manifolds (empty) → 0.1.0.0
raw patch · 7 files changed
+1607/−0 lines, 7 filesdep +MonadRandomdep +basedep +comonadsetup-changed
Dependencies added: MonadRandom, base, comonad, constrained-categories, containers, random, semigroups, transformers, vector, vector-algorithms, vector-space, void
Files
- COPYING +674/−0
- Data/List/FastNub.hs +43/−0
- Data/Manifold.hs +702/−0
- Setup.hs +6/−0
- Util/Associate.hs +57/−0
- Util/LtdShow.hs +64/−0
- manifolds.cabal +61/−0
+ COPYING view
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Of course, your program's commands+might be different; for a GUI interface, you would use an "about box".++ You should also get your employer (if you work as a programmer) or school,+if any, to sign a "copyright disclaimer" for the program, if necessary.+For more information on this, and how to apply and follow the GNU GPL, see+<http://www.gnu.org/licenses/>.++ The GNU General Public License does not permit incorporating your program+into proprietary programs. If your program is a subroutine library, you+may consider it more useful to permit linking proprietary applications with+the library. If this is what you want to do, use the GNU Lesser General+Public License instead of this License. But first, please read+<http://www.gnu.org/philosophy/why-not-lgpl.html>.
+ Data/List/FastNub.hs view
@@ -0,0 +1,43 @@+{-# LANGUAGE ConstraintKinds #-}+++module Data.List.FastNub where+++import Data.List+import Data.Function+++type FastNub a = (Eq a, Ord a) -- S̶h̶o̶u̶l̶d̶ ̶r̶e̶a̶l̶l̶y̶ ̶b̶e̶ ̶(̶E̶q̶ ̶a̶,̶̶ ̶H̶a̶s̶h̶a̶b̶l̶e̶ ̶a̶)̶+fastNub :: FastNub a => [a] -> [a]+fastNub = map head . group . sort++-- | Simply a merge sort that discards equivalent elements.+fastNubBy :: (a->a->Ordering) -> [a] -> [a]+fastNubBy _ [] = []+fastNubBy _ [e] = [e]+fastNubBy cmp es = merge(fastNubBy cmp lhs)(fastNubBy cmp rhs)+ where (lhs,rhs) = splitAt (length es `quot` 2) es+ merge [] rs = rs+ merge ls [] = ls+ merge (l:ls) (r:rs) = case cmp l r of+ LT -> l : merge ls (r:rs)+ GT -> r : merge (l:ls) rs+ EQ -> merge (l:ls) rs++-- | Like 'fastNubBy', but doesn't just discard duplicates but \"merges\" them.+-- @'fastNubBy' cmp = cmp `'fastNubByWith'` 'const'@.+fastNubByWith :: (a->a->Ordering) -> (a->a->a) -> [a] -> [a]+fastNubByWith _ _ [] = []+fastNubByWith _ _ [e] = [e]+fastNubByWith cmp cmb es = merge(fastNubByWith cmp cmb lhs)(fastNubByWith cmp cmb rhs)+ where (lhs,rhs) = splitAt (length es `quot` 2) es+ merge [] rs = rs+ merge ls [] = ls+ merge (l:ls) (r:rs) = case cmp l r of+ LT -> l : merge ls (r:rs)+ GT -> r : merge (l:ls) rs+ EQ -> merge (cmb l r : ls) rs++sfGroupBy :: (a->a->Ordering) -> [a] -> [[a]]+sfGroupBy cmp = fastNubByWith (cmp`on`head) (++) . map(:[])
+ Data/Manifold.hs view
@@ -0,0 +1,702 @@+-- |+-- Module : Data.Manifold+-- Copyright : (c) Justus Sagemüller 2013+-- License : GPL v3+-- +-- Maintainer : (@) sagemueller $ geo.uni-koeln.de+-- Stability : experimental+-- Portability : portable+-- +-- This is something of a first attempt at formalising manifolds and continuous+-- mappings thereon. They /work/+-- (check out <http://hackage.haskell.org/package/dynamic-plot-0.1.0.0> for a use case),+-- but aren't very efficient. The interface might well change considerably in the future.+++{-# LANGUAGE FlexibleInstances #-}+{-# LANGUAGE UndecidableInstances #-}+-- {-# LANGUAGE OverlappingInstances #-}+{-# LANGUAGE TypeFamilies #-}+{-# LANGUAGE FunctionalDependencies #-}+{-# LANGUAGE FlexibleContexts #-}+{-# LANGUAGE GADTs #-}+{-# LANGUAGE RankNTypes #-}+{-# LANGUAGE TupleSections #-}+{-# LANGUAGE ConstraintKinds #-}+{-# LANGUAGE PatternGuards #-}+{-# LANGUAGE TypeOperators #-}+{-# LANGUAGE ScopedTypeVariables #-}+{-# LANGUAGE RecordWildCards #-}+++module Data.Manifold where++import Data.List+import Data.Maybe+import Data.Semigroup+import Data.Function (on)++import Data.VectorSpace+import Data.AffineSpace+import Data.Basis+import Data.Complex hiding (magnitude)+import Data.Void++import qualified Prelude++import Control.Category.Constrained.Prelude hiding ((^))+import Control.Arrow.Constrained+import Control.Monad.Constrained+import Data.Foldable.Constrained++++-- | Continuous mapping.