📘 Introduction: Bilinear Interpolation Calculator

What is Bilinear Interpolation?

Bilinear interpolation is a mathematical method used to estimate values at a point within a rectangular area, based on the values at the four surrounding corner points. It is widely used in image processing, geographical mapping, and numerical simulations.

About This Calculator

This Bilinear Interpolation Calculator helps you quickly compute the interpolated value at any point (x, y) inside a rectangle, based on the known values at the four corner points:

• Q11 at (x1, y1)

• Q21 at (x2, y1)

• Q12 at (x1, y2)

• Q22 at (x2, y2)

Just input the coordinates and values, and the tool will instantly calculate the interpolated result. It also includes a visual 2D graph to help you understand how the interpolation is performed within the grid.

Features:

• Clean and responsive interface

• Grouped input fields for easy understanding

• Automatic interpolation calculation

• Ideal for students, engineers, and researchers

How to Use the Bilinear Interpolation Calculator

1. 🧭 Enter the Coordinates of the Rectangle

   • These define the corners of the grid (must form a rectangle):

     • X1 and X2 (horizontal boundaries)

     • Y1 and Y2 (vertical boundaries)
       👉 Make sure X1 ≠ X2 and Y1 ≠ Y2

2. 📊 Enter the Values at Each Corner

   • These are the known values (often from measurements or image pixels) at:

     • Q11 → Value at (X1, Y1)

     • Q21 → Value at (X2, Y1)

     • Q12 → Value at (X1, Y2)

     • Q22 → Value at (X2, Y2)

3. 📌 Enter the Target Point (x, y)

   • This is the point inside the rectangle where you want to estimate the value.

4. ▶️ Click “Calculate”

   • The tool will apply the bilinear interpolation formula and show:

     • The interpolated result

     • A 2D graph with your point and corners marked

📈 Example:

If you want to estimate a value inside a rectangle bounded by (0,0) and (10,10) with corner values:

Q11 = 5     Q21 = 20
Q12 = 15   Q22 = 30

Bilinear Interpolation Formula

Bilinear interpolation estimates the value of a point (x, y) inside a rectangle using the known values at the rectangle’s four corners.

 Given:

• Rectangle corners:

  • $(x_1, y_1)$(x1​,y1​) and

    $(x_2, y_2)$(x2​,y2​)

• Values at corners:

  • $Q_{11}$Q11​: at

    $(x_1, y_1)$(x1​,y1​)

  • $Q_{21}$Q21​: at

    $(x_2, y_1)$(x2​,y1​)

  • $Q_{12}$Q12​: at

    $(x_1, y_2)$(x1​,y2​)

  • $Q_{22}$Q22​: at

    $(x_2, y_2)$(x2​,y2​)

• Target point:

  $(x, y)$(x,y)

Formula:

$$f(x, y) = \frac{1}{(x_2 – x_1)(y_2 – y_1)} \cdot \Big[ Q_{11} (x_2 – x)(y_2 – y) + Q_{21} (x – x_1)(y_2 – y) + Q_{12} (x_2 – x)(y – y_1) + Q_{22} (x – x_1)(y – y_1) \Big]$$

Simplified Step-by-Step (Used in Calculator):

1. Interpolate in x direction (at

   $y = y_1$y=y1​ and

   $y = y_2$y=y2​):

$$f_{xy1} = \frac{(x_2 – x)}{(x_2 – x_1)} \cdot Q_{11} + \frac{(x – x_1)}{(x_2 – x_1)} \cdot Q_{21}$$

$$f_{xy2} = \frac{(x_2 – x)}{(x_2 – x_1)} \cdot Q_{12} + \frac{(x – x_1)}{(x_2 – x_1)} \cdot Q_{22}$$

2. Then interpolate in y direction:

$$f(x,y)=\frac{(y_2-y)}{(y_2-y_1)}\cdot f_{xy1}+\frac{(y-y_1)}{(y_2-y_1)}\cdot f_{xy2}$$

Note:

• This method assumes the point

  $(x, y)$(x,y) lies inside or near the rectangle.

• If

  $(x_1 = x_2)$(x1​=x2​) or

  $(y_1 = y_2)$(y1​=y2​), the formula breaks (division by zero).