(x₂ − x)(y − y₁) + Q₂₂(x − x₁)(y − y₁)(x₂ − x₁)(y₂ − y₁)

Where: Q₁₁ = f(x₁, y₁) is the value at the bottom-left corner. Q₂₁ = f(x₂, y₁) is the bottom-right. Q₁₂ = f(x₁, y₂) is the top-left. Q₂₂ = f(x₂, y₂) is the top-right. t and u are the normalized interpolation parameters, each ∈ [0, 1].

Reference Data

Interpolation Method	Dimensions	Required Points	Continuity	Typical Use Case	Error Order
Nearest Neighbor	Any	1	C⁻¹ (discontinuous)	Fast image scaling, classification	O(h)
Linear (1D)	1D	2	C⁰	Signal resampling, table lookup	O(h²)
Bilinear	2D	4	C⁰	Image scaling, terrain grids, CFD	O(h²)
Trilinear	3D	8	C⁰	Volumetric data, 3D textures	O(h²)
Bicubic	2D	16	C¹	Photo resampling, smooth surfaces	O(h⁴)
Cubic Spline (1D)	1D	4	C²	Smooth curve fitting, CAD	O(h⁴)
Lanczos	1D/2D	6-12	C⁰	High-quality image downscaling	O(h⁴)
Inverse Distance Weighting	Any	n	C⁰	Irregular point clouds, GIS	Data-dependent
Kriging	2D/3D	n	C⁰	Geostatistics, mineral exploration	Optimal (BLUE)
Radial Basis Function	Any	n	C^∞	Mesh-free methods, scattered data	Spectral
Barycentric (Triangle)	2D	3	C⁰	FEM, triangulated surfaces	O(h²)
Hermite	1D/2D	4 + derivatives	C¹	Animation, smooth motion paths	O(h⁴)
B-Spline (Quadratic)	1D/2D	9	C¹	CAD surfaces, font rendering	O(h³)
Shepard’s Method	Any	n	C⁰	Quick scattered data approx.	Data-dependent

Frequently Asked Questions

When the point lies on an edge, bilinear interpolation degenerates to standard linear interpolation along that edge. If the point coincides exactly with a corner, the result equals the corner value. Both are mathematically correct special cases of the general formula - the weights for the non-contributing corners become zero. This calculator handles these cases without additional logic because the formula naturally produces the correct result.

Bilinear interpolation uses 4 points and produces C⁰ continuity (continuous values, but discontinuous first derivatives at cell boundaries). Bicubic interpolation uses 16 points (a 4×4 neighborhood) and achieves C¹ continuity (continuous first derivatives). Use bilinear when speed matters or data is sparse. Use bicubic when visual smoothness is critical, such as photographic image scaling or smooth surface rendering in CAD.

Technically, the formula accepts any values of x and y, but extrapolation (when t or u falls outside [0, 1]) produces unreliable results. The linear assumption between grid points does not hold beyond the sampled region. This calculator warns you if the point lies outside the grid and flags the result as an extrapolation.

No. Bilinear interpolation is commutative with respect to axis order. Interpolating along x first then y yields the identical result as interpolating along y first then x. This is because the expanded formula is symmetric in the product terms (x₂ − x)(y₂ − y), etc.

The truncation error is O(h²), where h is the grid spacing. More precisely, the error bound is proportional to h_xh_y times the magnitude of the mixed second partial derivative ∂²f/∂x∂y. Halving the grid spacing reduces the error by a factor of 4. For functions with large curvature or sharp gradients, bilinear interpolation introduces visible artifacts.

It does not. Bilinear interpolation for points inside the cell always returns a value within the range [min(Q₁₁, Q₂₁, Q₁₂, Q₂₂), max(Q₁₁, Q₂₁, Q₁₂, Q₂₂)]. This is the convex combination property. If your result exceeds corner values, the interpolation point is outside the cell (extrapolation) or the inputs contain an error.