+data domain :--> codomain where+ Continuous :: ( Manifold d, Manifold c+ , v ~ TangentSpace d, u ~ TangentSpace c+ , δ ~ Metric v, ε ~ Metric u ) =>+ { runContinuous :: Chart d -> v -> (Chart c, u, ε->Option δ) }+ -> d :--> c+ +++++++continuous_id' :: Manifold m => m :--> m+continuous_id' = Continuous id'+ where id' chart v = (chart, v, return)+++const__ :: (Manifold c, Manifold d)+ => c -> d:-->c+const__ x = Continuous f+ where f _ _ = (tgtChart, w, const mzero)+ tgtChart = head $ localAtlas x+ w = case tgtChart of + IdChart -> x+ Chart _ tchOut _ -> fromJust (tchOut x) $ x+++flatContinuous :: ( FlatManifold v, FlatManifold w, δ~Metric v, ε~Metric w )+ => (v -> (w, ε -> Option δ)) -> (v:-->w)+flatContinuous f = Continuous cnt+ where cnt IdChart v = let (w, postEps) = f v + in (IdChart, w, postEps)+ cnt (Chart inMap _ _) v = let (v', preEps) = runFlatContinuous inMap v+ (w, postEps) = f v'+ in (IdChart, w, preEps>=>postEps)++runFlatContinuous :: ( FlatManifold v, FlatManifold w, δ~Metric v, ε~Metric w )+ => (v:-->w) -> v -> (w, ε -> Option δ)+runFlatContinuous (Continuous cnf) v = (w, preEps>=>postEps)+ where (cc', v', preEps) = cnf IdChart v+ (w, postEps) = case cc' of + IdChart -> (v', return)+ Chart inMap _ _ -> runFlatContinuous inMap v'+++instance Category (:-->) where+ type Object (:-->) t = Manifold t++ id = Continuous $ \c v -> (c, v, just)+ + Continuous f . Continuous g = Continuous h+ where h srcChart u = (tgtChart, w, q>=>p)+ where (interChart, v, p) = g srcChart u+ (tgtChart, w, q) = f interChart v+ +instance EnhancedCat (->) (:-->) where+ Continuous f `arr` x = y+ where (tch, v, _) = f sch u+ y = case tch of Chart tchIn _ _ -> tchIn $ v+ IdChart -> v+ u = case sch of Chart _ schOut _ -> fromJust (schOut x) $ x+ IdChart -> x+ sch = head $ localAtlas x+++instance Cartesian (:-->) where+ type PairObjects (:-->) a b = ( FlatManifold a, FlatManifold b, Manifold(a,b) )+ swap = Continuous $ \c t -> case c of+ IdChart -> let (v,w) = t in (IdChart, (w,v), return)+ Chart inMap _ _ -> let ((v,w), epsP) = runFlatContinuous inMap t + in (IdChart, (w,v), epsP)+ attachUnit = Continuous $ \c v -> case c of+ IdChart -> (IdChart, (v,()), return)+ Chart inMap _ _ -> let (v', epsP) = runFlatContinuous inMap v+ in (IdChart, (v',()), epsP)+ detachUnit = Continuous $ \c t -> case c of+ IdChart -> let (v,()) = t in (IdChart, v, return)+ Chart inMap _ _ -> let ((v,()), epsP) = runFlatContinuous inMap t+ in (IdChart, v, epsP)+ regroup = Continuous $ \c t -> case c of+ IdChart -> let (u,(v,w)) = t in (IdChart, ((u,v),w), return)+ Chart inMap _ _ -> let ((u,(v,w)), epsP) = runFlatContinuous inMap t+ in (IdChart, ((u,v),w), epsP)+ regroup' = Continuous $ \c t -> case c of+ IdChart -> let ((u,v),w) = t in (IdChart, (u,(v,w)), return)+ Chart inMap _ _ -> let (((u,v),w), epsP) = runFlatContinuous inMap t+ in (IdChart, (u,(v,w)), epsP)++instance Morphism (:-->) where+ first (Continuous f) = Continuous $ \c t -> case c of+ IdChart -> let (v,w) = t+ (IdChart, v', epsP) = f IdChart v+ in (IdChart, (v',w), (/ sqrt 2) >>> + \ε -> fmap getMin $ (fmap Min $ epsP ε)+ <>(just $ Min ε) )+ second (Continuous g) = Continuous $ \c t -> case c of+ IdChart -> let (v,w) = t+ (IdChart, w', epsP) = g IdChart w+ in (IdChart, (v,w'), (/ sqrt 2) >>> + \ε -> fmap getMin $ (just $ Min ε)+ <>(fmap Min $ epsP ε) )+ Continuous f *** Continuous g = Continuous $ \c t -> case c of+ IdChart -> let (v,w) = t+ (IdChart, v', epsPv) = f IdChart v+ (IdChart, w', epsPw) = g