Linear zoom function 2D

Question

I have a canvas and a scale value. The maximum scale is 1, the minimum scale value is something like 0.1 or above. Let’s say we have discrete time units. I’m looking for a function that zooms linear over an time interval I (lets say 100 time units), from a start zoom s to an end zoom e. Let 0 >= i < I be the current interval.

Example: Zoom from 0.2 to 1.0 in 100 time units.

Obviously zoom(i) = (e-s)/I * i does not produce a linear zoom. Because a step from 0.2 to 0.4 doubles the zoom, while the same amount from 0.8 to 1.0 only increases the zoom by 25%.

I was thinking that this function needs something logarithmic to base 2, but I’m stuck finding the right function.

Answer 1

To provide constant ratio with constant argument difference, you need exponential function (it is possible to use any base, e, 2, 10 and so on with corresponding logarithms)

F(x) = A * Exp(B * x)

To get coefficients A and B for given border conditions (argument x0 corresponds to function value F0):

F0 = A * Exp(B * x0)
F1 = A * Exp(B * x1)

divide the second equation by the first:

Exp(B * (x1 -x0) = F1 / F0
B * (x1 -x0) = ln(F1 / F0)

so

B = ln(F1 / F0) / (x1 - x0)

and

A = F0 * Exp(-B * x0)

For your example

x0=0, x1=100
zoom0 = 0.2, zoom1=1
B = ln(5) / 100 = 0.0161
A = 0.2 * Exp(0) = 0.2
zoom(i) = 0.2 * Exp(0.0161 * i)

zoom(0) = 0.2
zoom(50) = 0.447
zoom(100) = 1

note that 
zoom(50) / zoom(0) = zoom(100) / zoom(50)