{-# LANGUAGE DeriveFunctor             #-}
{-# LANGUAGE ExistentialQuantification #-}
{-# LANGUAGE ScopedTypeVariables       #-}
{-# LANGUAGE TypeFamilies              #-}
{-# LANGUAGE ViewPatterns              #-}

-----------------------------------------------------------------------------
-- |
-- Module      :  Diagrams.TwoD.Polygons
-- Copyright   :  (c) 2011 diagrams-lib team (see LICENSE)
-- License     :  BSD-style (see LICENSE)
-- Maintainer  :  diagrams-discuss@googlegroups.com
--
-- This module defines a general API for creating various types of
-- polygons.
--
-----------------------------------------------------------------------------

module Diagrams.TwoD.Polygons(
        -- * Polygons
          PolyType(..)
        , PolyOrientation(..)
        , PolygonOpts(..)

        , polygon
        , polyTrail

        -- ** Generating polygon vertices

        , polyPolarTrail
        , polySidesTrail
        , polyRegularTrail

        , orient

        -- * Star polygons
        , StarOpts(..)
        , star

        -- ** Function graphs
        -- $graphs
        , GraphPart(..)
        , orbits, mkGraph

    ) where

import           Control.Monad           (forM, liftM)
import           Control.Monad.ST        (ST, runST)
import           Data.Array.ST           (STUArray, newArray, readArray,
                                          writeArray)
import qualified Data.Foldable           as F
import           Data.List               (maximumBy, minimumBy)
import qualified Data.List.NonEmpty      as NEL
import           Data.Maybe              (catMaybes)
import           Data.Monoid             (mconcat, mempty)
import           Data.Ord                (comparing)

import           Data.AffineSpace        ((.+^), (.-.))
import           Data.Default.Class
import           Data.VectorSpace        (magnitude, normalized, project, (<.>),
                                          (^*))

import           Diagrams.Core
import           Diagrams.Located
import           Diagrams.Path
import           Diagrams.Points         (centroid)
import           Diagrams.Segment
import           Diagrams.Trail
import           Diagrams.TrailLike
import           Diagrams.TwoD.Transform
import           Diagrams.TwoD.Types
import           Diagrams.TwoD.Vector    (leftTurn, unitX, unitY, unit_Y)
import           Diagrams.Util           (tau, ( # ))

-- | Method used to determine the vertices of a polygon.
data PolyType = forall a. Angle a => PolyPolar [a] [Double]
                -- ^ A \"polar\" polygon.
                --
                --   * The first argument is a list of /central/
                --     /angles/ from each vertex to the next.
                --
                --   * The second argument is a list of /radii/ from
                --     the origin to each successive vertex.
                --
                --   To construct an /n/-gon, use a list of /n-1/
                --   angles and /n/ radii.  Extra angles or radii
                --   are ignored.
                --
                --   Cyclic polygons (with all vertices lying on a
                --   circle) can be constructed using a second
                --   argument of @(repeat r)@.

              | forall a. Angle a => PolySides [a] [Double]
                -- ^ A polygon determined by the distance between
                --   successive vertices and the angles formed by
                --   each three successive vertices.  In other
                --   words, a polygon specified by \"turtle
                --   graphics\": go straight ahead x1 units; turn by
                --   angle a1; go straght ahead x2 units; turn by
                --   angle a2; etc. The polygon will be centered at
                --   the /centroid/ of its vertices.
                --
                --   * The first argument is a list of /vertex/
                --     /angles/, giving the angle at each vertex
                --     from the previous vertex to the next.  The
                --     first angle in the list is the angle at the
                --     /second/ vertex; the first edge always starts
                --     out heading in the positive y direction from
                --     the first vertex.
                --
                --   * The second argument is a list of distances
                --     between successive vertices.
                --
                --   To construct an /n/-gon, use a list of /n-2/
                --   angles and /n-1/ edge lengths.  Extra angles or
                --   lengths are ignored.

              | PolyRegular Int Double
                -- ^ A regular polygon with the given number of
                --   sides (first argument) and the given radius
                --   (second argument).

