Interpolation

Date: 03/06/2003 at 16:26:08
From: Milt
Subject: Interpolation

I cannot remember the formula for interpolating between two numbers.
  
For example:

1    26.4
2     x
4    24.2

Thank you for your help.
Milt

Date: 03/07/2003 at 11:08:41
From: Doctor Douglas
Subject: Re: Interpolation

Hi, Milt,

You want a linear function that satisfies f(1) = 26.4, f(4) = 24.2,
and we then want to compute f(2).

You can start from the formula for a line: f(z) = mz + b, and solve 
for the constants m and b from the given data. That's one way to do it 
by brute force.

There is also a way to write down the function by constructing it
cleverly so that it goes through the data points when z takes on 
the appropriate values:

    f(z) = 26.4[(z-4)/(1-4)] + 24.2[(z-1)/(4-1)]

Now this is clearly a linear function, and we've built up the square
brackets from the given data such that when z=4, the first bracket
disappears, the second bracket is equal to 1, and f(4) = 24.2. 
When z=1, the first bracket is equal to 1 and the second bracket 
disappears, so f(1) = 26.4. You can "simplify" this expression by 
multiplying out the constants and combining things, if you wish.
And you can easily plug z=2 into this equation to find x = f(2).

I hope this helps.  Let me know if you'd like to talk about 
this some more, or if you have any other questions.

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/