Content

• To step
• Tips

Linear interpolation, also referred to simply as interpolation or "lerping", is the ability to derive a value between two values ​​explicitly stated in a table or graph. While many people can interpolate intuitively, the article below shows the formalized mathematical approach behind intuition.

To step

1. Identify the value for which you want to find a corresponding value. Interpolation can be used for something like finding a logarithm or the value of a trigonometric function, or for the corresponding gas pressure or volume at a given temperature in chemistry. Because scientific calculators have largely replaced logarithmic and trigonometric tables, we use as an example to determine an interpolated value, determining the pressure of a gas at a temperature not listed in the reference table, or as a point on a graph.
   • For the equation we will derive, we represent the value for which we want to find a corresponding value as X and the interpolated value we want to find as y. We use these labels because in a chart the values ​​we know are plotted on the horizontal or x axis, and the value we are trying to find on the vertical or y axis.
   • Our Xvalue becomes the temperature of the gas (37C in this example).
2. Find the values ​​closest below and above the value of x in the table or on the graph. Our reference table does not give gas pressure for 37C, but it does for 30C and 40C. The gas pressure at 30C is 3 kilopascals (kPa) and the pressure at 40C is 5 kPa.
   • As we indicated with 37C X, we will indicate a temperature of 30 degrees with X₁ and 40 degrees as X₂.
   • Because we indicate the pressure we are trying to find with y, we denote a pressure of 3 kPa at 30C with y₁ and a pressure of 5 kPa at 40C with y₂.
3. Determine the interpolated value mathematically. The equation for finding the interpolated value can be written as y = y₁ + ((X - x₁)/(X₂ - x₁) * (Y₂ - y₁))
   • Entering the values ​​for x, x₁ and x_/2 for the variables, returns (37 - 30) / (40 -30), simplifies to 7/10 or 0.7.
   • Entering the values ​​for y₁ and y₂ at the end of the equation gives (5 - 3) or 2.
   • Multiplying 0.7 by 2 gives the product 1.4. Tel 1.4 on at y₁ (or 3), gives a value of 4.4 kPa. After comparing this result with our original values, we see that 4.4 is between 3 kPa at 30C and 5 kPa at 40C, and since 37 is closer to 40 than 30, the result should be closer to 5 kPa than at 3 kPa.

Tips

• If you are good at estimating distances on graphs, you can do a rough interpolation by reading the position of a point on the x axis and finding the corresponding y value. If the above example were graphed with the x-axis divided in units of 10C and the y-axis in units of 1 kPa, you could find the approximate position of 37C and then on the y-axis a search for landmark not quite half way between 4 and 5 kPa. The above equation formalizes the thinking process and gives a more exact value.
• Related to interpolation is extrapolation, where you look for a matching value for a given value outside the range of values ​​in a table, or as shown in a graph.