Math to get relative scale from level stack

Question

Apologies for the horrible title. I spent 10 minutes trying to explain this in one sentence and failed.

Although the application prompting this question is in Java (And

Answer 1

Your "levels" can be defined by the following exponential function (with your levels starting at 1):

For simplicity, let's assume you have (scale) and and you're trying to find .

Seeing how is defined, you can see that

However, your scale is not given in neat powers of 2 like that, so you can find a with the following (rounding up as per your scenario):

Which means that and

Now that we have the new level, we have the following equation:

/edit
Yes, that's what 2^x-1 means.

• s_x is just some constant that goes with x (in this case it's your scale), likewise, s_y is some constant that goes with y.

• The weird square brackets with just the top bit (⌈ ⌉) are just the ceiling function (Math.ceil).

• Logarithms are the inverse of exponentiation. Logarithms are defined as the following: if a^c = b, then log_a(b) = c. For instance, log₂(8) = 3 because 2³ = 8. Is anything else confusing?

/java sample

int newLevel = oldLevel + (int) Math.ceil(Math.log(oldScale) / Math.log(2));
double newScale = oldScale * Math.pow(2, oldLevel - newLevel);

We have to use the change of base formula for logarithms because Math only provides loge and log10. It might be worthwhile to create a constant for Math.log(2) so you aren't constantly recalculating it.