Generalized Height Functions

While messing around with some height functions (mainly Weil height and Benedetti height), I noticed a few cool properties but didn't really think too much about their implications. Then Mike informed me that a month prior he had already formed an equation to represent the patterns we both noticed. Here is the paramaterized height Mike came up with:

F(k) = (log(n*d) + k*|log(n/d)|)

Now to simplify, lets require that n>d, and lets also say that G(k) = 2^(F(k)).

G(k) = n*d*(n/d)^k

Notice what happens when you put in various values for k:

G(-infinity) = d/n
G(-1) = min(n,d)^2
G(0) = n*d
G(1) = max(n,d)^2
G(infinity) = n/d

The last three are Benedetti/Tenney height, Weil height, and interval span
respectively. As long as k is in the open interval (-1,infinity), this function
is a valid height function.

Ryan