diff --git a/CHANGELOG.md b/CHANGELOG.md new file mode 100644 --- /dev/null +++ b/CHANGELOG.md @@ -0,0 +1,7 @@ +# Change log + +Tropical uses [Semantic Versioning][]. +The change log is available through the [releases on GitHub][]. + +[Semantic Versioning]: http://semver.org/spec/v2.0.0.html +[releases on GitHub]: https://github.com/ferynando7/Tropical/releases diff --git a/LICENSE b/LICENSE new file mode 100644 --- /dev/null +++ b/LICENSE @@ -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. 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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/README.md b/README.md new file mode 100644 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +# Tropical +Haskell Library for Tropical Algebra diff --git a/benchmark/Main.hs b/benchmark/Main.hs new file mode 100644 --- /dev/null +++ b/benchmark/Main.hs @@ -0,0 +1,6 @@ +-- You can benchmark your code quickly and effectively with Criterion. See its +-- website for help: . +import Criterion.Main + +main :: IO () +main = defaultMain [bench "const" (whnf const ())] diff --git a/executable/Main.hs b/executable/Main.hs new file mode 100644 --- /dev/null +++ b/executable/Main.hs @@ -0,0 +1,27 @@ +-- It is generally a good idea to keep all your business logic in your library +-- and only use it in the executable. Doing so allows others to use what you +-- wrote in their libraries. + +{-#LANGUAGE DataKinds #-} + +module Main where + +import Core + + +x, y :: Polynomial (Tropical Integer) Lex 2 +x = variable 0 +y = variable 1 + +f1,f2,f3,f4,f5 :: Polynomial (Tropical Integer) Lex 2 + +f1 = 1*x^2 + x*y + 1*y^2 + x + y + 2 +f2 = 3*x^2 + x*y + 3*y^2 + 1*x + 1*y + 0 +f3 = 3*x^3 + 1*x^2*y + 1*x*y^2 + 3*y^3 + 1*x^2 + x*y + 1*y^2 + 1*x + 1*y + 3 +f4 = x^3 + x^2*y + x*y^2 + y^3 + x^2 + x*y + y^2 + x + y + 0 +f5 = 2*x*y^^(-1) + 2*y^^(-1) + (-2) + +res = makeFigs [f1,f2] + +main :: IO () +main = putStrLn "Hello worls" diff --git a/library/Arithmetic/Matrix.hs b/library/Arithmetic/Matrix.hs new file mode 100644 --- /dev/null +++ b/library/Arithmetic/Matrix.hs @@ -0,0 +1,27 @@ +module Arithmetic.Matrix where + +import Numeric.Algebra hiding ((+), (*)) +import Data.List +import Data.Function (on) +-- import Data.Matrix hiding (transpose, zero) + +newtype TMatrix a = TMatrix {toList :: [[a]]} -- ^ "a" should be Tropical + deriving (Eq, Show) + +-- | This should be fixed with Functor instance +fmap' :: ([[a]] -> [[b]]) -> TMatrix a -> TMatrix b +fmap' f (TMatrix a) = TMatrix (f a) + +transp :: TMatrix a -> TMatrix a +transp = fmap' transpose + +msum :: (Num a) => TMatrix a-> TMatrix a -> TMatrix a +msum a b = TMatrix $ (zipWith (zipWith (+)) `on` toList) a b + +mmult :: (Rig a, Num a) => TMatrix a -> TMatrix a -> TMatrix a +mmult a b = TMatrix [[ foldr (+) zero $ zipWith (*) ar bc | bc <- transpose (toList b)] | ar <- toList a ] + + +instance (Num a, Rig a) => Num (TMatrix a) where + (+) = msum + (*) = mmult diff --git a/library/Arithmetic/Numbers.hs b/library/Arithmetic/Numbers.hs new file mode 100644 --- /dev/null +++ b/library/Arithmetic/Numbers.hs @@ -0,0 +1,94 @@ +{-# LANGUAGE FlexibleInstances, MultiParamTypeClasses #-} +{-# LANGUAGE UndecidableInstances #-} + +module Arithmetic.Numbers +( + -- * Types + Tropical(..) +) where + +import Data.Function +import Numeric.Algebra hiding (negate, fromInteger) +import Numeric.Algebra.Class as AC + + + + +--The set includes all the real numbers and the infinity +data Tropical a = Tropical {value :: a} | Inf + deriving (Eq) + + +instance (Show a) => Show (Tropical a) where + show Inf = "Inf" + show (Tropical a) = show a + + + +(.