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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/Math/Manifold/Core/PseudoAffine.hs b/Math/Manifold/Core/PseudoAffine.hs new file mode 100644 --- /dev/null +++ b/Math/Manifold/Core/PseudoAffine.hs @@ -0,0 +1,359 @@ +-- | +-- Module : Math.Manifold.Core.PseudoAffine +-- Copyright : (c) Justus Sagemüller 2016 +-- License : GPL v3 +-- +-- Maintainer : (@) sagemueller $ geo.uni-koeln.de +-- Stability : experimental +-- Portability : portable +-- + +{-# LANGUAGE FlexibleInstances #-} +{-# LANGUAGE UndecidableInstances #-} +{-# LANGUAGE TypeFamilies #-} +{-# LANGUAGE FlexibleContexts #-} +{-# LANGUAGE GADTs #-} +{-# LANGUAGE DefaultSignatures #-} +{-# LANGUAGE UnicodeSyntax #-} +{-# LANGUAGE ScopedTypeVariables #-} +{-# LANGUAGE CPP #-} + + +module Math.Manifold.Core.PseudoAffine where + +import Data.VectorSpace +import Data.AffineSpace + +import Data.Tagged +import Data.Fixed (mod') + +import Math.Manifold.Core.Types +import Math.Manifold.VectorSpace.ZeroDimensional + +import Control.Applicative + +data BoundarylessWitness m where + BoundarylessWitness :: (Semimanifold m, Interior m ~ m) + => BoundarylessWitness m + +-- | This is the reified form of the property that the interior of a semimanifold +-- is a manifold. These constraints would ideally be expressed directly as +-- superclass constraints, but that would require the @UndecidableSuperclasses@ +-- extension, which is not reliable yet. +-- +-- Also, if all those equality constraints are in scope, GHC tends to infer needlessly +-- complicated types like @'Interior' ('Interior' ('Needle' ('Interior' x)))@, which is +-- the same as just @'Needle' x@. +data SemimanifoldWitness x where + SemimanifoldWitness :: + ( Semimanifold (Needle x), Needle (Interior x) ~ Needle x + , Needle (Needle x) ~ Needle x + , Interior (Needle x) ~ Needle x ) + => BoundarylessWitness (Interior x) -> SemimanifoldWitness x + +data PseudoAffineWitness x where + PseudoAffineWitness :: + ( PseudoAffine (Interior x), PseudoAffine (Needle x) ) + => SemimanifoldWitness x -> PseudoAffineWitness x + +infix 6 .-~., .-~! +infixl 6 .+~^, .-~^ + +class AdditiveGroup (Needle x) => Semimanifold x where + {-# MINIMAL ((.+~^) | fromInterior), toInterior, translateP #-} + -- | The space of “natural” ways starting from some reference point + -- and going to some particular target point. Hence, + -- the name: like a compass needle, but also with an actual length. + -- For affine spaces, 'Needle' is simply the space of + -- line segments (aka vectors) between two points, i.e. the same as 'Diff'. + -- The 'AffineManifold' constraint makes that requirement explicit. + -- + -- This space should be isomorphic to the tangent space (and is in fact + -- used somewhat synonymously). + type Needle x :: * + + -- | Manifolds with boundary are a bit tricky. We support such manifolds, + -- but carry out most calculations only in “the fleshy part” – the + -- interior, which is an “infinite space”, so you can arbitrarily scale paths. + -- + -- The default implementation is @'Interior' x = x@, which corresponds + -- to a manifold that has no boundary to begin with. + type Interior x :: * + type Interior x = x + + -- | Generalised translation operation. Note that the result will always also + -- be in the interior; scaling up the needle can only get you ever /closer/ + -- to a boundary. + (.+~^) :: Interior x -> Needle x -> x + (.+~^) = addvp + where addvp :: ∀ x . Semimanifold x => Interior x -> Needle x -> x + addvp p = fromInterior . tp p + where (Tagged tp) = translateP :: Tagged x (Interior x -> Needle x -> Interior x) + + -- | 'id' sans boundary. + fromInterior :: Interior x -> x + fromInterior p = p .