Double Interpolation Calculator

Formula

The formula to calculate the interpolated value \( Z \) is:

\[ Z = \frac{Z_{11} (X_2 - X) (Y_2 - Y) + Z_{21} (X - X_1) (Y_2 - Y) + Z_{12} (X_2 - X) (Y - Y_1) + Z_{22} (X - X_1) (Y - Y_1)}{(X_2 - X_1) (Y_2 - Y_1)} \]

Where:

\( Z \) is the interpolated value
\( X \) and \( Y \) are the coordinates of the point to interpolate
\( X_1, X_2, Y_1, Y_2 \) are the coordinates of the known points
\( Z_{11}, Z_{12}, Z_{21}, Z_{22} \) are the values at the known points

What is Double Interpolation?

Double interpolation is a mathematical technique used to estimate values in a two-dimensional grid by interpolating between known values. It is particularly useful when the available data points are insufficient to determine a desired value directly. Imagine a scenario where you have a grid of data points with values given at specific coordinates.

Example Calculation

Let's assume the following values:

X: 2.5
Y: 2.5
X1: 2
X2: 3
Y1: 2
Y2: 3
Z at (X1, Y1): 4
Z at (X1, Y2): 6
Z at (X2, Y1): 5
Z at (X2, Y2): 7

Step 1: Calculate the interpolated value:

\[ Z = \frac{4 \times (3 - 2.5) \times (3 - 2.5) + 5 \times (2.5 - 2) \times (3 - 2.5) + 6 \times (3 - 2.5) \times (2.5 - 2) + 7 \times (2.5 - 2) \times (2.5 - 2)}{(3 - 2) \times (3 - 2)} = 5.5 \]