Bilinear Interpolation Calculator
Estimate function values using 2D bilinear interpolation on rectangular grids
Calculate Bilinear Interpolation
Rectangle Corner Coordinates
Rectangle corners: (x₁,y₁), (x₁,y₂), (x₂,y₁), (x₂,y₂)
Function Values at Corner Points
Known function values at the four corner points
Interpolation Point
Point (x,y) where you want to estimate the function value
Interpolation Results
Detailed Calculation
Formula: P = w₁₁×Q₁₁ + w₂₁×Q₂₁ + w₁₂×Q₁₂ + w₂₂×Q₂₂
Area of rectangle: (x₂-x₁)×(y₂-y₁) = 8
Weights:
- w₁₁ = (x₂-x)×(y₂-y) / area = 0.3750
- w₂₁ = (x-x₁)×(y₂-y) / area = 0.1250
- w₁₂ = (x₂-x)×(y-y₁) / area = 0.3750
- w₂₂ = (x-x₁)×(y-y₁) / area = 0.1250
Weighted terms:
- 0.3750 × 12 = 4.5000
- 0.1250 × 0 = 0.0000
- 0.3750 × -4 = -1.5000
- 0.1250 × 8 = 1.0000
Result: P = 4.5000 + 0.0000 + -1.5000 + 1.0000 = 4.000000
Step-by-Step Linear Interpolations
Step 1: Linear interpolation at y = y₁ = 1
R₁ = f(1, 1) = 9.000000
Step 2: Linear interpolation at y = y₂ = 3
R₂ = f(1, 3) = -1.000000
Step 3: Final interpolation between R₁ and R₂
P = f(1, 2) = 4.000000
Example Calculation
Temperature Distribution Example
Given: Temperature readings at four corners of a room
Corner (0,1): 12°C
Corner (0,3): -4°C
Corner (4,1): 0°C
Corner (4,3): 8°C
Find: Temperature at point (1,2)
Solution
Area = (4-0) × (3-1) = 8
w₁₁ = (4-1)×(3-2)/8 = 3/8
w₂₁ = (1-0)×(3-2)/8 = 1/8
w₁₂ = (4-1)×(2-1)/8 = 3/8
w₂₂ = (1-0)×(2-1)/8 = 1/8
P = (3/8)×12 + (1/8)×0 + (3/8)×(-4) + (1/8)×8
P = 4.5 + 0 - 1.5 + 1 = 4°C
Formula Reference
Bilinear Interpolation
P = w₁₁Q₁₁ + w₂₁Q₂₁ + w₁₂Q₁₂ + w₂₂Q₂₂
Weights:
w₁₁ = (x₂-x)(y₂-y)/A
w₂₁ = (x-x₁)(y₂-y)/A
w₁₂ = (x₂-x)(y-y₁)/A
w₂₂ = (x-x₁)(y-y₁)/A
A: Rectangle area = (x₂-x₁)(y₂-y₁)
Requirements
• x₁ < x₂ and y₁ < y₂
• x₁ ≤ x ≤ x₂ and y₁ ≤ y ≤ y₂
• Four function values at corners
Key Concepts
Used for 2D interpolation on rectangular grids
Weighted average based on distance to corners
Linear in each direction, quadratic overall
Common in image processing and computer graphics
Understanding Bilinear Interpolation
What is Bilinear Interpolation?
Bilinear interpolation is a method for estimating function values at any point within a rectangle, given the function values at the four corners. It performs linear interpolation first in one direction, then in the perpendicular direction.
How It Works
- Perform linear interpolation along the bottom edge (y₁)
- Perform linear interpolation along the top edge (y₂)
- Interpolate between these two results vertically
Applications
- •Image processing: Resizing and rotating images
- •Computer graphics: Texture mapping and rendering
- •Numerical analysis: Solving PDEs on grids
- •Geographic mapping: Interpolating elevation data
Key Properties
- • Exact at corner points
- • Linear along edges
- • Continuous across boundaries
- • Computationally efficient