+~^ zeroV + + toInterior :: x -> Maybe (Interior x) + + -- | The signature of '.+~^' should really be @'Interior' x -> 'Needle' x -> 'Interior' x@, + -- only, this is not possible because it only consists of non-injective type families. + -- The solution is this tagged signature, which is of course rather unwieldy. That's + -- why '.+~^' has the stronger, but easier usable signature. Without boundary, these + -- functions should be equivalent, i.e. @translateP = Tagged (.+~^)@. + translateP :: Tagged x (Interior x -> Needle x -> Interior x) + + -- | Shorthand for @\\p v -> p .+~^ 'negateV' v@, which should obey the /asymptotic/ law + -- + -- @ + -- p .-~^ v .+~^ v ≅ p + -- @ + -- + -- Meaning: if @v@ is scaled down with sufficiently small factors /η/, then + -- the difference @(p.-~^v.+~^v) .-~. p@ should scale down even faster: + -- as /O/ (/η/²). For large vectors, it will however behave differently, + -- except in flat spaces (where all this should be equivalent to the 'AffineSpace' + -- instance). + (.-~^) :: Interior x -> Needle x -> x + p .-~^ v = p .+~^ negateV v + + semimanifoldWitness :: SemimanifoldWitness x + default semimanifoldWitness :: + ( Semimanifold (Interior x), Semimanifold (Needle x) + , Interior (Interior x) ~ Interior x, Needle (Interior x) ~ Needle x + , Needle (Needle x) ~ Needle x + , Interior (Needle x) ~ Needle x ) + => SemimanifoldWitness x + semimanifoldWitness = SemimanifoldWitness BoundarylessWitness + + +-- | This is the class underlying manifolds. ('Manifold' only precludes boundaries +-- and adds an extra constraint that would be circular if it was in a single +-- class. You can always just use 'Manifold' as a constraint in your signatures, +-- but you must /define/ only 'PseudoAffine' for manifold types – +-- the 'Manifold' instance follows universally from this, if @'Interior x ~ x@.) +-- +-- The interface is (boundaries aside) almost identical to the better-known +-- 'AffineSpace' class, but we don't require associativity of '.+~^' with '^+^' +-- – except in an /asymptotic sense/ for small vectors. +-- +-- That innocent-looking change makes the class applicable to vastly more general types: +-- while an affine space is basically nothing but a vector space without particularly +-- designated origin, a pseudo-affine space can have nontrivial topology on the global +-- scale, and yet be used in practically the same way as an affine space. At least the +-- usual spheres and tori make good instances, perhaps the class is in fact equivalent to +-- manifolds in their usual maths definition (with an atlas of charts: a family of +-- overlapping regions of the topological space, each homeomorphic to the 'Needle' +-- vector space or some simply-connected subset thereof). +class Semimanifold x => PseudoAffine x where + {-# MINIMAL (.-~.) | (.-~!) #-} + -- | The path reaching from one point to another. + -- Should only yield 'Nothing' if + -- + -- * The points are on disjoint segments of a non–path-connected space. + -- + -- * Either of the points is on the boundary. Use '|-~.' to deal with this. + -- + -- On manifolds, the identity + -- + -- @ + -- p .+~^ (q.-~.p) ≡ q + -- @ + -- + -- should hold, at least save for floating-point precision limits etc.. + -- + -- '.-~.' and '.+~^' only really work in manifolds without boundary. If you consider + -- the path between two points, one of which lies on the boundary, it can't really + -- be possible to scale this path any longer – it would have to reach “out of the + -- manifold”. To adress this problem, these functions basically consider only the + -- /interior/ of the space. + (.-~.) :: x -> x -> Maybe (Needle x) + p.-~.q = return $ p.-~!q + + -- | Unsafe version of '.-~.'. If the two points lie in disjoint regions, + -- the behaviour is undefined. + (.