What’s the most natural/useful way of ranking Pythagorean triples “by size”? I want to “rank” [primitive] Pythagorean triples by some metric that could reasonably be referred to as “size”.
Naturally, there are a huge number of options: size of hypotenuse, size of smallest leg, perimeter, area, radius of incircle, etc. etc. etc. (Note: One thing I don’t want to use is the row index from the triple’s position in one of the infinite ternary trees.)
Is there a widely-accepted “sizing” of triples? What are the pros and cons of various metrics?
EDIT (inspired by Gerry Myerson’s comment): The “Holy Grail” in this investigation would be a strictly “linear” ordering of the Pythagorean triples. Does (or can) such a thing exist?
EDIT #2: Let $(p,q)$ be the Euclid generating pair for a primitive Pythagorean triple. Applying the Cantor pairing function with $p-q$ and $q$ gives a unique integer value for each triple, which generally correlates with “size”; and I have yet to find anything more compact (e.g., in a set of $38$ of the “smallest” triples, the area-divided-by-6 range is $1–1820$, while the Cantor function for the same set has a range of $4–106$). What’s clearly missing is any obvious way to “descend” through this ordered set.
EDIT #3: The inradius is the most obvious “size” ranking, since the only gaps in the sequence are the powers of $2$. The issue here is the fact that inradius isn’t unique.
 A: Concerning the so-called sign issue with the ternary tree:
The Wikipedia essay cited elsewhere on this page gives the three matrices $$A=\pmatrix{1&-2&2\cr2&-1&2\cr2&-2&3\cr},\qquad B=\pmatrix{1&2&2\cr2&1&2\cr2&2&3\cr},\qquad C=\pmatrix{-1&2&2\cr-2&1&2\cr-2&2&3\cr}$$ with the properties that 1) if $v$ is a (positive) primitive pythagorean triple then so are $Av$, $Bv$, and $Cv$, 2) every primitive pythagorean triple can be obtained from $v=(3,4,5)$ in exactly one way by multiplying by a (finite) sequence of matrices, each matrix in the sequence being $A$ or $B$ or $C$.
We calculate the inverses, $$A^{-1}=\pmatrix{1&2&-2\cr-2&-1&2\cr-2&-2&3\cr},\qquad B^{-1}=\pmatrix{1&2&-2\cr2&1&-2\cr-2&-2&3\cr},\qquad C^{-1}=\pmatrix{-1&-2&2\cr2&1&-2\cr-2&-2&3\cr}$$ It follows that given any (positive) primitive pythagorean triple $w$, exactly one of the three vectors $A^{-1}w,B^{-1}w,C^{-1}w$ is a positive pythagorean triple.
For example, for $w=(165,52,173)$, we get $A^{-1}w=(-77,-36,85)$, $B^{-1}w=(-77,36,85)$, and $C^{-1}w=(77,36,85)$, so $C^{-1}w$ is the direct ancestor of $w$ on the tree.
