Thanks for the replies, much appreciated. All the solutions work on the data I supplied in my original post. As tobyink says the segment samples I gave form a cycle but when I try the solutions on segments that don't form a cycle I run into a problem. For example, if I take out one of the segments and use choroba's solution I get three lines.

my @AoA = (
    # removed [11,29, 10,25],
    [15,35, 11,29],
    [15,15, 11,21],
    [10,25, 11,21],
    [15,35, 21,39],
    [25,40, 21,39],
    [21,11, 25,10],
    [15,15, 21,11],
    [35,35, 29,39],
    [29,39, 25,40],
    [35,15, 29,11],
    [25,10, 29,11],
    [40,25, 39,29],
    [35,35, 39,29],
    [39,21, 40,25],
    [35,15, 39,21],
);

# Result 

(15,35, 11,29)

(15,15, 11,21, 10,25)

(15,35, 21,39, 25,40, 29,39, 35,35, 39,29, 40,25, 39,21, 35,15, 29,11,
+ 25,10, 21,11, 15,15)
[download]

It seems to depend on which point I pick to start with, so if I pick a point at the end or start of the line it will work (where the break is), but I don't know in advance what these points are. Here is the same data with the segments re-arranged so the break point is at the start:

my @AoA = (
    # removed [11,29, 10,25],
    [10,25, 11,21],
    [15,35, 11,29],
    [15,15, 11,21],
    [15,35, 21,39],
    [25,40, 21,39],
    [21,11, 25,10],
    [15,15, 21,11],
    [35,35, 29,39],
    [29,39, 25,40],
    [35,15, 29,11],
    [25,10, 29,11],
    [40,25, 39,29],
    [35,35, 39,29],
    [39,21, 40,25],
    [35,15, 39,21],
);

# Result 

(10,25, 11,21, 15,15, 21,11, 25,10, 29,11, 35,15, 39,21, 40,25, 39,29,
+ 35,35, 29,39, 25,40, 21,39, 15,35, 11,29)
[download]