Moduli of polygons
moduli of triangles
slickest:
order edges $a \geq b \geq c$,
and set $a = 1$
Then the triangle inequality say
$b - c\leq a \leq b+c$
Hence $1/2 \leq b \leq 1$
and $1 - b \leq c \leq b$.
These are the only constraints
draw the triangle!
1,1 corresponds to equilateral
if you classify as $(a,b,c)$ w/o normalization,
get cone on this
(ah! we've *projectivized* via similar triangles)
if you don't order, then get an $S_3$-fold ($6$-fold)
cover, with equilateral and isoceles as orbifold points
wow! We've got an orbifold already!
parameter space of triangles:
3 points
(distinct, not co-linear!)
(open set)
ah, maybe you want 3 *ordered* points;
then it's simply a fibre bundle!