Linear Interpolation Formula
y = y₁ + (x - x₁) × (y₂ - y₁) / (x₂ - x₁). This is the slope-intercept form of the line through the two known points, evaluated at x. If x falls between x₁ and x₂ it is interpolation; if x is outside that range it is extrapolation, which is shown with a warning.
Frequently Asked Questions
Linear interpolation estimates a value between two known data points by assuming a straight-line relationship. Given points (x1, y1) and (x2, y2), the interpolated value at x is: y = y1 + (x - x1) x (y2 - y1) / (x2 - x1). It is one of the most common techniques in engineering, mathematics, and data analysis.
Interpolation estimates values within the range of known data points (between x1 and x2). Extrapolation estimates values outside that range (beyond x1 or x2). Extrapolation is less reliable because it assumes the linear trend continues beyond the observed data.
Engineers use interpolation constantly: reading values from steam tables, material property tables, and performance curves at conditions not exactly listed. Examples include finding efficiency at an intermediate load point, interpolating stress values from a fatigue curve, or reading a thermocouple table between listed temperatures.
Bilinear interpolation extends linear interpolation to 2D tables with two variables (e.g., efficiency as a function of both speed and load). It performs linear interpolation first in one direction, then in the other. This calculator performs 1D linear interpolation; bilinear interpolation requires four points.
Accuracy depends on how well the data follows a linear trend between the two points. For slowly-changing smooth functions, linear interpolation is very accurate. For rapidly changing or highly curved data, other methods (quadratic, cubic spline) are more accurate. Always check that your function is reasonably linear in the interpolation interval.
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