-~!) :: x -> x -> Needle x + p.-~!q = case p.-~.q of + Just v -> v + + pseudoAffineWitness :: PseudoAffineWitness x + default pseudoAffineWitness :: + ( PseudoAffine (Interior x), PseudoAffine (Needle x) ) + => PseudoAffineWitness x + pseudoAffineWitness = PseudoAffineWitness semimanifoldWitness + + + + + + + +-- | Interpolate between points, approximately linearly. For +-- points that aren't close neighbours (i.e. lie in an almost +-- flat region), the pathway is basically undefined – save for +-- its end points. +-- +-- A proper, really well-defined (on global scales) interpolation +-- only makes sense on a Riemannian manifold, as 'Data.Manifold.Riemannian.Geodesic'. +palerp :: ∀ x. (PseudoAffine x, VectorSpace (Needle x)) + => x -> x -> Maybe (Scalar (Needle x) -> x) +palerp p₀ p₁ = case (toInterior p₀, p₁.-~.p₀) of + (Just b, Just v) -> return $ \t -> b .+~^ t *^ v + _ -> Nothing + +-- | Like 'palerp', but actually restricted to the interval between the points, +-- with a signature like 'Data.Manifold.Riemannian.geodesicBetween' +-- rather than 'Data.AffineSpace.alerp'. +palerpB :: ∀ x. (PseudoAffine x, VectorSpace (Needle x), Scalar (Needle x) ~ ℝ) + => x -> x -> Maybe (D¹ -> x) +palerpB p₀ p₁ = case (toInterior p₀, p₁.-~.p₀) of + (Just b, Just v) -> return $ \(D¹ t) -> b .+~^ ((t+1)/2) *^ v + _ -> Nothing + +-- | Like 'alerp', but actually restricted to the interval between the points. +alerpB :: ∀ x. (AffineSpace x, VectorSpace (Diff x), Scalar (Diff x) ~ ℝ) + => x -> x -> D¹ -> x +alerpB p1 p2 = case p2 .-. p1 of + v -> \(D¹ t) -> p1 .+^ ((t+1)/2) *^ v + + + +hugeℝVal :: ℝ +hugeℝVal = 1e+100 + +#define deriveAffine(c,t) \ +instance (c) => Semimanifold (t) where { \ + type Needle (t) = Diff (t); \ + fromInterior = id; \ + toInterior = pure; \ + translateP = Tagged (.+^); \ + (.+~^) = (.+^) }; \ +instance (c) => PseudoAffine (t) where { \ + a.-~.b = pure (a.-.b); } + +deriveAffine((),Double) +deriveAffine((),Rational) + + +instance Semimanifold (ZeroDim k) where + type Needle (ZeroDim k) = ZeroDim k + fromInterior = id + toInterior = pure + Origin .+~^ Origin = Origin + Origin .-~^ Origin = Origin + translateP = Tagged (.+~^) +instance PseudoAffine (ZeroDim k) where + Origin .-~. Origin = pure Origin + +instance ∀ a b . (Semimanifold a, Semimanifold b) => Semimanifold (a,b) where + type Needle (a,b) = (Needle a, Needle b) + type Interior (a,b) = (Interior a, Interior b) + (a,b).+~^(v,w) = (a.+~^v, b.+~^w) + (a,b).-~^(v,w) = (a.-~^v, b.-~^w) + fromInterior (i,j) = (fromInterior i, fromInterior j) + toInterior (a,b) = (,) <$> toInterior a <*> toInterior b + translateP = Tagged $ \(a,b) (v,w) -> (ta a v, tb b w) + where Tagged ta = translateP :: Tagged a (Interior a -> Needle a -> Interior a) + Tagged tb = translateP :: Tagged b (Interior b -> Needle b -> Interior b) + semimanifoldWitness = case ( semimanifoldWitness :: SemimanifoldWitness a + , semimanifoldWitness :: SemimanifoldWitness b ) of + (SemimanifoldWitness BoundarylessWitness, SemimanifoldWitness BoundarylessWitness) + -> SemimanifoldWitness BoundarylessWitness +instance (PseudoAffine a, PseudoAffine b) => PseudoAffine (a,b) where + (a,b).-~.(c,d) = liftA2 (,) (a.-~.c) (b.-~.d) + pseudoAffineWitness = case ( pseudoAffineWitness :: PseudoAffineWitness a + , pseudoAffineWitness :: PseudoAffineWitness b ) of + ( PseudoAffineWitness (SemimanifoldWitness BoundarylessWitness) + , PseudoAffineWitness (SemimanifoldWitness BoundarylessWitness) ) + ->PseudoAffineWitness (SemimanifoldWitness BoundarylessWitness) + +instance ∀ a b c . (Semimanifold a, Semimanifold b, Semimanifold c) + => Semimanifold (a,b,c) where + type Needle (a,b,c) = (Needle a, Needle b, Needle c) + type Interior (a,b,c) = (Interior a, Interior b, Interior c) + (a,b,c).