Interpolation Reference Page

Author

DOFPro group

  • Video links go directly to the YouTube video.
  • JTF (Just the Facts) videos are the streamlined versions: greenscreen LaTeX equations, clean graphics, minimal narrative. Think efficient and to the point. Each has a companion TFS video.
  • TFS (The Full Story) videos include interviews, additional explanation, and equations written on whiteboards. Same math, more context, more personality. Each has a matching JTF version.
  • Info Page links lead to definitions, expanded explanations, and related material—because sometimes you really do need to explain it.
  • Visuals links contain the greenscreen or whiteboard materials used in the video, for those who like to see the scaffolding.
  • Wondering about the titles? See Appendix B: If you have to explain it, it’s no longer funny.
  • Videos marked This is NOT a DOFPro video were not produced by DOFPro but are included because they are relevant to the topic. They are shown in red so no one calls the academic integrity police.

Purpose of the Interpolation Videos

Before computers became widely available, most engineering data were published as tabulations in handbooks, textbooks, and journal articles. While computer-based calculations are now common, tabulated data are still widely used in engineering.

The videos Interpol Is Late to a Mind-Blowing Math Hack and No Clear Interpretation a Mind-Blowing Math Hack explain the principles of linear interpolation and nonlinear interpolation, respectively, and demonstrate how to estimate values between tabulated data points.

Interpol Is Late to a Mind-Blowing Math Hack

This video introduces both one-dimensional (1-D) and two-dimensional (2-D) linear interpolation and demonstrates examples of each.

Just the Facts

The Just the Facts video explains how to perform both 1-D linear interpolation and 2-D (double) linear interpolation.

Visuals

The Full Story

The Full Story video discusses both the principles of linear interpolation and common misconceptions about its use. It explains how to perform both 1-D interpolation and 2-D (double) interpolation, with additional conceptual discussion and worked examples.

Visuals

No Clear Interpretation a Mind-Blowing Math Hack

This video introduces nonlinear interpolation and demonstrates how a change of variables can transform nonlinear data into a form that can be interpolated more accurately.

Visuals

Examples and Definitions

Definitions

Interpolation
The process of estimating the value of a function between known tabulated data points.
Linear Interpolation
Estimating an intermediate value between two data points by assuming the relationship between them is linear.

For two tabulated points ((x_1,y_1)) and ((x_2,y_2)), the interpolated value at (x) is

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

Nonlinear Interpolation
Estimating intermediate values when the relationship between variables is nonlinear. This often involves fitting a nonlinear function or transforming variables before applying interpolation.
2-D Interpolation
Interpolation performed when the data depend on two independent variables. Values are estimated using surrounding grid points in both coordinate directions.
Steam Table
A tabulation of thermodynamic properties of water and steam, typically arranged by temperature or pressure for saturated states and by both temperature and pressure for superheated states.
Antoine Equation
An empirical equation used to estimate vapor pressure as a function of temperature:

\[ \log_{10} P = A - \frac{B}{C + T} \]

where (A), (B), and (C) are experimentally determined constants.

Clausius-Clapeyron Equation
A thermodynamic relation describing how vapor pressure varies with temperature during phase change:

\[ \frac{dP}{dT} = \frac{\Delta H_{vap}}{T\Delta V} \]