force-layout-0.1.0.0: src/Physics/ForceLayout.hs
{-# LANGUAGE TemplateHaskell
, ScopedTypeVariables
, FlexibleContexts
#-}
-----------------------------------------------------------------------------
-- |
-- Module : Physics.ForceLayout
-- Copyright : (c) 2011 Brent Yorgey
-- License : BSD-style (see LICENSE)
-- Maintainer : byorgey@cis.upenn.edu
--
-- A simple, Haskell-native simulator for doing force-directed layout,
-- /e.g./ of trees or graphs.
--
-- To use, just create an 'Ensemble' like so:
--
-- > e = Ensemble [ (edges, hookeForce 0.05 4)
-- > , (allPairs, coulombForce 1)
-- > ]
-- > particleMap
-- > where edges = [(1,2), (2,3), (2,5), (3,5), (3,4), (4,5)]
-- > allPairs = [(x,y) | x <- [1..4], y <- [x+1..5]]
-- > particleMap = M.fromList . zip [1..]
-- > . map (initParticle . P)
-- > $ [ (2.0, 3.1), (6.3, 7.2)
-- > , (0.3, 4.2), (1.6, -1.1)
-- > , (4.8, 2.9)
-- > ]
--
-- Then run a simulation using either 'simulate' (to get the list of
-- all intermediate states) or 'forceLayout' (to get only the ending
-- state):
--
-- > e' = forceLayout (FLOpts { damping = 0.8
-- > , energyLimit = Just 0.001
-- > , stepLimit = Nothing
-- > }
-- > )
-- > e
--
-- See the diagrams-contrib package
-- (<http://patch-tag.com/r/byorgey/diagrams-contrib/home>) for more
-- examples.
-----------------------------------------------------------------------------
module Physics.ForceLayout
( -- * Data structures
Particle(..), pos, vel, force
, initParticle
, PID
, Edge
, Ensemble(..), forces, particles
-- * Pre-defined forces
, hookeForce
, coulombForce
, distForce
-- * Running simulations
, ForceLayoutOpts(..)
, simulate
, forceLayout
-- * Internals
, ensembleStep
, particleStep
, recalcForces
, kineticEnergy
) where
import Control.Monad
import Control.Newtype
import Data.AffineSpace
import Data.AffineSpace.Point
import Data.Foldable (foldMap)
import qualified Data.Foldable as F
import Data.Label (mkLabels)
import qualified Data.Label as L
import qualified Data.Map as M
import Data.Maybe
import Data.Monoid
import Data.VectorSpace hiding (Sum)
import Prelude
------------------------------------------------------------
-- Particles
------------------------------------------------------------
-- | A particle has a current position, current velocity, and current
-- force acting on it.
data Particle v = Particle { _pos :: Point v
, _vel :: v
, _force :: v
}
deriving (Eq, Show)
mkLabels [''Particle]
-- | Create an initial particle at rest at a particular location.
initParticle :: AdditiveGroup v => Point v -> Particle v
initParticle p = Particle p zeroV zeroV
------------------------------------------------------------
-- Ensembles
------------------------------------------------------------
-- | Used to uniquely identify particles.
type PID = Int
-- | An edge is a pair of particle identifiers.
type Edge = (PID, PID)
-- | An @Ensemble@ is a physical configuration of particles. It
-- consists of a mapping from particle IDs (unique integers) to
-- particles, and a list of forces that are operative. Each force
-- has a list of edges to which it applies, and is represented by a
-- function giving the force between any two points.
data Ensemble v = Ensemble { _forces :: [([Edge], Point v -> Point v -> v)]
, _particles :: M.Map PID (Particle v)
}
mkLabels [''Ensemble]
------------------------------------------------------------
-- Simulation internals
------------------------------------------------------------
-- | Simulate one time step for an entire ensemble, with the given
-- damping factor.
ensembleStep :: VectorSpace v => Scalar v -> Ensemble v -> Ensemble v
ensembleStep d = (L.modify particles . M.map) (particleStep d) . recalcForces
-- | Simulate one time step for a particle (assuming the force acting
-- on it has already been computed), with the given damping factor.
particleStep :: VectorSpace v => Scalar v -> Particle v -> Particle v
particleStep d = stepPos . stepVel
where stepVel p = L.set vel (d *^ (L.get vel p ^+^ L.get force p)) p
stepPos p = L.modify pos (.+^ L.get vel p) p
-- | Recalculate all the forces acting in the next time step of an
-- ensemble.
