Interpolation

Definition

Interpolation is a mathematical method of estimating unknown values that fall strictly within the range of a known set of data points.

Why It Matters

Interpolation is highly reliable because it is bounded on all sides by confirmed, observed data. It is a critical tool for “filling in the gaps” in empirical models, data fitting, computer graphics rendering, and statistics, providing safe estimates without introducing wild speculation.

Core Concepts

• Bounded Estimation: Unlike extrapolation, interpolation never projects into the unknown. The estimated value is always constrained by neighboring known values.
• Linear Interpolation (Lerp): The simplest form, which assumes a straight line between two adjacent data points to estimate values: $y = y_1 + \frac{x - x_1}{x_2 - x_1}(y_2 - y_1)$
  • How to read: “y equals y one plus the quantity x minus x one divided by the quantity x two minus x one, all times the quantity y two minus y one.”
  • Meaning: Find the proportional distance of $x$ between $x_1$ and $x_2$ and use it to find the corresponding value of $y$.
• Spline Interpolation: Using piecewise polynomials (like cubic splines) to construct smooth curves through all data points, avoiding the sharp changes of linear estimation.

Connected Concepts

• Extrapolation
• Empirical Models, Data Fitting
• Map-Territory Relation