P.Mean: Exponential interpolation (created 2010-02-11).

Someone wanted an exponential interpolation formula. It's not quite a statistics question, but it caught my interest. Any interpolation starts with a range for x and for y. Let x be between a and b. Let y be between c and d. In a linear interpolation, an x-value halfway between a and b produces a y value halfway between c and d. The formula for any x would be

y = (x-a) / (b-a) * (d-c) + c

The exponential interpolation assumes a multiplicative relationship throughout the range. If x is halfway between a and b, then y has to be a factor of the square root of d/c between c and d. The formula  for any x would be

y = c * (d/c)^((x-a) / (b-a))

Note that both terms have a factor

(x-a) / (b-a)

which shows the proportion of the distance from a to b. In an exponential model, you use this proportion as an exponent. Here's a plot with circles showing the exponential interpolation and X's showing the linear interpolation.

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