Bilinear Interpolation Calculator

Enter the four corner coordinates (x₁, x₂, y₁, y₂) and their corresponding known values (Q₁₁, Q₁₂, Q₂₁, Q₂₂), then specify your target point (x, y) to get the bilinearly interpolated value P. This Bilinear Interpolation Calculator performs two-dimensional grid interpolation across a rectangular region — useful in engineering, image processing, and numerical analysis.

The smaller x-coordinate of the rectangle.

The x-coordinate of the point to interpolate. Must be between x₁ and x₂.

The larger x-coordinate of the rectangle.

The smaller y-coordinate of the rectangle.

The y-coordinate of the point to interpolate. Must be between y₁ and y₂.

The larger y-coordinate of the rectangle.

Known function value at the bottom-left corner (x₁, y₁).

Known function value at the bottom-right corner (x₂, y₁).

Known function value at the top-left corner (x₁, y₂).

Known function value at the top-right corner (x₂, y₂).

Results

Interpolated Value P

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Intermediate R₁ (interpolated along x at y₁)

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Intermediate R₂ (interpolated along x at y₂)

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Weight for x₁ side

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Weight for x₂ side

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Corner Values vs. Interpolated Result

Frequently Asked Questions

What is bilinear interpolation?

Bilinear interpolation is a technique for estimating the value of a function at any point inside a rectangle when you know the function's values at the four corners. It works by performing linear interpolation first in one direction (e.g. x), then in the other direction (e.g. y), hence the prefix 'bi-linear'.

What is the bilinear interpolation formula?

The formula computes two intermediate values: R₁ = Q₁₁ + (x − x₁)/(x₂ − x₁) × (Q₂₁ − Q₁₁) at y₁, and R₂ = Q₁₂ + (x − x₁)/(x₂ − x₁) × (Q₂₂ − Q₁₂) at y₂. The final result is P = R₁ + (y − y₁)/(y₂ − y₁) × (R₂ − R₁). This effectively blends all four corner values proportionally.

When should I use bilinear interpolation instead of nearest-neighbor interpolation?

Bilinear interpolation is preferred when you need a smooth, continuous estimate that takes into account all four surrounding data points. Nearest-neighbor simply picks the closest known value, which can produce blocky or discontinuous results. Bilinear interpolation is standard in image resizing, GIS mapping, and scientific data grids.

Does the target point (x, y) have to lie inside the rectangle?

For true interpolation, yes — x should be between x₁ and x₂, and y should be between y₁ and y₂. If the point lies outside the rectangle, the calculation becomes extrapolation, which can produce less reliable results. Always verify your target coordinates fall within the known grid bounds.

What do Q₁₁, Q₁₂, Q₂₁, and Q₂₂ represent?

These are the known function values at the four corners of the rectangle: Q₁₁ is the value at (x₁, y₁), Q₂₁ at (x₂, y₁), Q₁₂ at (x₁, y₂), and Q₂₂ at (x₂, y₂). You read these values from your existing data table or grid before performing the interpolation.

Is bilinear interpolation the same as 2D linear interpolation?

Yes — bilinear interpolation is often called double interpolation or 2D linear interpolation. It is linear in each of the two coordinate directions (x and y) separately, making it a natural extension of standard 1D linear interpolation to two dimensions.

What are common real-world applications of bilinear interpolation?

Bilinear interpolation is widely used in image processing (scaling and rotating images), GIS (estimating elevations between known survey points), engineering tables (finding values between tabulated data), texture mapping in computer graphics, and numerical simulation grids.

What happens if x₁ equals x₂ or y₁ equals y₂?

If x₁ = x₂ or y₁ = y₂, the denominator in the interpolation formula becomes zero, making the calculation undefined. This means the four points do not form a proper rectangle. Make sure x₁ ≠ x₂ and y₁ ≠ y₂ when using the calculator.

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