Linear Interpolation Calculator
Estimate intermediate values between two known coordinate points (x₀, y₀) and (x₁, y₁) using the standard linear interpolation formula.
How to Use the Linear Interpolation Calculator
Linear interpolation is a fundamental numerical method used across engineering, mathematics, physics, and computer graphics to estimate an unknown value that falls between two known discrete data points. By assuming a constant rate of change (straight line) between the points, linear interpolation provides an intuitive, instantaneous approximation without requiring complex curve-fitting or polynomial algorithms.
Formula and Calculation Methodology
- (x₀, y₀): Coordinates of the first known reference point
- (x₁, y₁): Coordinates of the second known reference point
- x: Target independent variable value where y is estimated
- y: Interpolated dependent variable outcome
- m: Linear slope equal to (y₁ – y₀) / (x₁ – x₀)
When applying this formula, it is critical to confirm whether your target value falls inside the range of known points (interpolation) or outside of it (extrapolation). While interpolation within closely spaced points exhibits minimal error, extrapolating beyond known bounds assumes the physical process remains strictly linear, which can introduce compounding inaccuracy in non-linear systems such as temperature-dependent viscosity or non-linear asset pricing curves.
Common Linear Interpolation Applications & Typical Ranges
| Field | Input X | Output Y | Typical Accuracy |
|---|---|---|---|
| Steam & Thermodynamics | Pressure (kPa) | Enthalpy / Specific Volume | High (< 0.5% error) |
| Fluid Dynamics | Temperature (°C) | Dynamic Viscosity | Moderate (1 – 2% error) |
| Civil Engineering | Span Length (ft) | Beam Deflection (in) | High (< 0.2% error) |
| Finance & Yield Curves | Maturity (Years) | Bond Yield (%) | Good (Linear approximation) |
| Sensor Calibration | Voltage Output (mV) | Physical Reading (°C / psi) | Very High (Linearized) |
Frequently Asked Questions
What is the difference between linear interpolation and extrapolation?
Interpolation estimates values located within the bounds of two known data points (between x₀ and x₁). Extrapolation projects values beyond the known domain (less than x₀ or greater than x₁), carrying higher uncertainty.
Can x₀ and x₁ have the same value?
No. If x₀ equals x₁, the change in x is zero, resulting in division by zero and an undefined vertical line. The two known points must have distinct x coordinates.
When should I avoid using linear interpolation?
Linear interpolation should be avoided when the relationship between variables exhibits high curvature, exponential growth, or abrupt phase transitions (such as boiling water or structural buckling). In those cases, cubic spline or polynomial interpolation is preferred.