IdChart w+ in (IdChart, (v',w'), (/ sqrt 2) >>> + \ε -> fmap getMin $ (fmap Min $ epsPv ε)+ <>(fmap Min $ epsPw ε) )++instance PreArrow (:-->) where+ terminal = const__ ()+ Continuous f &&& Continuous g = Continuous $ \c v -> case c of+ IdChart -> let (IdChart, v', epsPv) = f IdChart v+ (IdChart, w', epsPw) = g IdChart v+ in (IdChart, (v',w'), (/ sqrt 2) >>> + \ε -> fmap getMin $ (fmap Min $ epsPv ε)+ <>(fmap Min $ epsPw ε) )+ fst = Continuous $ \c t -> case c of+ IdChart -> let (v,_) = t+ in (IdChart, v, return)+ snd = Continuous $ \c t -> case c of+ IdChart -> let (_,v) = t+ in (IdChart, v, return)+ ++++type EuclidSpace v = (HasBasis v, EqFloating(Scalar v), Eq v)+type EqFloating f = (Eq f, Ord f, Floating f)+++-- | A chart is a homeomorphism from a connected, open subset /Q/ ⊂ /M/ of+-- an /n/-manifold /M/ to either the open unit disk /Dⁿ/ ⊂ /V/ ≃ ℝ/ⁿ/, or+-- the half-disk /Hⁿ/ = {/x/ ∊ /Dⁿ/: x₀≥0}. In e.g. the former case, 'chartInMap'+-- is thus defined ∀ /v/ ∊ /V/ : |/v/| < 1, while 'chartOutMap p' will yield @Just x@+-- with /x/ ∊ /Dⁿ/ provided /p/ is in /Q/, and @Nothing@ otherwise.+-- Obviously, @fromJust . 'chartOutMap' . 'chartInMap'@ should be equivalent to @id@+-- on /Dⁿ/, and @'chartInMap' . fromJust . 'chartOutMap'@ to @id@ on /Q/.+data Chart :: * -> * where+ IdChart :: (FlatManifold v) => Chart v+ Chart :: (Manifold m, v ~ TangentSpace m, FlatManifold v) =>+ { chartInMap :: v :--> m+ , chartOutMap :: m -> Maybe (m:-->v)+ , chartKind :: ChartKind } -> Chart m+data ChartKind = LandlockedChart -- ^ A /M/ ⇆ /Dⁿ/ chart, for ordinary manifolds+ | RimChart -- ^ A /M/ ⇆ /Hⁿ/ chart, for manifolds with a rim+++type FlatManifold v = (MetricSpace v, Manifold v, v~TangentSpace v)+++++isInUpperHemi :: EuclidSpace v => v -> Bool+isInUpperHemi v = (snd . head) (decompose v) >= 0++-- rimGuard :: EuclidSpace v => ChartKind -> v -> Maybe v+-- rimGuard LandlockedChart v = Just v+-- rimGuard RimChart v+-- | isInUpperHemi v = Just v+-- | otherwise = Nothing+-- +-- chartEnv :: Manifold m => Chart m+-- -> (TangentSpace m->TangentSpace m)+-- -> m -> Maybe m+-- chartEnv IdChart f x = Just $ f x+-- chartEnv (Chart inMap outMap chKind) f x = do+-- vGet <- outMap x+-- let v = vGet $ x+-- v' <- rimGuard chKind v+-- Just $ inMap $ v'+-- + ++ ++type Atlas m = [Chart m]++class (MetricSpace(TangentSpace m), Metric(TangentSpace m) ~ ℝ) => Manifold m where+ type TangentSpace m :: *+ type TangentSpace m = m -- For \"flat\", i.e. vector space manifolds.+ + localAtlas :: m -> Atlas m+++vectorSpaceAtlas :: FlatManifold v => v -> Atlas v+vectorSpaceAtlas _ = [IdChart]+++ +instance Manifold () where+ type TangentSpace () = ()+ localAtlas = vectorSpaceAtlas++instance Manifold Double where+ localAtlas = vectorSpaceAtlas+ +instance ( FlatManifold v₁, FlatManifold v₂, Scalar v₁~Scalar v₂+ , MetricSpace (Scalar v₁), Metric (Scalar v₁)~ℝ+ , VectorSpace (v₁,v₂), Scalar (v₁,v₂) ~ Scalar v₁+ ) => Manifold (v₁,v₂) where+ localAtlas = vectorSpaceAtlas+++++++++type Representsℝ r = (EqFloating r, FlatManifold r, r~Scalar r, r~Metric r)++continuousFlatFunction :: ( FlatManifold d, FlatManifold c, ε~Metric c, δ~Metric d ) + => (d -> (c, ε->Option δ)) -> d:-->c+continuousFlatFunction f = Continuous f'+ where f' IdChart x = (IdChart, y, eps2Delta)+ where (y, eps2Delta) = f x+ f' (Chart inMap _ _) v = (IdChart, y, postEps>=>preEps)+ where (v', preEps) = runFlatContinuous inMap v+ (y, postEps) = f v'++type CntnRealFunction = Representsℝ r => r :--> r++sin__, cos__, atan__ , exp__ , sinh__, cosh__, tanh__, asinh__ :: CntnRealFunction+sin__ = continuousFlatFunction sin'+ where sin' x = (sinx, eps2Delta)+ where eps2Delta ε+ | ε > 1 + abs sinx = nothing+ | otherwise = just $ ε / (dsinx + sqrt ε)+ dsinx = abs $ cos x+ sinx = sin x+cos__ = continuousFlatFunction cos'+ where cos' x = (cosx, eps2Delta)+ where eps2Delta ε+ | ε > 1 + abs cosx = nothing+ | otherwise = just $ ε / (dcosx + sqrt ε)+ dcosx = abs $ sin x+ cosx = cos x+atan__ = continuousFlatFunction atan'+ where atan' x = (atanx, eps2Delta)+ where eps2Delta ε+ | ε >= pi/2 + abs atanx = nothing+ | otherwise = just $ abs x - tan (abs atanx - ε)+ atanx = atan x++exp__ = continuousFlatFunction exp'+ where exp' x = (expx, eps2Delta)+ where expx = exp x+ eps2Delta ε + | x>0, expx*2 == expx = just 0 -- "Infinity" in floating-point+ | otherwise = just $ log (expx + ε) - x+-- exp x + ε = exp (x + δ) = exp x * exp δ+-- δ = ln ( (exp x + ε)/exp x )++sinh__ = continuousFlatFunction sinh'+ where sinh' x = (sinhx, eps2Delta)+ where eps2Delta ε = just $ asinh (abs sinhx + ε) - abs x+ sinhx = sinh x+cosh__ = continuousFlatFunction cosh'+ where cosh' x = (coshx, eps2Delta)+ where eps2Delta ε = just $ acosh (coshx + ε) - abs x+ coshx = cosh x+tanh__ = continuousFlatFunction tanh'+ where tanh' x = (tanhx, eps2Delta)+ where eps2Delta ε+ | ε >= 1 + abs tanhx = nothing+ | otherwise = just $ abs x - atanh (abs tanhx - ε)+ tanhx = tanh x+asinh__ = continuousFlatFunction asinh'+ where asinh' x = (asinhx, eps2Delta)+ where eps2Delta ε = just $ abs x - sinh (abs asinhx - ε)+ asinhx = asinh x+ ++cntnFuncsCombine :: forall d v c c' c'' ε ε' ε''. + ( FlatManifold c, FlatManifold c', FlatManifold c''+ , ε ~ Metric c , ε' ~ Metric c' , ε'' ~ Metric c'', ε~ε', ε~ε'' )+ => (c'->c''->(c, ε->(ε',ε''))) -> (d:-->c') -> (d:-->c'') -> d:-->c+cntnFuncsCombine cmb (Continuous f) (Continuous g) = Continuous h+ where h ζd u = case (ζc', ζc'') of + (IdChart, IdChart) + -> let (y, epsSplit) = cmb fu gu+ fullEps ε = fmap getMin $ (fmap Min $ fEps ε') + <>(fmap Min $ gEps ε'')+ where (ε', ε'') = epsSplit ε+ in (IdChart, y, fullEps)+ (IdChart, Chart c''In _ _)+ -> let (y'', c''Eps) = runFlatContinuous c''In gu+ (y, epsSplit) = cmb fu y''+ fullEps ε = fmap getMin $ (fmap Min $ fEps ε')+ <>(fmap Min $ gEps =<< c''Eps ε'')+ where (ε', ε'') = epsSplit ε+ in (IdChart, y, fullEps)+ (Chart c'In _ _, IdChart)+ -> let (y', c'Eps) = runFlatContinuous c'In fu + (y, epsSplit) = cmb y' gu+ fullEps ε = fmap getMin $ (fmap Min $ fEps =<< c'Eps ε') + <>(fmap Min $ gEps ε'')+ where (ε', ε'') = epsSplit ε+ in (IdChart, y, fullEps)+ (Chart c'In _ _, Chart c''In _ _)+ -> let (y', c'Eps) = runFlatContinuous c'In fu + (y'', c''Eps) = runFlatContinuous c''In gu + (y, epsSplit) = cmb y' y'' + fullEps ε = fmap getMin $ (fmap Min $ fEps =<< c'Eps ε') + <>(fmap Min $ gEps =<< c''Eps ε'')+ where (ε', ε'') = epsSplit ε+ in (IdChart, y, fullEps)+ where (ζc', fu, fEps) = f ζd u+ (ζc'',gu, gEps) = g ζd u+++data CntnFuncValue d c = CntnFuncValue { runCntnFuncValue :: d :--> c }+ | CntnFuncConst c++instance HasProxy (:-->) where+ type ProxyVal (:-->) d c = CntnFuncValue d c+ alg f = case f $ CntnFuncValue id of + CntnFuncValue q -> q+ CntnFuncConst c -> const__ c+ f $~ CntnFuncValue g = CntnFuncValue $ f . g+ f $~ CntnFuncConst c = CntnFuncConst $ f $ c++instance PointProxy CntnFuncValue (:-->) d c where+ point = CntnFuncConst++instance CartesianProxy (:-->) where+ alg1to2 f = case f $ CntnFuncValue id of+ (CntnFuncConst c₁, CntnFuncConst c₂) -> const__ (c₁, c₂)+ (CntnFuncConst c₁, CntnFuncValue f₂)+ -> Continuous $ \IdChart x -> let (fx, epsP) = runFlatContinuous f₂ x+ in (IdChart, (c₁, fx), epsP) + (CntnFuncValue f₁, CntnFuncConst c₂)+ -> Continuous $ \IdChart x -> let (fx, epsP) = runFlatContinuous f₁ x+ in (IdChart, (fx, c₂), epsP) + (CntnFuncValue f₁, CntnFuncValue f₂) -> f₁ &&& f₂ + alg2to1 f = case f (CntnFuncValue fst) (CntnFuncValue snd) of+ CntnFuncConst c -> const__ c+ CntnFuncValue f -> f+ alg2to2 f = case f (CntnFuncValue fst) (CntnFuncValue snd) of+ (CntnFuncConst c₁, CntnFuncConst c₂) -> const__ (c₁, c₂)+ (CntnFuncConst c₁, CntnFuncValue