Linear Interpolation Calculator
Interpolate a y-value for a given x between two known points using linear interpolation (LERP). Supports extrapolation, multi-point error analysis, and shows slope, equation, and interpolation para...
Bilinear Interpolation Calculator
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.
Grid Corners (coordinates)
Corner Values
Interpolation Point
Interpolated Value⧉
0.000000
The bilinear interpolation result f(x, y) at the target point.
Verification (alt path)⧉
0.000000
Same result computed by interpolating in the opposite order (y first, then x). Should match.
Normalized Coords (t, u)⧉
t = 0.5000, u = 0.5000
t = (x − x₁)/(x₂ − x₁), u = (y − y₁)/(y₂ − y₁). Both should be in [0, 1].
Value Range⧉
0.00 – 0.00
The interpolated value always falls within the range of the four corner values.
∂f/∂x ≈⧉
0.000000
Approximate partial derivative in the x-direction at the target point.
∂f/∂y ≈⧉
0.000000
Approximate partial derivative in the y-direction at the target point.
Corner Weights
Q(0,0)
25.0%
Q(1,0)
25.0%
Q(0,1)
25.0%
Q(1,1)
25.0%
Step-by-Step Breakdown
| Step | Operation | Formula | Value |
|---|---|---|---|
| 1 | Normalize x | t = (x − x₁) / (x₂ − x₁) | 0.500000 |
| 2 | Normalize y | u = (y − y₁) / (y₂ − y₁) | 0.500000 |
| 3 | Interp along y₁ | R₁ = Q₁₁·(1−t) + Q₂₁·t | 0.000000 |
| 4 | Interp along y₂ | R₂ = Q₁₂·(1−t) + Q₂₂·t | 0.000000 |
| 5 | Interp along y | f = R₁·(1−u) + R₂·u | 0.000000 |
Weight Matrix & Contribution
| Corner | Value | Weight | Contribution |
|---|---|---|---|
| (0,0) | 0.00 | 0.2500 | 0.000000 |
| (1,0) | 0.00 | 0.2500 | 0.000000 |
| (0,1) | 0.00 | 0.2500 | 0.000000 |
| (1,1) | 0.00 | 0.2500 | 0.000000 |
| Total (interpolated value) | 0.000000 | ||
Planning notes, formulas, and examples
About the Bilinear Interpolation Calculator
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 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.
When This Page Helps
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.
How to Use the Inputs
- Enter x₁ and y₁ in the input fields.
- Select the mode, method, or precision options that match your bilinear interpolation problem.
- Read Interpolated Value first, then use Verification (alt path) to confirm your setup is correct.
- Try a preset such as "Simple unit square" to test a known case quickly.
Formula used
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).Example Calculation
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.
Tips & Best Practices
- The interpolated value always lies within the range of the four corner values — it cannot extrapolate beyond them.
- If the target point sits exactly on a corner, the result equals that corner's value.
- The order of interpolation (x-first vs y-first) does not affect the result — bilinear interpolation is commutative.
- For higher accuracy on curved data, consider bicubic interpolation, which uses 16 surrounding points instead of 4.
- Use the partial derivative outputs to estimate how sensitive the interpolated value is to small shifts in x or y.
How This Bilinear Interpolation Calculator Works
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.
Interpreting Results
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.
Study Strategy
Sources & Methodology
Last updated:
Frequently Asked Questions
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.
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.
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.
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