A mesh defining a surface has uniquely numbered vertices, and named,
simple-polygonal faces described usually by an ordered list of edge numbers
going around the face,



For example:
External image of two faces

Rough textual version without edges:


      1         
                        17
 7   A
             B

       11                     
                  23


 A is the triangle (1, 11, 7), or equally (7, 11, 1), going anti-clockwise, or
any of all the rotations of those ordered vertices.

      1         
                        
 7   A
            

       11
 B is the four-sided face (1, 17, 23, 11), or equally (23, 17, 1, 11) or any 
of their rotations.

1         
                  17

       B

 11                     
            23
Let's call the above the perimeter format as it traces around the perimeter.

A second format

A separate algorithm returns polygonal faces consisting of a face name and an unordered
set of edge definitions for each face.

 A single edge is described by the vertex numbers at its two ends, always in 
ascending order.

 All edges for the face are given, but in an undefined order.
For example face A could be described by the edges (1, 11), (7, 11), and (1, 7)
(The order of each vertex number in an edge is ascending, but the order in
which the edges are stated is arbitrary).

Similarly face B could be described by the edges (11, 23), (1, 17), (17, 23),
and (1, 11) in arbitrary order of the edges.

Let's call this second format the edge format.



Task

1. Write a routine to check if two perimeter formatted faces have the same perimeter. Use it to check if the following pairs of perimeters are the same:

 Q: (8, 1, 3)
 R: (1, 3, 8)

 U: (18, 8, 14, 10, 12, 17, 19)
 V: (8, 14, 10, 12, 17, 19, 18)

2. Write a routine and use it to transform the following faces from edge to perimeter format.

 E: {(1, 11), (7, 11), (1, 7)}
 F: {(11, 23), (1, 17), (17, 23), (1, 11)}
 G: {(8, 14), (17, 19), (10, 12), (10, 14), (12, 17), (8, 18), (18, 19)}
 H: {(1, 3), (9, 11), (3, 11), (1, 11)}

Show your output here.

import Data.List (find, delete, (\\))
import Control.Applicative ((<|>))
 
------------------------------------------------------------
 
newtype Perimeter a = Perimeter [a]
  deriving Show
 
instance Eq a => Eq (Perimeter a) where
  Perimeter p1 == Perimeter p2 =
    null (p1 \\ p2)
    && ((p1 `elem` zipWith const (iterate rotate p2) p1)
        || Perimeter p1 == Perimeter (reverse p2))
 
rotate lst = zipWith const (tail (cycle lst)) lst
 
toEdges :: Ord a => Perimeter a -> Maybe (Edges a)
toEdges (Perimeter ps)
  | allDifferent ps = Just . Edges $ zipWith ord ps (tail (cycle ps))
  | otherwise = Nothing
  where
    ord a b = if a < b then (a, b) else (b, a)
 
allDifferent [] = True
allDifferent (x:xs) = all (x /=) xs && allDifferent xs
 
------------------------------------------------------------
 
newtype Edges a = Edges [(a, a)]
  deriving Show
 
instance Eq a => Eq (Edges a) where
  e1 == e2 = toPerimeter e1 == toPerimeter e2
 
toPerimeter :: Eq a => Edges a -> Maybe (Perimeter a)
toPerimeter (Edges ((a, b):es)) = Perimeter . (a :) <$> go b es
  where
    go x rs
      | x == a = return []
      | otherwise = do
          p <- find ((x ==) . fst) rs <|> find ((x ==) . snd) rs
          let next = if fst p == x then snd p else fst p
          (x :) <$> go next (delete p rs)