+.) :: (Ord a) => Tropical a -> Tropical a -> Tropical a +Inf .+. t = t +t .+. Inf = t +t1 .+. t2 = Tropical $ (min `on` value) t1 t2 + +(.*.) :: (Num a) => Tropical a -> Tropical a -> Tropical a +t .*. Inf = Inf +Inf .*. t = Inf +t1 .*. t2 = Tropical $ ((Prelude.+) `on` value) t1 t2 + + +instance (Num a) => Unital (Tropical a) where + one = Tropical 0 + + +instance (Num a, Ord a, Rig a) => LeftModule Natural (Tropical a) where + a .* t1 = Tropical (fromNatural a) AC.* t1 + +instance (Num a, Ord a, Rig a) => RightModule Natural (Tropical a) where + (*.) = flip (.*) + + +instance (Num a, Ord a, Rig a) => Monoidal (Tropical a) where + zero = Inf + +instance (Num a) => Multiplicative (Tropical a) where + (*) = (.*.) + +instance (Num a) => Commutative (Tropical a) + +instance (Rig a, Num a, Ord a) => Rig (Tropical a) where + fromNatural a = Tropical $ fromNatural a + +instance (Ord a) => Additive (Tropical a) where + (+) = (.+.) +instance (Ord a) => Abelian (Tropical a) +instance (Ord a, Num a) => Semiring (Tropical a) + +instance (Ord a, Num a) => Num (Tropical a) where + (+) = (.+.) + (*) = (.*.) + fromInteger a = Tropical $ Prelude.fromInteger a + negate = fmap negate + +-- | Not neccesary to implement since it is already defined +instance (Num a, Ord a, Rig a) => DecidableZero (Tropical a) +instance Functor Tropical where + fmap f (Tropical a) = Tropical (f a) + + +instance Integral (Tropical Integer) where + toInteger (Tropical a) = a + toInteger Inf = 0 + +instance Enum (Tropical Integer) where + toEnum a = Tropical (toInteger a) + fromEnum (Tropical a) = fromInteger a + fromEnum Inf = 0 + +instance Real (Tropical Integer) where + toRational (Tropical a) = toRational a + toRational Inf = 0 + +instance Ord (Tropical Integer) where + compare Inf a = LT + compare a Inf = GT + compare (Tropical a) (Tropical b) = compare a b diff --git a/library/Core.hs b/library/Core.hs new file mode 100644 --- /dev/null +++ b/library/Core.hs @@ -0,0 +1,23 @@ +module Core ( + module Polynomial.Prelude, + module Polynomial.Monomial, + module Polynomial.Hypersurface, + module Arithmetic.Numbers, + module Arithmetic.Matrix, + module Geometry.ConvexHull2, + module Geometry.ConvexHull3, + module Graphics.Drawings + + ) + +where + + import Polynomial.Prelude + import Arithmetic.Numbers + import Arithmetic.Matrix + import Polynomial.Monomial + import Geometry.ConvexHull2 + import Geometry.ConvexHull3 + import Polynomial.Hypersurface + import Graphics.Drawings + diff --git a/library/Geometry/ConvexHull2.hs b/library/Geometry/ConvexHull2.hs new file mode 100644 --- /dev/null +++ b/library/Geometry/ConvexHull2.hs @@ -0,0 +1,117 @@ +{-#LANGUAGE FlexibleInstances#-} + +module Geometry.ConvexHull2 where + +import Polynomial.Monomial +--import Geometry.ConvexHull3 + + +import Data.List +import Data.Matrix +import qualified Data.Map.Strict as MS +import qualified Data.Sized.Builtin as DS + + +-- | The algorithm