đ Bilinear Interpolation Calculator
Interpolate values between four corner points
Grid Corners
Corner Values f(x, y)
Interpolation Point
How to Use This Calculator
Define the Grid Rectangle
Enter the corner coordinates: xâ, xâ (left/right) and yâ, yâ (bottom/top) that define your rectangular grid.
Enter Corner Values
Input the function values at the four corners: f(xâ,yâ), f(xâ,yâ), f(xâ,yâ), and f(xâ,yâ).
Enter Interpolation Point
Specify the point (x, y) where you want to interpolate. Make sure xâ ⤠x ⤠xâ and yâ ⤠y ⤠yâ.
Calculate
Click "Calculate Interpolation" to get the interpolated value at your point.
Formula
f(x, y) = (1-w_x)(1-w_y)f(xâ,yâ) + w_x(1-w_y)f(xâ,yâ) + (1-w_x)w_y f(xâ,yâ) + w_x w_y f(xâ,yâ)
where w_x = (x - xâ)/(xâ - xâ), w_y = (y - yâ)/(yâ - yâ)
Alternative Method (two-step):
- Interpolate along x: fâ = f(xâ,yâ) + [f(xâ,yâ) - f(xâ,yâ)]w_x
- Interpolate along x: fâ = f(xâ,yâ) + [f(xâ,yâ) - f(xâ,yâ)]w_x
- Interpolate along y: f(x,y) = fâ + (fâ - fâ)w_y
Example: Interpolate at (0.5, 0.5) with corners (0,0), (1,0), (0,1), (1,1) having values 0, 1, 2, 3 respectively.
w_x = (0.5-0)/(1-0) = 0.5, w_y = (0.5-0)/(1-0) = 0.5
f(0.5, 0.5) = 0.25(0) + 0.25(1) + 0.25(2) + 0.25(3) = 1.5
About Bilinear Interpolation Calculator
The Bilinear Interpolation Calculator estimates values at points within a rectangular grid using the known values at the four corners. It performs linear interpolation in two dimensions, first along x-axis, then along y-axis (or vice versa).
When to Use This Calculator
- Image Processing: Resize or scale images using bilinear interpolation
- Digital Elevation Models: Estimate terrain height at arbitrary points
- Temperature Maps: Interpolate temperature values from weather station data
- Computer Graphics: Texture mapping and UV mapping in 3D graphics
- Data Analysis: Fill gaps in 2D data grids
- Numerical Methods: Interpolate function values in 2D space
Why Use Our Calculator?
- â Two-Step Process: See how interpolation works in stages
- â Visual Weights: Understand the weighting factors w_x and w_y
- â Works with All Numbers: Handles decimals, fractions, and negatives
- â 100% Accurate: Precise mathematical calculations
- â Educational: Helps understand 2D interpolation concepts
- â Completely Free: No registration required
Understanding Bilinear Interpolation
Bilinear interpolation extends linear interpolation to two dimensions. It combines four corner values using weighted averages, where weights depend on the position of the interpolation point relative to the corners.
- Works on rectangular grids (xâ to xâ, yâ to yâ)
- Requires values at all four corners
- Produces smooth transitions between corner values
- Weights are normalized (0 to 1) based on position
- At corners: returns exact corner value
- At edges: reduces to linear interpolation
Real-World Applications
Image Scaling: When resizing an image, new pixel values are calculated using bilinear interpolation from nearby original pixels, producing smoother results than nearest-neighbor interpolation.
Terrain Mapping: Digital elevation models use bilinear interpolation to estimate ground height at any GPS coordinate based on elevation data at grid points.
Scientific Data: In climate modeling, bilinear interpolation estimates temperature or pressure at locations between weather stations using data from surrounding stations.
Frequently Asked Questions
What is bilinear interpolation?
Bilinear interpolation estimates values at points within a rectangle using known values at the four corners. It performs linear interpolation in two dimensions, creating a smooth surface.
How does bilinear interpolation work?
It uses weighted averages of the four corner values. The weights depend on the interpolation point's position: closer corners have higher influence. The calculation can be done in two steps (interpolate x first, then y) or all at once.
What if my point is outside the grid?
Bilinear interpolation requires the point to be within the rectangle (xâ ⤠x ⤠xâ, yâ ⤠y ⤠yâ). For extrapolation (outside the grid), use other methods like bilinear extrapolation or higher-order interpolation.
What's the difference from linear interpolation?
Linear interpolation works in 1D (between two points). Bilinear interpolation works in 2D (within a rectangle using four corner points). It's the 2D extension of linear interpolation.
Can I use this for 3D interpolation?
This calculator is for 2D (bilinear). For 3D interpolation, you'd need trilinear interpolation, which uses 8 corner values of a cube. That requires a different calculator.
Is bilinear interpolation accurate?
Bilinear interpolation provides smooth, continuous results suitable for most applications. Accuracy depends on how well the underlying function can be approximated linearly. For highly nonlinear functions, consider higher-order methods like bicubic interpolation.
How are the weights calculated?
Weights w_x and w_y are calculated based on the interpolation point's position: w_x = (x - xâ)/(xâ - xâ) and w_y = (y - yâ)/(yâ - yâ). These weights range from 0 to 1, indicating how close the point is to each corner.