Absolute ratios I'm curious about the following idea:
suppose we have two values $P$ and $Q$, and the magnitude of the ratio $\frac{P}{Q}$ is between $0$ and $\infty$. If $P$ is smaller, then it's between $0$ and $1$.  If $Q$ is smaller, it's between $1$ and $\infty$ (but the ratio $\frac{P}{Q}$ is between $0$ and $1$).  
Is there a way to denote the "absolute ratio" (my term) that is always the ratio that is between $0$ and $1$ (either $\frac{P}{Q}$ or $\frac{Q}{P}$)?
As an example, the $\operatorname{absratio}(10,1) = \operatorname{absratio}(1,10) = 0.1.$
 A: Because we're not supposed to start sentences with mathematical symbols, I'm presenting the answer this way: $$\displaystyle \min \biggl(\left\{\frac{P}{Q} ,\frac{Q}{P}\right\}\biggr).$$
A: I frequently find myself using this function. The prettiest statement, to my taste, is: $$absratio(A,B) = exp\{{|log(A)-log(B)|\}}$$ 
A: Here are a couple of candidates:
$$
\frac12\left(\frac PQ+\frac QP-\left|\frac PQ-\frac QP\right|\right)
$$
and
$$
\frac{|P+Q|-|P-Q|}{|P+Q|+|P-Q|}
$$
A: Cool!  I'll build on Andrei B.'s and Mark Melville's comment (because otherwise I couldn't make nice notation), 
I'm trying to use this to judge the similarity of aspect ratios of images regardless of scale - so 640:480 and 800:600 have an aspect ratio of 1.333 and the inverse orientation is .75.  I'm thinking that these aspect ratios are not good for fair comparisons.
I like this function:
$$absratio(a, b) = \begin{cases} 
      e^{(-|log(a)-log(b)|)}-1 & b \lt a \\
      1-e^{(-|log(a)-log(b)|)} & a \leq b \\
\end{cases}$$
which gives -0.11745 for 640:480, 800:600 and gives 0.11745 for 480:640, 600:800.  $absratio_1 - absratio_2$ close to zero means that the ratios are close.  So it's still absolute in the sense that it is symmetrical in an important way around 1:1, but it differentiates between 2:1 vs 1:2 as negative and positive.  And if we really want only positive values, we can use regular old $abs()$ on that.
But is there a continuous representation of that piece-wise surface??  I would rather use that formula but I don't know how to do it.
Also, I don't know if I like all the properties of this function .. from 100:100 (or even 1:1) to 100:110 represents is an increase in area(?) of 10%.  From 300:100 to 300:110 is also an increase in area of 10% but this function doesn't grow consistently for both.  Could it?