-- | Determine how a polygon should be oriented.
data PolyOrientation = NoOrient     -- ^ No special orientation; the first
                                    --   vertex will be at (1,0).
                                    --   This is the default.
                     | OrientH      -- ^ Orient /horizontally/, so the
                                    --   bottommost edge is parallel to
                                    --   the x-axis.
                     | OrientV      -- ^ Orient /vertically/, so the
                                    --   leftmost edge is parallel to the
                                    --   y-axis.
                     | OrientTo R2  -- ^ Orient so some edge is
                                    --   /facing/ /in/ /the/ /direction/
                                    --   /of/, that is, perpendicular
                                    --   to, the given vector.
  deriving (Eq, Ord, Show, Read)

-- | Options for specifying a polygon.
data PolygonOpts = PolygonOpts
                   { polyType   :: PolyType
                     -- ^ Specification for the polygon's vertices.

                   , polyOrient :: PolyOrientation
                     -- ^ Should a rotation be applied to the
                     --   polygon in order to orient it in a
                     --   particular way?

                   , polyCenter :: P2
                     -- ^ Should a translation be applied to the
                     --   polygon in order to place the center at a
                     --   particular location?
                   }

-- | The default polygon is a regular pentagon of radius 1, centered
--   at the origin, aligned to the x-axis.
instance Default PolygonOpts where
    def = PolygonOpts (PolyRegular 5 1) OrientH origin

-- | Generate a polygon.  See 'PolygonOpts' for more information.
polyTrail :: PolygonOpts -> Located (Trail R2)
polyTrail po = transform ori tr
    where
        tr = case polyType po of
            PolyPolar ans szs -> polyPolarTrail ans szs
            PolySides ans szs -> polySidesTrail ans szs
            PolyRegular n r   -> polyRegularTrail n r
        ori = case polyOrient po of
            OrientH      -> orient unit_Y tr
            OrientV      -> orient unitX  tr
            OrientTo v   -> orient v      tr
            NoOrient     -> mempty

-- | Generate the polygon described by the given options.
polygon :: (TrailLike t, V t ~ R2) => PolygonOpts -> t
polygon = trailLike . polyTrail

-- | Generate the located trail of a polygon specified by polar data
--   (central angles and radii). See 'PolyPolar'.
polyPolarTrail :: Angle a => [a] -> [Double] -> Located (Trail R2)
polyPolarTrail [] _ = emptyTrail `at` origin
polyPolarTrail _ [] = emptyTrail `at` origin
polyPolarTrail ans (r:rs) = tr `at` p1
  where
    p1 = p2 (1,0) # scale r
    tr = closeTrail . trailFromVertices $
           zipWith
             (\a l -> rotate a . scale l $ p2 (1,0))
             (scanl (+) 0 ans)
             (r:rs)

-- | Generate the vertices of a polygon specified by side length and
--   angles, and a starting point for the trail such that the origin
--   is at the centroid of the vertices.  See 'PolySides'.
polySidesTrail :: Angle a => [a] -> [Double] -> Located (Trail R2)
polySidesTrail ans ls = tr `at` (centroid ps # scale (-1))
  where
    ans'    = scanl (+) 0 ans
    offsets = zipWith rotate ans' (map (unitY ^*) ls)
    ps      = scanl (.+^) origin offsets
    tr      = closeTrail . trailFromOffsets $ offsets

-- | Generate the vertices of a regular polygon.  See 'PolyRegular'.
polyRegularTrail :: Int -> Double -> Located (Trail R2)
polyRegularTrail n r = polyPolarTrail
                         (take (n-1) . repeat $ (tau::Rad) / fromIntegral n)
                         (repeat r)

-- | Generate a transformation to orient a trail.  @orient v t@
--   generates the smallest rotation such that one of the segments
--   adjacent to the vertex furthest in the direction of @v@ is
--   perpendicular to @v@.
orient :: R2 -> Located (Trail R2) -> T2
orient v = orientPoints v . trailVertices

orientPoints :: R2 -> [P2] -> T2
orientPoints v xs = rotation a
  where
    (n1,x,n2) = maximumBy (comparing (distAlong v . sndOf3))
                  (zip3 (tail (cycle xs)) xs (last xs : init xs))
    distAlong w ((.-. origin) -> p) = signum (w <.> p) * magnitude (project w p)
    sndOf3 (_,x,_) = x
    a = minimumBy (comparing abs) . map (angleFromNormal . (.-. x)) $ [n1,n2]
    v' = normalized v
    angleFromNormal o
      | leftTurn o' v' = phi
      | otherwise      = negate phi
      where
        o' = normalized o
        theta = acos (v' <.> o')
        phi
          | theta <= tau/4 = Rad $ tau/4 - theta
          | otherwise      = Rad $ theta - tau/4

