Linear Interpolation Calculator | Find Points on a Line

What is Linear Interpolation?

Linear interpolation is a method of estimating a value between two known data points. It assumes a straight line between these points and finds values along this line. This technique is widely used in various fields, including mathematics, engineering, and computer graphics.

How to Calculate Linear Interpolation

To perform linear interpolation, we use the equation of a straight line passing through two points. This allows us to estimate values between these points or even extend beyond them (extrapolation).

Formula

The formula for linear interpolation is:

\[ y = y_1 + \frac{(x - x_1)(y_2 - y_1)}{(x_2 - x_1)} \]

Where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the known points
• \((x, y)\) is the point we're trying to find
• \(x\) is the known x-value for which we want to find the corresponding y-value

Calculation Steps

1. Identify the two known points \((x_1, y_1)\) and \((x_2, y_2)\)
2. Determine the x-value for which you want to find the y-value
3. Plug these values into the formula
4. Solve for y

Example and Visual Representation

Let's interpolate between the points (2, 4) and (6, 10) to find y when x = 4.

Using our formula:

\[ y = 4 + \frac{(4 - 2)(10 - 4)}{(6 - 2)} = 4 + \frac{2 \cdot 6}{4} = 4 + 3 = 7 \]

So, when x = 4, y = 7

Here's a visual representation of this linear interpolation:

In this graph, the blue dots represent our known points (2, 4) and (6, 10). The green line shows the linear interpolation between these points. The red dot at (4, 7) is our interpolated point. As you can see, it falls exactly on the line between our two known points, demonstrating the principle of linear interpolation.