Bilinear Interpolation Calculator - 2D Grid Weighted Average

A bilinear interpolation calculator estimates the value of a function at any point (x, y) inside a rectangle when you already know the function's value at the four rectangle corners. It applies the bilinear interpolation formula as a weighted average, so the closer a corner is to the target, the more it pulls the result. The same routine is used to resample image pixels, fill gaps in 2D measurement grids, and smooth a rectangular table of lookup values.

• • Resample image pixels: Estimate the color or intensity at fractional pixel coordinates when scaling or rotating a digital image.
• • Estimate tabular lookups: Pull a value from a rectangular table of measurements when the desired row or column is between two known entries.
• • Smooth a 2D field: Fill a denser grid from a coarser grid of sensor readings, weather data, or simulation outputs.
• • Verify textbook examples: Check the algebra of homework and exam problems that ask for an interpolated value on a rectangle.

Bilinear interpolation is a multivariate method that generalizes one-dimensional linear interpolation to a 2D grid. It draws a small bilinear surface patch through four corner values.

Because the result is a weighted average of four corner values, you can use the Weighted Average Calculator to sanity-check how the four weights combine to form P.

The bilinear interpolation formula computes a single weighted average of the four corner values Q11, Q12, Q21, and Q22. Each weight is the product of the opposite horizontal and vertical distances divided by the rectangle area, so the four weights always sum to exactly 1 for any real (x, y). Inside the rectangle every weight is in the 0..1 range; outside it, some go negative or above 1, which is how you tell you are extrapolating.

• x1, x2: Left and right x-coordinates of the rectangle's edges.
• y1, y2: Bottom and top y-coordinates of the rectangle's edges.
• Q11, Q12, Q21, Q22: Known function values at the four corners; the first index is the x corner, the second is the y corner.
• x, y: Target coordinates where the function is being interpolated.
• P: The estimated function value at (x, y) returned by the calculator.

The same formula can be derived by performing two linear interpolations in the x-direction and one in the y-direction. The order of axes does not change the answer, because the final expression collapses to the same closed form regardless of which axis you interpolate first.

When x and y both lie on the rectangle edges, two of the four weights become 0 and the formula collapses to ordinary 1D linear interpolation. When the four corner values are constant, the result is that constant, which is a quick sanity check that the implementation is correct.

According to Wikipedia - Bilinear interpolation, the bilinear interpolation result can be written as a weighted average w11*Q11 + w12*Q12 + w21*Q21 + w22*Q22 where the weights are the products of the opposite horizontal and vertical distances divided by the rectangle area (x2-x1)(y2-y1).

When the four corners come from a triangle or polygon laid out on the same coordinate plane, the Area Triangle Coordinates Calculator shows how the same corner coordinates drive area and side-length calculations.

Four ideas make the result straightforward to read and verify.

These four concepts line up with the calculator's output panel: the four weights and the rectangle area are the diagnostic numbers that confirm the algebra without re-deriving the formula.

The same arithmetic shows up as a 2x2 weighted blend of neighboring texels, which is why bilinear interpolation is sometimes called bilinear filtering.

Scan a single edge of the rectangle and the bilinear patch collapses to ordinary 1D linear interpolation, so the Slope Percentage Calculator gives the matching one-dimensional answer for a line through two of the corner values.

Five quick steps take you from raw corner values to a verified interpolated estimate.

1. 1 Enter the rectangle edges: Type x1 and x2 for the left and right edges, then y1 and y2 for the bottom and top edges. The rectangle must have non-zero width and height.
2. 2 Enter the four corner values: Provide Q11 at (x1, y1), Q12 at (x1, y2), Q21 at (x2, y1), and Q22 at (x2, y2). These are the known function values that anchor the surface.
3. 3 Enter the target point: Type the x and y where you want the interpolated value. The point can be inside or outside the rectangle; the calculator will compute the result and the matching weights.
4. 4 Read the result and weights: Check P, the four corner weights, the rectangle area, and the weight-sum check. The weights always add up to exactly 1; when an individual weight is negative or above 1, the target is outside the rectangle and you are extrapolating.
5. 5 Iterate with new corners or targets: Change any input to update the result on the fly to scan a rectangle or build a denser grid. The individual corner weights are the cleanest signal of whether each new target is inside the rectangle.

Try the Omni Calculator example: x1=0, x2=4, y1=1, y2=3, Q11=12, Q12=-4, Q21=0, Q22=8, target (1, 2). The calculator returns P=4 with weights 0.375, 0.375, 0.125, 0.125, matching the published result.

If you want to draw the rectangle and target point before interpolating, the Coordinate Plane Calculator gives you a quick way to plot the corners and the (x, y) target on the same grid.

Six practical reasons to use this calculator instead of recomputing the formula by hand.

• • Get the answer in seconds: Skip the handwritten algebra; the calculator applies the closed-form formula on every keystroke.
• • See every weight: Each of the four corner weights is shown so you can understand exactly how P is built from Q11..Q22.
• • Catch rectangle errors early: Validation stops the calculation when x2 equals x1 or y2 equals y1, so you never see a divide-by-zero error.
• • Check the algebra with a weight sum: The weight-sum line is always exactly 1.000000 for any real target, giving a free sanity check that the formula is wired up right.
• • Resample any rectangle: Use the same tool to upscale image regions, smooth sensor grids, and rescale lookup tables without writing code.
• • Learn the method step by step: Worked examples and visible weights make this calculator a study aid for multivariate interpolation homework.

Together, these benefits turn the bilinear interpolation formula into a one-line operation. Chain the calculator with the related tools below to extend the result into weighted averages, rectangle geometry, and slope work. When an individual weight falls outside the 0..1 range, the target is outside the rectangle and a different method is more appropriate.

When you want to sanity-check the rectangle dimensions (x2-x1 and y2-y1) that drive the corner weights, the Length Width Area Rectangle Calculator computes the width, height, and area of the same rectangle in one step.

Five factors determine how reliable the result is.

• • Bilinear interpolation is linear in each variable separately and carries only a single xy cross term, so it cannot follow sharp bends. If the true function bends sharply inside the rectangle, switch to bicubic or trilinear interpolation for a closer fit.
• • Bilinear interpolation is not invertible: many different (x, y) points can produce the same P inside a constant-valued region, so the method cannot be used to back out a unique location from a value.

These factors line up with the same limitations that appear in image processing. Bilinear filtering is fast and gentle, but it cannot recover sharp edges the way a higher-order resampler can.

Move to a finer grid, a smaller rectangle, or a bicubic interpolant when the field curves faster than bilinear can follow.

According to Omni Calculator - Bilinear Interpolation, P is a degree-2 polynomial in (x, y) whose only non-linear term is the xy cross term, which is why bilinear interpolation is reliable on smooth fields but can lose accuracy where the true function curves sharply.

When the corner data lives in lat/lon, polar, or image coordinates and you need to map it into the (x, y) rectangle first, the Coordinates Converter handles the conversion between the source and target coordinate systems.