recalcForces :: forall v. AdditiveGroup v => Ensemble v -> Ensemble v
recalcForces = calcForces . zeroForces
where zeroForces = L.modify particles . M.map $ L.set force zeroV
calcForces (Ensemble fs ps)
= Ensemble fs
(ala Endo foldMap (concatMap (\(es, f) -> (map (mkForce f) es)) fs) ps)
mkForce :: (Point v -> Point v -> v) -> Edge -> M.Map Int (Particle v) -> M.Map Int (Particle v)
mkForce f (i1, i2) m
= case (M.lookup i1 m, M.lookup i2 m) of
(Just p1, Just p2) ->
( M.adjust (L.modify force (^+^ f (L.get pos p1) (L.get pos p2))) i1
. M.adjust (L.modify force (^-^ f (L.get pos p1) (L.get pos p2))) i2)
m
_ -> m
-- | Compute the total kinetic energy of an ensemble.
kineticEnergy :: (InnerSpace v, Num (Scalar v)) => Ensemble v -> Scalar v
kineticEnergy = ala Sum F.foldMap . fmap (magnitudeSq . L.get vel) . L.get particles
------------------------------------------------------------
-- Simulation
------------------------------------------------------------
-- | Options for customizing a simulation.
data ForceLayoutOpts v =
FLOpts
{ damping :: Scalar v -- ^ Damping factor to be
-- applied at each step.
-- Should be between 0 and 1.
, energyLimit :: Maybe (Scalar v) -- ^ Kinetic energy below which
-- simulation should stop. If
-- @Nothing@, pay no attention
-- to kinetic energy.
, stepLimit :: Maybe Int -- ^ Maximum number of
-- simulation steps. If
-- @Nothing@, pay no attention
-- to the number of steps.
}
-- | Simulate a starting ensemble according to the given options,
-- producing a list of all the intermediate ensembles. Useful for,
-- /e.g./, making an animation. Note that the resulting list could
-- be infinite, if a 'stepLimit' is not specified and either the
-- kinetic energy never falls below the specified threshold, or no
-- energy threshold is specified.
simulate :: (InnerSpace v, Ord (Scalar v), Num (Scalar v))
=> ForceLayoutOpts v -> Ensemble v -> [Ensemble v]
simulate opts e
= (e:)
. takeWhile (maybe (const True) (<) (energyLimit opts) . kineticEnergy)
. maybe id take (stepLimit opts)
. drop 1
. iterate (ensembleStep (damping opts))
$ e
-- | Run a simluation from a starting ensemble, yielding either the
-- first ensemble to have kinetic energy below the 'energyLimit' (if
-- given), or the ensemble that results after a number of steps
-- equal to the 'stepLimit' (if given), whichever comes first.
-- Otherwise @forceLayout@ will not terminate.
forceLayout :: (InnerSpace v, Ord (Scalar v), Num (Scalar v))
=> ForceLayoutOpts v -> Ensemble v -> Ensemble v
forceLayout opts = last . simulate opts
------------------------------------------------------------
-- Standard forces
------------------------------------------------------------
-- | @distForce f p1 p2@ computes the force between two points as a
-- multiple of the unit vector in the direction from @p1@ to @p2@,
-- given a function @f@ which computes the force's magnitude as a
-- function of the distance between the points.
distForce :: (InnerSpace v, Floating (Scalar v))
=> (Scalar v -> Scalar v) -> Point v -> Point v -> v
distForce f p1 p2 = withLength (f (distance p1 p2)) (p2 .-. p1)
where withLength s v = s *^ normalized v
-- | @hookeForce k l p1 p2@ computes the force on @p1@, assuming that
-- @p1@ and @p2@ are connected by a spring with equilibrium length @l@
-- and spring constant @k@.
hookeForce :: (InnerSpace v, Floating (Scalar v))
=> Scalar v -> Scalar v -> Point v -> Point v -> v
hookeForce k l = distForce (\d -> k * (d - l))
-- | @coulombForce k@ computes the electrostatic repulsive force
-- between two charged particles, with constant of proportionality
-- @k@.
coulombForce :: (InnerSpace v, Floating (Scalar v))
=> Scalar v -> Point v -> Point v -> v
coulombForce k = distForce (\d -> -k * 1/(d*d))