+~^(v,w,x) = (a.+~^v, b.+~^w, c.+~^x) + (a,b,c).-~^(v,w,x) = (a.-~^v, b.-~^w, c.-~^x) + fromInterior (i,j,k) = (fromInterior i, fromInterior j, fromInterior k) + toInterior (a,b,c) = liftA3 (,,) (toInterior a) (toInterior b) (toInterior c) + translateP = Tagged $ \(a,b,c) (v,w,x) -> (ta a v, tb b w, tc c x) + where Tagged ta = translateP :: Tagged a (Interior a -> Needle a -> Interior a) + Tagged tb = translateP :: Tagged b (Interior b -> Needle b -> Interior b) + Tagged tc = translateP :: Tagged c (Interior c -> Needle c -> Interior c) + semimanifoldWitness = case ( semimanifoldWitness :: SemimanifoldWitness a + , semimanifoldWitness :: SemimanifoldWitness b + , semimanifoldWitness :: SemimanifoldWitness c ) of + ( SemimanifoldWitness BoundarylessWitness + ,SemimanifoldWitness BoundarylessWitness + ,SemimanifoldWitness BoundarylessWitness ) + -> SemimanifoldWitness BoundarylessWitness +instance (PseudoAffine a, PseudoAffine b, PseudoAffine c) => PseudoAffine (a,b,c) where + (a,b,c).-~.(d,e,f) = liftA3 (,,) (a.-~.d) (b.-~.e) (c.-~.f) + pseudoAffineWitness = case ( pseudoAffineWitness :: PseudoAffineWitness a + , pseudoAffineWitness :: PseudoAffineWitness b + , pseudoAffineWitness :: PseudoAffineWitness c ) of + ( PseudoAffineWitness (SemimanifoldWitness BoundarylessWitness) + , PseudoAffineWitness (SemimanifoldWitness BoundarylessWitness) + , PseudoAffineWitness (SemimanifoldWitness BoundarylessWitness) ) + ->PseudoAffineWitness (SemimanifoldWitness BoundarylessWitness) + + + + +instance Semimanifold S⁰ where + type Needle S⁰ = ZeroDim ℝ + fromInterior = id + toInterior = pure + translateP = Tagged (.+~^) + p .+~^ Origin = p + p .-~^ Origin = p +instance PseudoAffine S⁰ where + PositiveHalfSphere .-~. PositiveHalfSphere = pure Origin + NegativeHalfSphere .-~. NegativeHalfSphere = pure Origin + _ .-~. _ = Nothing + +instance Semimanifold S¹ where + type Needle S¹ = ℝ + fromInterior = id + toInterior = pure + translateP = Tagged (.+~^) + S¹ φ₀ .+~^ δφ + | φ' < 0 = S¹ $ φ' + tau + | otherwise = S¹ $ φ' + where φ' = toS¹range $ φ₀ + δφ +instance PseudoAffine S¹ where + S¹ φ₁ .-~. S¹ φ₀ + | δφ > pi = pure (δφ - 2*pi) + | δφ < (-pi) = pure (δφ + 2*pi) + | otherwise = pure δφ + where δφ = φ₁ - φ₀ + +instance Semimanifold D¹ where + type Needle D¹ = ℝ + type Interior D¹ = ℝ + fromInterior = D¹ . tanh + toInterior (D¹ x) | abs x < 1 = return $ atanh x + | otherwise = empty + translateP = Tagged (+) +instance PseudoAffine D¹ where + D¹ 1 .-~. _ = empty + D¹ (-1) .-~. _ = empty + D¹ x .-~. D¹ y + | abs x < 1, abs y < 1 = return $ atanh x - atanh y + | otherwise = empty + + + + + + + + +tau :: ℝ +tau = 2 * pi + +toS¹range :: ℝ -> ℝ +toS¹range φ = (φ+pi)`mod'`tau - pi + + + + diff --git a/Math/Manifold/Core/Types.hs b/Math/Manifold/Core/Types.hs new file mode 100644 --- /dev/null +++ b/Math/Manifold/Core/Types.hs @@ -0,0 +1,83 @@ +-- | +-- Module : Math.Manifold.Core.Types +-- Copyright : (c) Justus Sagemüller 2016 +-- License : GPL v3 +-- +-- Maintainer : (@) sagemueller $ geo.uni-koeln.de +-- Stability : experimental +-- Portability : portable +-- +-- Several low-dimensional manifolds, represented in some simple way as Haskell +-- data types. All these are in the 'PseudoAffine' class. +-- + + +{-# LANGUAGE TypeFamilies #-} + + +module Math.Manifold.Core.Types where + + +import Data.VectorSpace +import Math.Manifold.VectorSpace.ZeroDimensional +import Data.AffineSpace +import Data.Basis +import Data.Void +import Data.Monoid + +import Control.Applicative (Const(..), Alternative(..)) + + + + +-- | The zero-dimensional sphere is actually just two points. Implementation might +-- therefore change to @ℝ⁰ 'Control.Category.Constrained.