How this instrument works
Bilinear interpolation estimates a value at a point (x, y) sitting inside a rectangle when only the four corner readings are known — the values at (x₁,y₁), (x₂,y₁), (x₁,y₂), and (x₂,y₂). It works by running ordinary straight-line interpolation twice: once across x to blend the bottom edge and the top edge separately, then again across y to blend those two results into the final figure. Nothing here is stranger than reading a value off a mark between two ticks on a ruler — this formula just repeats that trick along a second axis.
The name describes exactly what happens: hold y fixed and the surface is linear in x; hold x fixed and it's linear in y — linear along each axis taken on its own, hence bi-linear. Run the two passes in either order, x first or y first, and the arithmetic lands on the same figure both times, because the whole expression multiplies out into one polynomial carrying a shared xy term rather than two competing recipes. Strung between four corners that aren't already coplanar, though, the surface isn't flat — picture a saddle, not a tabletop.
A telling edge case: set q11 equal to q21 and set q12 equal to q22, so the left and right readings agree along the bottom and again along the top, and the x-step of the formula multiplies by a matched pair every time and contributes nothing — the whole thing collapses to plain one-dimensional interpolation in y. The formula also keeps computing outside the rectangle: push x or y past the [x₁,x₂] or [y₁,y₂] bounds and it still returns a number, only now the fractional weights exceed 1 or drop below 0. That result is an extrapolation riding on four points that no longer bracket it, a common slip when resizing an image or resampling an elevation grid near the edge of the surveyed area.
- Enter the rectangle's horizontal edges in x₁ (left) and x₂ (right), and its vertical edges in y₁ (bottom) and y₂ (top).
- Enter the four known readings, one per corner: Q₁₁ value (x₁,y₁), Q₂₁ value (x₂,y₁), Q₁₂ value (x₁,y₂), and Q₂₂ value (x₂,y₂).
- Enter the point you need a value for in x and y — it works best inside the rectangle, which is what the formula is built to interpolate across.
- Read Interpolated value for Q(x,y), the blend of the four corner readings weighted by how close (x, y) sits to each one.
Worked example — the center of a unit square
Take a grid cell with x₁ = 0, x₂ = 1, y₁ = 0, and y₂ = 1, carrying corner readings Q₁₁ = 0, Q₂₁ = 1, Q₁₂ = 1, and Q₂₂ = 2 — values that climb steadily toward the top-right corner. Asked for the value at the exact center, x = 0.5 and y = 0.5: blend along the bottom edge first, (1 − 0.5)×0 + 0.5×1 = 0.5, then along the top edge, (1 − 0.5)×1 + 0.5×2 = 1.5. Blending those two along y at 0.5 gives (1 − 0.5)×0.5 + 0.5×1.5 = 1.0 — Interpolated value reads 1.0 exactly, which happens to be the plain average of all four corners, since the center sits equally close to each one.
Run the passes in the other order and nothing changes. Blend the left edge along y at 0.5 first — (1 − 0.5)×0 + 0.5×1 = 0.5 — then the right edge the same way — (1 − 0.5)×1 + 0.5×2 = 1.5 — then blend those two along x at 0.5: (1 − 0.5)×0.5 + 0.5×1.5 = 1.0. Whichever axis goes first, the four readings 0, 1, 1, and 2 settle on the identical 1.0 at the center of the square, which is exactly what a formula symmetric in x and y should do.
Questions
What does bilinear interpolation actually compute?
A weighted average of four known corner values, where the weights depend on how close the query point (x, y) sits to each corner along both axes. Equivalently, it's two ordinary linear interpolations run in sequence — first across x, then across y, or the reverse — with both orders landing on the identical number.
How is this different from plain linear interpolation?
Linear interpolation works along a single line between two known points; bilinear interpolation works across a rectangle between four known points, one at each corner. It's built directly from linear interpolation rather than a separate idea: blend along one edge, blend along the opposite edge, then blend those two results together.
Does interpolating x before y change the answer?
No. Blending across x first and y second, or y first and x second, produces the same figure every time, because the full formula multiplies out to a single polynomial with a shared xy term rather than two independent procedures. That means you can pick whichever axis is easier to read off your data first.
What's the most common mistake with this formula?
Treating the answer as a simple average of the four corners. That's only true exactly at the rectangle's center; everywhere else, the nearer corners count for more. A second common mistake is feeding in an (x, y) outside the [x₁,x₂] by [y₁,y₂] rectangle and calling the result an interpolation — past the edges it is an extrapolation, and its accuracy is not guaranteed the way interpolation's is.
Why is it called bilinear rather than just linear?
Because the result is linear in x for any fixed y, and linear in y for any fixed x — linear along each axis on its own, hence bi-linear. The joint surface stretched across all four corners together is generally curved into a saddle shape rather than a flat plane, unless the four values happen to already lie in one plane.
Where does this formula actually get used?
Anywhere a value sits on a grid and one is needed in between: resizing a digital image estimates each new pixel from its four nearest original pixels, and reading a height off a surveyed elevation grid estimates it from the four nearest survey points. Both lean on this same four-corner formula, one grid cell at a time.