On a single hexagonal tile a line may take one of three distinct paths--- either straight across, to one of the adjancent sides, or one of the "opposite" sides. These correspond to a change in direction of 0 degrees, 60 degrees, or 120 degrees. A complete loop must have bends that total to a multiple of 360 degrees. So, for a given combination of tiles, we should be able to limit the search for a loop by enumerating what possibilities of "left turns" and "right turns" the solution must have. (Of course, two 60 degree turns end up in a different location than a single 120 degree turn.)
I think that for this puzzle restrictions to exactly 360 degrees are sufficient--- the solution cannot loop (and it is obvious at least for small numbers of tiles that summing to zero will not produce a solution.) But right now neither my geometric intuition nor algebraic skills are sufficient to show that this is so.
3 tiles: 3x 120 only
4 tiles: 2x60 + 2x120 only, all right hand turns
5 tiles: 4x60 + 1x120; or 60 + -60 (left hand turn) + 3x120; or 5x120 + -120
Given this model of the problem it is trivial to write python code which returns all the possible curves. If you have python 3.1+ then builtins are all you need; otherwise you will have to copy the code from the manual. This version shows all curves for a given number of tiles, while the Tantrix problems are simpler since there's a fixed set of tiles!
tileSet = [ 60, -60, 120, -120 ] def cycles( numTiles ): return filter( lambda x : sum( x ) == 360, combinations_with_replacement( tileSet, numTiles ) )