+' ℝ⁰@: the disjoint sum of two +-- single-point spaces. +data S⁰ = PositiveHalfSphere | NegativeHalfSphere deriving(Eq, Show) + +otherHalfSphere :: S⁰ -> S⁰ +otherHalfSphere PositiveHalfSphere = NegativeHalfSphere +otherHalfSphere NegativeHalfSphere = PositiveHalfSphere + +-- | The unit circle. +newtype S¹ = S¹ { φParamS¹ :: Double -- ^ Must be in range @[-π, π[@. + } deriving (Show) + + + + +type ℝP¹ = S¹ + + + + +-- | The “one-dimensional disk” – really just the line segment between +-- the two points -1 and 1 of 'S⁰', i.e. this is simply a closed interval. +newtype D¹ = D¹ { xParamD¹ :: Double -- ^ Range @[-1, 1]@. + } deriving (Show) +fromIntv0to1 :: ℝ -> D¹ +fromIntv0to1 x | x<0 = D¹ (-1) + | x>1 = D¹ 1 + | otherwise = D¹ $ x*2 - 1 + + + +type ℝ = Double +type ℝ⁰ = ZeroDim ℝ + + + + +instance VectorSpace () where + type Scalar () = ℝ + _ *^ () = () + +instance HasBasis () where + type Basis () = Void + basisValue = absurd + decompose () = [] + decompose' () = absurd +instance InnerSpace () where + () <.> () = 0 + + diff --git a/Math/Manifold/VectorSpace/ZeroDimensional.hs b/Math/Manifold/VectorSpace/ZeroDimensional.hs new file mode 100644 --- /dev/null +++ b/Math/Manifold/VectorSpace/ZeroDimensional.hs @@ -0,0 +1,44 @@ +-- | +-- Module : Math.Manifold.VectorSpace.ZeroDimensional +-- Copyright : (c) Justus Sagemüller 2016 +-- License : GPL v3 +-- +-- Maintainer : (@) sagemueller $ geo.uni-koeln.de +-- Stability : experimental +-- Portability : portable +-- +{-# LANGUAGE TypeFamilies #-} + +module Math.Manifold.VectorSpace.ZeroDimensional ( + ZeroDim (..) + ) where + +import Data.AffineSpace +import Data.VectorSpace +import Data.Basis +import Data.Void + + + +data ZeroDim s = Origin + +instance Monoid (ZeroDim s) where + mempty = Origin + mappend Origin Origin = Origin + +instance AffineSpace (ZeroDim s) where + type Diff (ZeroDim s) = ZeroDim s + Origin .+^ Origin = Origin + Origin .-. Origin = Origin +instance AdditiveGroup (ZeroDim s) where + zeroV = Origin + Origin ^+^ Origin = Origin + negateV Origin = Origin +instance VectorSpace (ZeroDim s) where + type Scalar (ZeroDim s) = s + _ *^ Origin = Origin +instance HasBasis (ZeroDim s) where + type Basis (ZeroDim k) = Void + basisValue = absurd + decompose Origin = [] + decompose' Origin = absurd diff --git a/Setup.hs b/Setup.hs new file mode 100644 --- /dev/null +++ b/Setup.hs @@ -0,0 +1,2 @@ +import Distribution.Simple +main = defaultMain diff --git a/manifolds-core.cabal b/manifolds-core.cabal new file mode 100644 --- /dev/null +++ b/manifolds-core.cabal @@ -0,0 +1,30 @@ +-- Initial manifolds-core.cabal generated by cabal init. For further +-- documentation, see http://haskell.org/cabal/users-guide/ + +name: manifolds-core +version: 0.4.0.0 +synopsis: The basic classes for the manifolds hierarchy. +description: The basic classes for the + hierarchy. +homepage: https://github.com/leftaroundabout/manifolds +license: GPL-3 +license-file: LICENSE +author: Justus Sagemüller +maintainer: (@) jsagemue $ uni-koeln.de +-- copyright: +category: Math +build-type: Simple +-- extra-source-files: +cabal-version: >=1.10 + +library + exposed-modules: Math.Manifold.Core.Types + , Math.Manifold.Core.PseudoAffine + , Math.Manifold.VectorSpace.ZeroDimensional + -- other-modules: + other-extensions: FlexibleInstances, UndecidableInstances, ExplicitNamespaces, TypeFamilies, FunctionalDependencies, FlexibleContexts, GADTs, RankNTypes, TupleSections, ConstraintKinds, PatternGuards, TypeOperators, ScopedTypeVariables, RecordWildCards, DataKinds, StandaloneDeriving, DefaultSignatures, UnicodeSyntax, MultiWayIf + build-depends: base >=4.5 && <5 + , vector-space >=0.8 + , tagged + -- hs-source-dirs: + default-language: Haskell2010