------------------------------------------------------------
-- Function graphs
------------------------------------------------------------

-- $graphs
-- These functions are used to implement 'star', but are exported on
-- the offchance that someone else finds them useful.

-- | Pieces of a function graph can either be cycles or \"hairs\".
data GraphPart a = Cycle [a]
                 | Hair  [a]
  deriving (Show, Functor)

-- | @orbits f n@ computes the graph of @f@ on the integers mod @n@.
orbits :: (Int -> Int) -> Int -> [GraphPart Int]
orbits f n = runST genOrbits
  where
    f_n i = f i `mod` n

    genOrbits :: ST s [GraphPart Int]
    genOrbits = newArray (0,n-1) False >>= genOrbits'

    genOrbits' :: STUArray s Int Bool -> ST s [GraphPart Int]
    genOrbits' marks = liftM (concat . catMaybes) (forM [0 .. n-1] (genPart marks))

    genPart :: STUArray s Int Bool -> Int -> ST s (Maybe [GraphPart Int])
    genPart marks i = do
      tr <- markRho i marks
      case tr of
        [] -> return Nothing
        _  -> return . Just . splitParts $ tr

    markRho :: Int -> STUArray s Int Bool -> ST s [Int]
    markRho i marks = do
      isMarked <- readArray marks i
      case isMarked of
        True  -> return []
        False -> writeArray marks i True >>
                 liftM (i:) (markRho (f_n i) marks)

    splitParts :: [Int] -> [GraphPart Int]
    splitParts tr = hair ++ cyc
      where hair | not (null tl)   = [Hair $ tl ++ [f_n (last tl)]]
                 | otherwise       = []
            cyc  | not (null body) = [Cycle body]
                 | otherwise       = []
            l            = last tr
            (tl, body) = span (/= f_n l) tr

-- | Generate a function graph from the given function and labels.
mkGraph :: (Int -> Int) -> [a] -> [GraphPart a]
mkGraph f xs = (map . fmap) (xs!!) $ orbits f (length xs)

------------------------------------------------------------
--  Star polygons
------------------------------------------------------------

-- | Options for creating \"star\" polygons, where the edges connect
--   possibly non-adjacent vertices.
data StarOpts = StarFun (Int -> Int)
                -- ^ Specify the order in which the vertices should be
                --   connected by a function that maps each vertex
                --   index to the index of the vertex that should come
                --   next.  Indexing of vertices begins at 0.

              | StarSkip Int
                -- ^ Specify a star polygon by a \"skip\".  A skip of
                --   1 indicates a normal polygon, where edges go
                --   between successive vertices.  A skip of 2 means
                --   that edges will connect every second vertex,
                --   skipping one in between.  Generally, a skip of
                --   /n/ means that edges will connect every /n/th
                --   vertex.

-- | Create a generalized /star/ /polygon/.  The 'StarOpts' are used
--   to determine in which order the given vertices should be
--   connected.  The intention is that the second argument of type
--   @[P2]@ could be generated by a call to 'polygon', 'regPoly', or
--   the like, since a list of vertices is 'TrailLike'.  But of course
--   the list can be generated any way you like.  A @'Path' 'R2'@ is
--   returned (instead of any 'TrailLike') because the resulting path
--   may have more than one component, for example if the vertices are
--   to be connected in several disjoint cycles.
star :: StarOpts -> [P2] -> Path R2
star sOpts vs = graphToPath $ mkGraph f vs
  where f = case sOpts of
              StarFun g  -> g
              StarSkip k -> (+k)
        graphToPath = mconcat . map partToPath

        partToPath (Cycle ps) = pathFromLocTrail
                              . mapLoc closeTrail
                              . fromVertices
                              $ ps

        partToPath (Hair ps)  = fromVertices ps