What is this equation?

Question

I ran across this equation for use in web code here and am desperately wanting to know if any portion of it or the whole thing is a standard equation somewhere. This is the best I could do mathematically. I'm sorry if the symbols aren't correct.

Answer 1

So let $f_i$ be the font size of the $i^{th}$ tag, $t_i$ be the number of of times the $i^{th}$ tag is used, and $t_{max}$ be the largest number of times a tag is used. $M$ will be the maximum font size you wish to use, and $m$ will be the minimum font size. We assume that there is some tag that has been used at least twice.

This gives a more clear equation (your use of $\theta$ for most of the variables makes it quite confusing):

$$f_i = \frac{\ln(t_i)}{\ln(t_{max})}(M-m) + m, \textrm{ $t_{i} > 1$}$$

Every tag should be at least the minimum font size, and no tag should be larger than the maximum font size. Thus, the first term, $\frac{\ln(t_i)}{\ln(t_{max})}(M-m)$ distributes the $M-m$ potential font size to the $i^{th}$ tag, based on its size.

This could be done with a linear scale as well:

$$f_i = \frac{t_i}{t_{max}}(M-m) + m$$

The problem with doing it in this way (though it might be easier to see why this one works) is that a website is quite likely to have several tags that are significantly more popular than others, particularly if it generates content on some sort of timeline within a reasonably specific area. A linear timeline could leave the less frequently used tags significantly smaller. A logarithmic scale will not suffer from this problem (think of geological timelines, for instance, which employ a logarithmic scale to better show the events).

Answer 2

It's a simple mapping function that takes a value of $\theta$ and maps it to a value between $\alpha$ and $\beta$, with $\theta_{max}$ corresponding to $\beta$.

The logarithm is used so that the output decreases more rapidly when $\theta$ is smaller compared to $\theta_{max}$. For instance, if $\theta_{max} = 10^4$, a theta of 1000 would give a value 3/4 along the scale from $\alpha$ to $\beta$, a theta of 100 would give 1/2, and 10 would give 1/4. The change in theta is decreasing, but the amount that the scaled value changes increases.

You could substitute any other type of function for the log (such as exponential functions, sine, quadratic) to change the way the original values are mapped to the output.