Understanding Interpolation Methods: Simple Explanation

Let me explain the three interpolation methods in simple terms, without complex formulas:

Linear Interpolation

What it is: The simplest method that connects two data points with a straight line.

How it works:

• Imagine you have two known points on a graph
• To find a value between these points, just draw a straight line connecting them
• The interpolated value is the height of that line at your desired x-position

Example:
If you know that at 9:00 AM the temperature was 50°F and at 12:00 PM it was 68°F, and you want to estimate the temperature at 10:00 AM:

• 10:00 AM is 1/3 of the way between 9:00 AM and 12:00 PM
• So you’d estimate the temperature rose by 1/3 of the total increase
• That’s 1/3 of (68°F – 50°F) = 1/3 of 18°F = 6°F
• Add this to the starting temperature: 50°F + 6°F = 56°F
• The estimated temperature at 10:00 AM is 56°F

When to use it:

• When you need a quick, simple estimation
• When your data points are close together
• When the relationship between data points is approximately linear

Polynomial Interpolation

What it is: A method that creates a smooth curve passing through all your data points.

How it works:

• Instead of using straight lines, it creates a curved line (a polynomial)
• This single curve is shaped to pass exactly through all your data points
• The more points you have, the more complex the curve becomes

Example:
If you have yearly population data for a city (2000: 25,000 people, 2005: 27,500 people, 2010: 32,000 people), and want to estimate the population in 2007:

• Polynomial interpolation creates a curved line through all three points
• This accounts for the accelerating growth rate between 2005-2010
• The estimated 2007 population might be around 29,800 people (not just a straight-line estimate)

When to use it:

• When you have relatively few data points
• When you expect the relationship to follow a smooth curve
• When all your data points are reliable (no outliers)

Caution: This method can create wild, unrealistic curves when you have many points or outliers in your data.

Cubic Spline Interpolation

What it is: A method that creates a series of connected curves, each between adjacent data points.

How it works:

• Instead of one big curve, it creates separate curves between each pair of adjacent points
• These curves are carefully designed to connect smoothly at the data points
• The connections are so smooth that you can’t see where one curve ends and the next begins

Example:
If you’re tracking a runner’s position on a track at different times, cubic spline helps you visualize their smooth path:

• Between each pair of known positions, it creates a curve that represents natural movement
• The curves join perfectly at each known position
• When the runner changes direction or speed, the transition appears natural
• This gives you a realistic estimation of where they were at any moment during the run

When to use it:

• When you need a very smooth, natural-looking curve
• When you have many data points
• When the relationship between points changes throughout your data range
• When you want to avoid the wild swings that can happen with polynomial interpolation

How to Choose the Right Method

1. Use Linear Interpolation when:

• You need something quick and simple
• Your data points are close together
• The relationship appears to be roughly straight between points

1. Use Polynomial Interpolation when:

• You have just a few data points (2-5)
• You need a single smooth function for the entire range
• All your data points are accurate and important

1. Use Cubic Spline Interpolation when:

• You have many data points
• You need a smooth, natural-looking curve
• You want to avoid oscillations and unrealistic behavior
• The relationship between variables changes throughout your data

What’s Happening in Our Calculator

The calculator takes your data points and:

1. For linear interpolation: draws straight lines between adjacent points
2. For polynomial interpolation: creates one big curve that passes through all points
3. For cubic spline interpolation: creates a series of connected smooth curves

When you enter a specific x-value, the calculator finds the y-value on the appropriate curve and displays the result. The visualization helps you see how the different methods produce different curves and interpolated values.