{-# LANGUAGE CPP #-}

{- |
   Copyright  : Copyright (C) 2012 Joachim Breitner
   License    : BSD3

   Maintainer : Joachim Breitner <mail@joachim-breitner.de>
   Stability  : stable
   Portability: portable
-}
module Optimisation.CirclePacking (
    packCircles
    -- * Example
    -- $example
    ) where

#if defined(__GLASGOW_HASKELL__ )
import Data.List (sortBy, find)
import Data.Ord (comparing)
import Data.Maybe (fromMaybe)
#else
import Language.Fay.Prelude

-- Not provided in Fay set, it seems
comparing :: (b -> Double) -> b -> b -> Ordering
comparing p x y = compare (p x) (p y)

fromMaybe     :: a -> Maybe a -> a
fromMaybe d x = case x of {Nothing -> d;Just v  -> v}
#endif

type Radius = Double
type Circle a = (Double, a)
type Coordinate = (Double, Double)
type PlacedCircle a = ((Double, a), Coordinate)
type TouchingCircles a = [(PlacedCircle a, PlacedCircle a)]

{- | 'packCircles' takes a list of circles and a function that yields the
   radius of the circle.
  
   It returns a list of all circles, in unspecified order, together with 
   coordinates such that they do not overlap but sit as tight as possible,
   filling a large circle.

   
   Finding the optimal solution to this is NP hard, so only
   heuristics are feasible. This particular
   implementation is neither very good nor very fast,
   compared to the state of the art in research. Nevertheless
   it is simple to use and gives visually acceptable results.

   The heuristics begins by placing the largest circle first, and the
   next-to-largest next to it. From then on it adds circles by considering all
   points where the circle to be added would touch two circles but overlap with
   none, and picks the one that is closest to the center of mass of the current
   placements.

-}
packCircles :: (a -> Double) -> [a] -> [(a, (Double, Double))]
packCircles radiusFunction =
    -- Forget the cached radius
    map (\t -> case t of (x,p) -> (snd x, p)) .
    -- Make sure we center up the circles
    centerUp .
    -- Forget the pairs of touching circles
    fst .
    -- Run the main algorithm
    go . 
    -- Look at large circles last
    sortBy (comparing radius) .
    -- Cache the radius
    map (\x -> (radiusFunction x, x))

-- Just for a nicer name
radius :: Circle a -> Radius
radius = fst

-- Place the tail, then try to place the head
go :: [Circle a] -> ([PlacedCircle a], TouchingCircles a)
go  [] = ([],[])
go  (c:cs) = case go cs of
               (placed, pairs) -> case place c placed pairs of
                 (cp, newPairs) -> (cp : placed, newPairs ++ pairs)

place :: Circle a -> [PlacedCircle a] -> TouchingCircles a ->
            (PlacedCircle a, TouchingCircles a)
place c [] _ = ((c, (0,0)), [])

place c [cp'@(c',(x,y))] _ = 
        let cp = (c, (x + radius c + radius c', y))
        in (cp, [(cp, cp')])

place c placed pairs = (cp, newPairs)
  where
    newPairs = [ (cp, cp') | cp' <- placed, touching cp cp' ]
    cp = (c, p)
    p = fromMaybe (error "packCircles: The end of the real plane has been reached?") $
        find (\p' -> all (valid p') placed) $
        sortBy (comparing centerDistance)
                        [ p' | (c1, c2) <- pairs,
                              touching c1 c2,
                              p' <- near c1 c2
                        ]
    centerDistance (x,y) = sqrt ((centerx - x)**2  + (centery - y)**2)

    centerx = sum [x * (radius c')**2 | (c',(x,_)) <- placed] / area
    centery = sum [y * (radius c')**2 | (c',(_,y)) <- placed] / area
    area = sum [(radius c')**2 | (c',_) <- placed] 

    valid (x1,y1) (c2,(x2,y2)) =
        sqrt ((x2 - x1)**2  + (y2-y1)**2) >= (radius c + radius c2) * (1-eps)
    
    touching (c1,(x1,y1)) (c2, (x2, y2)) = 
        sqrt ((x2 - x1)**2  + (y2-y1)**2) <= (radius c1 + radius c2) * (1+eps)

    near (c1,(x1,y1)) (c2, (x2, y2)) = [(c1x,c1y), (c2x,c2y)]
      where
        base = sqrt ((x2 - x1)**2  + (y2 - y1)**2)
        lat1 = radius c1 + radius c
        lat2 = radius c2 + radius c
        --From http://stackoverflow.com/a/11356687/946226
        ad_length = (base**2 + lat1**2 - lat2**2)/(2 * base)
        h = sqrt (abs (lat1**2 - ad_length**2))
        dx = x1 + ad_length * (x2 - x1)/base
        dy = y1 + ad_length * (y2 - y1)/base
        
        c1x = dx + h * (y2 - y1) / base
        c1y = dy - h * (x2 - x1) / base

        c2x = dx - h * (y2 - y1) / base
        c2y = dy + h * (x2 - x1) / base

centerUp :: [PlacedCircle a] -> [PlacedCircle a]
centerUp placed = map (\(o,(x,y)) -> (o, (x-centerx, y-centery))) placed
  where
    centerx = sum [x * (radius c')**2 | (c',(x,_)) <- placed] / area
    centery = sum [y * (radius c')**2 | (c',(_,y)) <- placed] / area
    area = sum [(radius c')**2 | (c',_) <- placed] 

eps :: Double
eps = 0.00001

{-$example

   The following code demonstrates how one can use 'packCircles' together with
   the diagrams library:

>import Diagrams.Prelude
>import Diagrams.Backend.SVG.CmdLine
>
>import Optimisation.CirclePacking
>
>colorize = zipWith fc $
>    cycle [red,blue,yellow,magenta,cyan,bisque,firebrick,indigo]
>
>objects = colorize $
>    [ circle r  | r <- [0.1,0.2..1.6] ] ++
>    [ hexagon r | r <- [0.1,0.2..0.7] ] ++
>    [ decagon r | r <- [0.1,0.2..0.7] ]
>
>-- Just a approximation, diagram objects do not have an exact radius
>radiusApproximation o = maximum [ radius (e (CircleFrac alpha)) o | alpha <- [0,0.1..1.0]]
>
>main = defaultMain $
>    position $ map (\(o,(x,y)) -> (p2 (x,y),o)) $
>    packCircles radiusApproximation objects

This generates the following SVG file (if your browser manages to display it):

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