Interpolation with constant rate of increase

Question

Let's assume we have data for two points ( X_Left -> Y_Left, X_Right -> Y_Right ), and we break the interval between X_Left, and X_Right, into equaly distant sub intervals.

In case I wanted to do a linear interpolation, for either of the equaly distant points, between X_Left and X_Right, I could simply use the following equation,

Y = Y_Left + ( X - X_Left ) . ( Y_Right - Y_Left )/( X_Right - X_Left )

Where X was the point I wanted to find (linearly interpolate) the value of.

Now, let's say that instead of using the assumption of similar increments beteween X_Left, and X_Right, we want to assume that between each consecutive interval their rate of increase will be constant.

So, for example, in case X_Left = 0, Y_Left = 1, X_Right = 3, Y_Right = 8, we would have 3 equaly distant intervals. And the following condition would need to be satisfied,

( Y1 - Y_Left )/Y_Left = ( Y2 - Y1 )/Y1 = ( Y_Right - Y2 )/Y2

What would be the mathematical equation (equivalent to the one I shared for the linear interpolation), in this scenario of constant increase\decrease rate?

Answer 1

To interpolate from $y = A$ at $x = a$ to $y= B$ at $x=b$ in $n$ steps the overall change (expressed multiplicatively) is $R = B/A$ so at each step you have to multiply the current value by $r = R^{1/n}$, the $n$th root of $R$.

That means that at the $i$th step, when $x = a + i(b-a)/n$ you have $$ y = Ar^i . $$

It's easy to see then that when $i=0$ you have $y = A$, when $i=n$ you have $y = B$, and that at each step the ratio of successive values is $r$,

$R$ is the equivalent of the slope of the line joining the points that you use for linear interpolation.

This will work only when $A$ and $B$ have the same sign (usually positive).