Linear Interpolation Calculator

What is Linear Interpolation (Lerp)?

Linear interpolation, often abbreviated as lerp, is a fundamental mathematical method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points.

If two known coordinates are given by \((x_1, y_1)\) and \((x_2, y_2)\), the linear interpolant is the straight line connecting these two endpoints. For any intermediate value \(x\) in the interval \([x_1, x_2]\), the corresponding estimated value \(y\) along the straight line is determined by:

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

Understanding the Interpolation Parameter \(t\)

In computer graphics, physics simulations, and game development, linear interpolation is frequently expressed in normalized parametric form:

$$\text{lerp}(y_1, y_2, t) = (1 - t)y_1 + t y_2 = y_1 + t(y_2 - y_1)$$

where \(t = \frac{x - x_1}{x_2 - x_1}\) represents the fractional distance from \(x_1\) to \(x_2\).

• When \(t = 0\), \(y = y_1\) (at the starting point).
• When \(t = 0.5\), \(y\) is the exact midpoint between \(y_1\) and \(y_2\).
• When \(t = 1\), \(y = y_2\) (at the end point).
• When \(0 \le t \le 1\), the process is interpolation.
• When \(t < 0\) or \(t > 1\), the estimation lies outside the original data range and is known as linear extrapolation.

Inverse Linear Interpolation

If you know the target value \(y\) and need to determine the corresponding parameter \(x\), you can rearrange the linear equation to solve for \(x\):

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

This inverse interpolation is particularly helpful when reading tabular calibration charts, engineering psychrometric data, and steam tables.