f₂)+ -> Continuous $ \IdChart x -> let (fx, epsP) = runFlatContinuous f₂ x+ in (IdChart, (c₁, fx), epsP) + (CntnFuncValue f₁, CntnFuncConst c₂)+ -> Continuous $ \IdChart x -> let (fx, epsP) = runFlatContinuous f₁ x+ in (IdChart, (fx, c₂), epsP) + (CntnFuncValue f₁, CntnFuncValue f₂) -> f₁ &&& f₂ ++++cntnFnValsFunc :: ( FlatManifold c, FlatManifold c', Manifold d+ , ε~Metric c, ε~Metric c' )+ => (c' -> (c, ε->Option ε)) -> CntnFuncValue d c' -> CntnFuncValue d c+cntnFnValsFunc = ($~) . continuousFlatFunction++cntnFnValsCombine :: forall d c c' c'' ε ε' ε''. + ( FlatManifold c, FlatManifold c', FlatManifold c'', Manifold d+ , ε ~ Metric c , ε' ~ Metric c' , ε'' ~ Metric c'', ε~ε', ε~ε'' )+ => ( c' -> c'' -> (c, ε -> (ε',(ε',ε''),ε'')) )+ -> CntnFuncValue d c' -> CntnFuncValue d c'' -> CntnFuncValue d c+cntnFnValsCombine cmb (CntnFuncValue f) (CntnFuncValue g) + = CntnFuncValue $ cntnFuncsCombine (second (>>> \(_,splε,_)->splε) .: cmb) f g+cntnFnValsCombine cmb (CntnFuncConst p) (CntnFuncConst q) + = CntnFuncConst . fst $ cmb p q+cntnFnValsCombine cmb f (CntnFuncConst q) + = cntnFnValsFunc (\c' -> second (>>> \(ε',_,_)->return ε') $ cmb c' q) f+cntnFnValsCombine cmb (CntnFuncConst p) g+ = cntnFnValsFunc (second (>>> \(_,_,ε'')->return ε'') . cmb p) g++instance (Representsℝ r, Manifold d) => Num (CntnFuncValue d r) where+ fromInteger = point . fromInteger+ + (+) = cntnFnValsCombine $ \a b -> (a+b, \ε -> (ε, (ε/2,ε/2), ε))+ (-) = cntnFnValsCombine $ \a b -> (a-b, \ε -> (ε, (ε/2,ε/2), ε))+ + (*) = cntnFnValsCombine $ \a b -> (a*b, + \ε -> ( ε/b+ , (ε / (2 * sqrt(2*b^2+ε)), ε / (2 * sqrt(2*a^2+ε)))+ , ε/a ))+ -- |δa| < ε / 2·sqrt(2·b² + ε) ∧ |δb| < ε / 2·sqrt(2·a² + ε)+ -- ⇒ | (a+δa) · (b+δb) - a·b | = | a·δb + b·δa + δa·δb | + -- ≤ | a·δb | + | b·δa | + | δa·δb |+ -- ≤ | a·ε/2·sqrt(2·a² + ε) | + | b·ε/2·sqrt(2·b² + ε) | + | ε² / 4·sqrt(2·b² + ε)·sqrt(2·a² + ε) |+ -- ≤ | a·ε/2·sqrt(2·a²) | + | b·ε/2·sqrt(2·b²) | + | ε² / 4·sqrt(ε)·sqrt(ε) |+ -- ≤ | ε/sqrt(8) | + | ε/sqrt(8) | + | ε / 4 |+ -- ≈ .96·ε < ε++ negate = cntnFnValsFunc $ \x -> (negate x, return)+ abs = cntnFnValsFunc $ \x -> (abs x, return)+ signum = cntnFnValsFunc $ \x -> (signum x, \ε -> if ε>2 then nothing else just $ abs x)++instance (Representsℝ r, Manifold d) => Fractional (CntnFuncValue d r) where+ fromRational = point . fromRational+ recip = cntnFnValsFunc $ \x -> let x¹ = recip x+ in (x¹, \ε -> just $ abs x - recip(ε + abs x¹))+ -- Readily derived from the worst-case of ε = 1 / (|x| – δ) – 1/|x|.++instance (Representsℝ r, Manifold d) => Floating (CntnFuncValue d r) where+ pi = point pi+ + exp x = exp__$~ x+ sin x = sin__$~ x+ cos x = cos__$~ x+ atan x = atan__$~ x+ sinh x = sinh__$~ x+ cosh x = cosh__$~ x+ tanh x = tanh__$~ x+ asinh x = asinh__$~ x+ + log x = continuousFlatFunction ln' $~ x+ where ln' x = (lnx, eps2Delta)+ where lnx = log x+ eps2Delta ε = just $ x - exp (lnx - ε)+ asin x = continuousFlatFunction asin' $~ x+ where asin' x = (asinx, eps2Delta)+ where asinx = asin x+ eps2Delta ε = just $ + if ε > pi/2 - abs asinx+ then 1 - abs x+ else sin (abs asinx + ε) - abs x+ acos x = continuousFlatFunction acos' $~ x+ where acos' x = (acosx, eps2Delta)+ where acosx = acos x+ eps2Delta ε = just $ + if ε > pi/2 - abs (acosx - pi/2)+ then 1 - abs x+ else cos (abs acosx + ε) - abs x+ acosh x = continuousFlatFunction acosh' $~ x+ where acosh' x = (acoshx, eps2Delta)+ where acoshx = acosh x+ eps2Delta ε = just $ + if ε > acoshx+ then x - 1+ else x - cosh (acoshx - ε)+ atanh x = continuousFlatFunction atanh' $~ x+ where atanh' x = (atanhx, eps2Delta)+ where atanhx = atanh x+ eps2Delta ε = just $ tanh (abs atanhx + ε) - abs x+++instance (FlatManifold v, Manifold d) => AdditiveGroup (CntnFuncValue d v) where+ zeroV = point zeroV+ (^+^) = cntnFnValsCombine $ \a b -> (a^+^b, \ε -> (ε, (ε/2,ε/2), ε))+ negateV = cntnFnValsFunc $ \x -> (negateV x, return)++instance ( FlatManifold v, MetricSpace v, Metric v~ℝ, FlatManifold (Scalar v)+ , MetricSpace (Scalar v), Metric (Scalar v) ~ ℝ, Manifold d ) + => VectorSpace (CntnFuncValue d v) where+ type Scalar (CntnFuncValue d v) = CntnFuncValue d (Scalar v)+ (*^) = cntnFnValsCombine + $ \λ v -> ( λ*^v+ , \ε -> let l = metric v+ λ' = metric λ+ in ( ε/l+ , ( ε / (2 * sqrt(2 * l^2 + ε))+ , ε / (2 * sqrt(2 * λ'^2 + ε)))+ , ε / λ' ))+ + +++++++++++data GraphWindowSpec = GraphWindowSpec {+ lBound, rBound, bBound, tBound :: Double+ , xResolution, yResolution :: Int+ }++finiteGraphContinℝtoℝ :: GraphWindowSpec -> (Double:-->Double) -> [(Double, Double)]+finiteGraphContinℝtoℝ (GraphWindowSpec{..}) fc+ = connect [(x, f x, δyG) | x<-[lBound, rBound] ] [(rBound, fst (f rBound))]+ where connect [(x₁, (y₁, eps₁), ε₁), (x₂, (y₂, eps₂), ε₂)]+ = case (getOption $ eps₁ ε₁, getOption $ eps₂ ε₂) of+ (Nothing, Nothing) -> done+ (Just δ₁, Nothing) | δ₁>δxS -> done+ | otherwise -> refine+ (Nothing, Just δ₂) | δ₂>δxS -> done+ | otherwise -> refine+ (Just δ₁, Just δ₂) | δ₁>δxS, δ₂>δxS -> done+ | otherwise -> refine+ where δxS = x₂-x₁+ m = x₁ + δxS/2+ fm@(ym, _) = f m+ done = ((x₁, y₁) :)+ refine = connect [(x₁, (y₁, eps₁), ε₁), (m, fm, ε')]+ . connect [(m, fm, ε'), (x₂, (y₂, eps₂), ε₂)]+ ε' = (if δxS < δxG then max (min (abs $ ym - y₁) (abs $ ym - y₂)) else id)+ $ max ε₁ ε₂+ f = runFlatContinuous fc+ δxG = (rBound - lBound) / fromIntegral xResolution+ δyG = (tBound - bBound) / fromIntegral yResolution+++finiteGraphContinℝtoℝ² :: GraphWindowSpec -> (Double:-->(Double, Double)) -> [[(Double, Double)]]+finiteGraphContinℝtoℝ² (GraphWindowSpec{..}) fc+ = map (\(tl, tu) -> reCoarsen $ connect (tl, f tl) (tu, f tu) [fst (f tu)]) segments+ where connect n₁@(t₁, (p₁, eps₁)) n₂@(t₂, (p₂, eps₂)) + | and . catMaybes $ map (getOption . fmap( > t₂ - t₁ ) . ($reso)) [eps₁, eps₂] + = (p₁ : )+ | m <- (id &&& f) $ midBetween [t₁, t₂] = connect n₁ m . connect m n₂++ segments = do+ (start, dir) <- [ (Just 0 , -1)+ , (go (\_ -> not . inRange) reasonable 1 0, 1 ) ]+ foldMap (`explore`dir) start+ where explore t₀ dir+ | Just ti <- go (\_ -> inRange) reasonable dir t₀+ , Just tb <- exitWindow (-dir) ti+ , Just te <- exitWindow dir ti+ = (if dir > 0 then (tb, te) else (te, tb)) : explore te dir+ | otherwise = []+ where exitWindow = go (\t p -> not $ reasonable t && inRange p) (const True)+ go isDone hasHope dir t+ | not $ hasHope t = Nothing+ | isDone t p = Just t+ | Just s <- getOption(epsP $ mobility p)+ = go isDone hasHope dir $ t + dir * s+ | otherwise = Nothing+ where (p, epsP) = f t++ f = runFlatContinuous fc+ inRange (x, y) = x > lBound && x < rBound && y > bBound && y < tBound+ reasonable = (< 1e+250) . abs+ mobility = \p -> sqrt $ max (distanceSq p cp₁) (distanceSq p cp₂) + where cp₁ = ( midBetween[lBound, rBound, rBound], midBetween[bBound, tBound, tBound] )+ cp₂ = ( midBetween[lBound, lBound, rBound], midBetween[bBound, bBound, tBound] )+ resoSq = reso ^ 2+ reso = min ( (rBound - lBound) / fromIntegral xResolution )+ ( (tBound - bBound) / fromIntegral yResolution ) * 2+ firstJust = head . catMaybes++ reCoarsen (p₁ : p₂ : ps)+ | distanceSq p₁ p₂ > resoSq = p₁ : reCoarsen (p₂ : ps)+ | otherwise = reCoarsen (p₁ : ps)+ reCoarsen ps = ps+++ + + +midBetween :: (VectorSpace v, Fractional(Scalar v)) => [v] -> v+midBetween vs = sumV vs ^/ (fromIntegral $ length vs)+++++data S2 = S2 { ϑParamS2 :: Double -- [0, π[+ , φParamS2 :: Double -- [0, 2π[+ }+ ++-- instance Manifold S2 where+-- type TangentSpace S2 = (Double, Double)+-- localAtlas (S2 ϑ φ)+-- | ϑ<pi-2 = [ Chart (\(x,y)+-- -> S2(2 * sqrt(x^2+y^2)) (atan2 y x) )+-- (\(S2 ϑ' φ')+-- -> let r=ϑ'/2+-- in guard (r<1) >> Just (r * cos φ', r * sin φ') )+-- LandlockedChart ]+-- | ϑ>2 = [ Chart (\(x,y)+-- -> S2(pi - 2*sqrt(x^2+y^2)) (atan2 y x) )+-- (\(S2 ϑ' φ')+-- -> let r=(pi-ϑ')/2+-- in guard (r<1) >> Just (r * cos φ', r * sin φ') )+-- LandlockedChart ]+-- | otherwise = localAtlas(S2 0 φ) ++ localAtlas(S2 (2*pi) φ)+-- ++++++type Endomorphism a = a->a+++(.:) :: (c->d) -> (a->b->c) -> a->b->d +(.:) = (.) . (.)