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.
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)