Bilinear Interpolation Calculator โ 2D Grid Value Estimation
Interpolate f(x,y) from four corner values of a rectangle. Weighted average of corner values based on distance โ used in image resizing, terrain mapping, and scientific data.
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Weights sum to 1 โ convex combination of corner values Linear in each direction โ bilinear = linear in x and y Used in image resizing (e.g., texture sampling)
Ready to run the numbers?
Why: Bilinear interpolation estimates values between known grid points โ essential for image processing, terrain mapping, data analysis, texture sampling, and any 2D spatial interpolation.
How: Each corner contributes a weight proportional to the area of the opposite sub-rectangle. f(x,y) = [f11(x2-x)(y2-y) + f21(x-x1)(y2-y) + f12(x2-x)(y-y1) + f22(x-x1)(y-y1)] / D.
Run the calculator when you are ready.
Bilinear Interpolation
Enter rectangle corners (x1,y1)-(x2,y2), four corner values f(Q11), f(Q12), f(Q21), f(Q22), and interpolation point (x,y).
Quick Examples โ Click to Load
Inputs
Rectangle (x1,y1) to (x2,y2)
Corner values f(Q)
Interpolation point
Corner Values
Weight Distribution
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
๐งฎ Fascinating Math Facts
Image resizing uses bilinear interpolation for smooth scaling
โ Computer Graphics
Terrain elevation data is interpolated with bilinear for smooth surfaces
โ GIS
Key Takeaways
- โข Bilinear interpolation estimates f(x,y) from four corner values of a rectangle
- โข Formula: f(x,y) = [f(Q11)(x2-x)(y2-y) + f(Q21)(x-x1)(y2-y) + f(Q12)(x2-x)(y-y1) + f(Q22)(x-x1)(y-y1)] / [(x2-x1)(y2-y1)]
- โข Weights sum to 1; each weight is the area fraction of the opposite sub-rectangle
- โข Used in image resizing, GIS, texture mapping, scientific data
Did You Know?
How It Works
Given a rectangle with corners (x1,y1), (x2,y1), (x1,y2), (x2,y2) and values f11, f21, f12, f22 at those corners, bilinear interpolation computes f(x,y) as a weighted average. Each weight is proportional to the area of the rectangle opposite that corner when (x,y) divides the cell.
Formula
f(x,y) = f(Q11)ยท(x2-x)(y2-y) + f(Q21)ยท(x-x1)(y2-y) + f(Q12)ยท(x2-x)(y-y1) + f(Q22)ยท(x-x1)(y-y1), all divided by (x2-x1)(y2-y1)
Expert Tips
Corner convention
Q11=(x1,y1), Q21=(x2,y1), Q12=(x1,y2), Q22=(x2,y2). Ensure x1โ x2 and y1โ y2.
Bounds check
(x,y) must lie inside the rectangle. Extrapolation is not supported.
Applications
Image resizing, texture mapping, elevation data, temperature maps, numerical PDEs.
Alternatives
Bicubic and spline interpolation give smoother results for higher-quality resizing.
Interpolation Methods Comparison
| Method | Smoothness | Speed | Use Case |
|---|---|---|---|
| Nearest neighbor | Step | Fast | Pixel art |
| Bilinear | C1 continuous | Medium | General purpose |
| Bicubic | C2 continuous | Slower | Photo resizing |
| Spline | Very smooth | Slow | Scientific data |
FAQ
What if (x,y) is outside the rectangle?
Bilinear interpolation is defined only inside the rectangle. For extrapolation, use other methods or extend the grid.
Why do weights sum to 1?
The weights are normalized area fractions. The four sub-rectangles partition the cell, so their areas sum to the total cell area.
When is bilinear interpolation appropriate?
When you have a regular grid of values and need smooth estimates between grid points. Common in images and spatial data.
How does this relate to linear interpolation?
Bilinear is the 2D extension: interpolate in x, then in y (or vice versa). The result is the same.
Can I use this for 3D data?
For 3D you would use trilinear interpolation with 8 corner values of a cube.
Infographic Stats
Disclaimer: This calculator is for educational purposes. For production image processing or scientific applications, verify results against established libraries (e.g., OpenCV, SciPy).
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