+++just = Option . Just+nothing = Option Nothing++++type ℝ = Double++instance VectorSpace () where+ type Scalar () = ℝ+ _ *^ () = ()++instance HasBasis () where+ type Basis () = Void+ basisValue = absurd+ decompose () = []+ decompose' () = absurd+instance InnerSpace () where+ () <.> () = 0++class (RealFloat (Metric v), InnerSpace v) => MetricSpace v where+ type Metric v :: *+ type Metric v = ℝ+ metric :: v -> Metric v+ metric = sqrt . metricSq+ metricSq :: v -> Metric v+ metricSq = (^2) . metric+ (|*^) :: Metric v -> v -> v+ μ |*^ v = metricToScalar v μ *^ v + metricToScalar :: v -> Metric v -> Scalar v+ ++instance MetricSpace () where+ metric = const 0+ metricToScalar = const id+instance MetricSpace ℝ where+ metric = id+ metricToScalar = const id+instance ( RealFloat r, MetricSpace r, Scalar (Complex r)~Metric r ) + => MetricSpace (Complex r) where+ type Metric (Complex r) = Metric r+ metricSq (a :+ b) = metricSq a + metricSq b+ metricToScalar = const id+instance ( MetricSpace v, MetricSpace (Scalar v)+ , MetricSpace w, Scalar v~Scalar w+ , Metric v~Metric (Scalar v), Metric w~Metric v+ , Metric(Scalar w)~Metric v, RealFloat (Metric v)+ ) => MetricSpace (v,w) where+ type Metric (v,w) = Metric v+ metricSq (v,w) = metric (magnitudeSq v) + metric (magnitudeSq w)+ metricToScalar (v,_) = metricToScalar v++++++++(^) :: Num a => a -> Int -> a+(^) = (Prelude.^)+
+ Setup.hs view
@@ -0,0 +1,6 @@+module Main (main) where++import Distribution.Simple++main :: IO ()+main = defaultMain
+ Util/Associate.hs view
@@ -0,0 +1,57 @@+module Util.Associate where++++-- | Unlike the related 'zipWith', 'associateWith' \"spreads out\" the shorter+-- list by duplicating elements, before merging, to minimise the number of+-- elements from the longer list which aren't used.+associateWith :: (a->b->c) -> [a] -> [b] -> [c]+associateWith f a b+ | lb>la = spreadn(lb`quot`la) f a b+ | otherwise = spreadn(la`quot`lb) (flip f) b a+ where la = length a; lb = length b+ spreadn n f' = go+ where go (e:es) t+ | (et, tr) <- splitAt n t + = foldr((:) . f' e) (go es tr) et+ go _ _ = []+ +-- | @associate = associateWith (,)@.+associate :: [a] -> [b] -> [(a,b)]+associate = associateWith (,)++associaterSectorsWith :: (a->[b]->c) -> [a] -> [b] -> [c]+associaterSectorsWith f a b = spreadn(lb`quot`la) a b+ where la = length a; lb = length b+ spreadn n (e:es) t+ | (et, tr) <- splitAt n t = f e et : spreadn n es tr+ spreadn _ _ _ = []++associaterSectors :: [a] -> [b] -> [(a,[b])]+associaterSectors = associaterSectorsWith (,)+ +associatelSectorsWith :: ([a]->b->c) -> [a] -> [b] -> [c]+associatelSectorsWith f = flip (associaterSectorsWith $ flip f)++associatelSectors :: [a] -> [b] -> [([a],b)]+associatelSectors = associatelSectorsWith (,)++partitions :: Int -> [a] -> [[a]]+partitions n = go+ where go [] = []+ go l | (chunk,rest) <- splitAt n l = chunk : go rest++divide :: Int -> [a] -> [[a]]+divide n ls = partitions(length ls`div`n) ls+ + +mapOnNth :: (a->a) -> Int -> [a] -> [a]+mapOnNth f 0 (l:ls) = f l : ls+mapOnNth f n (l:ls) = l : mapOnNth f (n-1) ls+mapOnNth _ _ [] = []++mapExceptOnNth :: (a->a) -> Int -> [a] -> [a]+mapExceptOnNth f 0 (l:ls) = l : map f ls+mapExceptOnNth f n (l:ls) = f l : mapOnNth f (n-1) ls+mapExceptOnNth _ _ [] = []+
+ Util/LtdShow.hs view
