diff --git a/COPYING b/COPYING new file mode 100644 --- /dev/null +++ b/COPYING @@ -0,0 +1,674 @@ + GNU GENERAL PUBLIC LICENSE + Version 3, 29 June 2007 + + Copyright (C) 2007 Free Software Foundation, Inc. + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + Preamble + + The GNU General Public License is a free, copyleft license for +software and other kinds of works. + + The licenses for most software and other practical works are designed +to take away your freedom to share and change the works. By contrast, +the GNU General Public License is intended to guarantee your freedom to +share and change all versions of a program--to make sure it remains free +software for all its users. We, the Free Software Foundation, use the +GNU General Public License for most of our software; it applies also to +any other work released this way by its authors. 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Interpretation of Sections 15 and 16. + + If the disclaimer of warranty and limitation of liability provided +above cannot be given local legal effect according to their terms, +reviewing courts shall apply local law that most closely approximates +an absolute waiver of all civil liability in connection with the +Program, unless a warranty or assumption of liability accompanies a +copy of the Program in return for a fee. + + END OF TERMS AND CONDITIONS + + How to Apply These Terms to Your New Programs + + If you develop a new program, and you want it to be of the greatest +possible use to the public, the best way to achieve this is to make it +free software which everyone can redistribute and change under these terms. + + To do so, attach the following notices to the program. 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If not, see . + +Also add information on how to contact you by electronic and paper mail. + + If the program does terminal interaction, make it output a short +notice like this when it starts in an interactive mode: + + Copyright (C) + This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'. + This is free software, and you are welcome to redistribute it + under certain conditions; type `show c' for details. + +The hypothetical commands `show w' and `show c' should show the appropriate +parts of the General Public License. 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 +. + + 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 +. diff --git a/Data/List/FastNub.hs b/Data/List/FastNub.hs new file mode 100644 --- /dev/null +++ b/Data/List/FastNub.hs @@ -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(:[]) diff --git a/Data/Manifold.hs b/Data/Manifold.hs new file mode 100644 --- /dev/null +++ b/Data/Manifold.hs @@ -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 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 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.^) + diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,6 @@ +module Main (main) where + +import Distribution.Simple + +main :: IO () +main = defaultMain diff --git a/Util/Associate.hs b/Util/Associate.hs new file mode 100644 --- /dev/null +++ b/Util/Associate.hs @@ -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 _ _ [] = [] + diff --git a/Util/LtdShow.hs b/Util/LtdShow.hs new file mode 100644 --- /dev/null +++ b/Util/LtdShow.hs @@ -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 + + diff --git a/manifolds.cabal b/manifolds.cabal new file mode 100644 --- /dev/null +++ b/manifolds.cabal @@ -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