@@ -0,0 +1,64 @@+{-# LANGUAGE FlexibleInstances #-}+{-# LANGUAGE ConstraintKinds #-}++++module Util.LtdShow (LtdShow(..)) where++import qualified Data.Vector as V+import Data.Vector(fromList, toList, (!), singleton)++type Array = V.Vector+++class LtdShow s where+ ltdShow :: Int -> s -> String++ltdShows :: LtdShow s => Int -> s -> ShowS+ltdShows n o s = ltdShow n o ++ s++ltdPrint :: LtdShow s => Int -> s -> IO()+ltdPrint n = putStrLn . ltdShow n++newtype LtdShowT a = LtdShow { runLtdShow :: a }++instance (Show a) => LtdShow ( LtdShowT a ) where+ ltdShow n = go "" (n*16) . show . runLtdShow where+ go ('{':um) 0 _ = "..}" ++ go um 0 []+ go ('[':um) 0 _ = "..]" ++ go um 0 []+ go ('(':um) 0 _ = "..)" ++ go um 0 []+ go [] n _ | n<=0 = "..."+ go unmatched n (c:cs)+ | c `elem` "([{" = c : go (c:unmatched) (n-8) cs+ go ('{':um) n ('}':cs) = '}' : go um (n-1) cs+ go ('[':um) n (']':cs) = ']' : go um (n-1) cs+ go ('(':um) n (')':cs) = ')' : go um (n-1) cs+ go unmatched n (c:cs) = c : go unmatched n' cs+ where n' | c`elem`(['a'..'z']++['A'..'Z']++['0'..'9']) = n-1+ | otherwise = n-8+ go [] _ "" = ""+ ++instance (LtdShow s) => LtdShow (Array s) where+ ltdShow n arr + | n<=1, l>0 = "[∘∘{" ++ show l ++ "}∘∘]"+ | otherwise = ('[':) . V.foldr (("∘ "++).) " ∘]"+ . V.imap(\i -> ltdShows $ round(+ fromIntegral n + * 2**(-1 - sqrt(fromIntegral i)) ))+ $ arr+ where l = V.length arr+ +instance (LtdShow l, LtdShow r) => LtdShow (l,r) where+ ltdShow n (l, r) = "(" ++ pShow l ++ ", " ++ pShow r ++ ")"+ where pShow :: LtdShow s => s->String+ pShow = ltdShow $ n`quot`2+++instance (Show p) => LtdShow [p] where+ ltdShow n l = "[" ++ lsh' n l "]"+ where lsh' 0 _ = ("... "++)+ lsh' _ [] = id+ lsh' n (x:xs) = ((show x ++ ", ") ++) . lsh' (n-1) xs++
+ manifolds.cabal view
@@ -0,0 +1,61 @@+Name: manifolds+Version: 0.1.0.0+Category: Math+Synopsis: Working with manifolds in a direct, embedding-free way.+Description: Manifolds, a generalisation of the notion of \"smooth curves\" or sufaces,+ are topological spaces /locally homeomorphic to a vector space/. This gives+ rise to what is actually the most natural / mathematically elegant way of dealing+ with them: calculations can be carried out locally, in connection with Riemannian+ products etc., in a vector space, the tangent space / tangent bundle.+ .+ However, this does not trivially translate to non-local operations. Common+ ways to carry those out include using a single affine map to cover (almost) all of the manifold+ (in general not possible homeomorphically, which leads to both topological and geometrical+ problems), to embed the manifold into a larger-dimensional vector space (which tends+ to distract from the manifold's own properties and is often not friendly to computations)+ or approximating the manifold by some kind of finite simplicial mesh (which intrinsically+ introduces non-differentiability issues and leads to the question of what precision+ is required).+ .+ This library tries to mitigate these problems by using Haskell's+ functional nature to keep the representation close to the mathematical ideal of+ local linearity with homeomorphic coordinate transforms, and, where it is+ necessary to recede to the less elegant alternatives, exploiting lazy evaluation+ etc. to optimise the compromises that have to be made.+License: GPL-3+License-file: COPYING+Author: Justus Sagemüller+Maintainer: (@) sagemueller $ geo.uni-koeln.de+Build-Type: Simple+Cabal-Version: >=1.10++Library+ Build-Depends: base>=4.5 && < 6+ , transformers+ , vector-space>=0.8+ , vector+ , vector-algorithms+ , containers+ , random+ , MonadRandom+ , comonad+ , semigroups+ , void+ , constrained-categories+ other-extensions: FlexibleInstances+ , TypeFamilies+ , FlexibleContexts+ , GADTs+ , RankNTypes+ , ConstraintKinds+ , PatternGuards+ , ScopedTypeVariables+ , RecordWildCards+ , TupleSections+ ghc-options: -O2+ Exposed-modules: Data.Manifold+ -- Data.Manifold.Visualisation.R3.GLUT+ Other-modules: Data.List.FastNub+ Util.Associate+ Util.LtdShow+ default-language: Haskell2010