performed in this module is Graham Scan + +type Point2D = (Int, Int) + + +(>*<) :: Int -> (Int, Int) -> (Int, Int) +(>*<) num (px, py) = (num*px, num*py) + +instance Num (Int, Int) where + (x1,y1) + (x2,y2) = (x1+x2, y1+y2) + negate (x,y) = (-x,-y) + + +lift2To3 :: Point2D -> (Int, Int, Int) +lift2To3 (a,b) = (a,b,1) + + +isColinear :: Point2D -> Point2D -> Point2D -> Bool +isColinear a b c = let + matrix = fromLists (map (\(x,y)-> map toRational [x,y,1]) [a,b,c]) + res = detLU matrix + in res == 0 + +isColinearFromList :: [Point2D] -> Bool +isColinearFromList [a,b,c] = isColinear a b c + + + +project3To2 :: (Int, Int, Int) -> Point2D +project3To2 (a,b,_) = (a,b) + + +determinant :: (Int, Int, Int) -> (Int, Int, Int) -> (Int, Int, Int) -> Rational +determinant p1 p2 p3 = detLU $ fromLists $ map toList [p1,p2,p3] + where + toList (a,b,c) = map toRational [a,b,c] + +semiHullUp :: [Point2D] -> Maybe Point2D -> [Point2D] +semiHullUp [a] _= [a] +semiHullUp [a,b] _ = [a,b] +semiHullUp points@(x:y:z:w) Nothing = if determinant x3 y3 z3 < 0 then x:semiHullUp (y:z:w) (Just x) + else semiHullUp (x:z:w) Nothing + where + (x3:y3:z3:w3) = map lift2To3 points +semiHullUp points@(x:y:z:w) (Just prev) = if determinant x3 y3 z3 < 0 then x:semiHullUp (y:z:w) (Just x) + else semiHullUp (prev:x:z:w) Nothing + where + (x3:y3:z3:w3) = map lift2To3 points + +semiHullDown :: [Point2D] -> Maybe Point2D -> [Point2D] +semiHullDown [a] _ = [a] +semiHullDown [a,b] _ = [a,b] +semiHullDown points@(x:y:z:w) Nothing = if determinant x3 y3 z3 > 0 then x:semiHullDown (y:z:w) (Just x) + else semiHullDown (x:z:w) Nothing + where + (x3:y3:z3:w3) = map lift2To3 points +semiHullDown points@(x:y:z:w) (Just prev) = if determinant x3 y3 z3 > 0 then x:semiHullDown (y:z:w) (Just x) + else semiHullDown (prev:x:z:w) Nothing + where + (x3:y3:z3:w3) = map lift2To3 points + + +leftMost :: [Point2D] -> Point2D +leftMost = foldl1 (\(a,b) (c,d) -> if c < a then (c,d) else (a,b)) + +rightMost :: [Point2D] -> Point2D +rightMost = foldl1 (\(a,b) (c,d) -> if c > a then (c,d) else (a,b)) + +isPointUp :: Point2D -> Point2D -> Point2D -> Bool +isPointUp (l1,l2) (r1,r2) (p1,p2) = rP2 >= (m * rP1) + b + where + [rL1, rL2, rR1,rR2, rP1, rP2] = map toRational [l1, l2, r1, r2, p1, p2] + m = (rR2- rL2) / (rR1-rL1) + b = rL2 - m * rL1 + +isPointDown :: Point2D -> Point2D -> Point2D -> Bool +isPointDown (l1,l2) (r1,r2) (p1,p2) = rP2 <= (m * rP1) + b + where + [rL1, rL2, rR1,rR2, rP1, rP2] = map toRational [l1, l2, r1, r2, p1, p2] + m = (rR2- rL2) / (rR1-rL1) + b = rL2 - m * rL1 + + +dropColinearPoints :: [Point2D] -> [Point2D] +dropColinearPoints [] = [] +dropColinearPoints [a] = [a] +dropColinearPoints [a,b] = [a,b] +dropColinearPoints (x:y:z:w) + | isColinear x y z = dropColinearPoints (x:z:w) + | otherwise = x:dropColinearPoints (y:z:w) + + +convexHull2 :: [Point2D] -> [Point2D] +convexHull2 points = lowerHull `union` reverse upperHull + where + lst = sort $ nub points + left = leftMost lst + right = rightMost lst + lowerHull = dropColinearPoints $ semiHullDown (filter (isPointDown left right) lst) Nothing + upperHull = dropColinearPoints $ semiHullUp (filter (isPointUp left right) lst) Nothing + + + diff --git a/library/Geometry/ConvexHull3.hs b/library/Geometry/ConvexHull3.hs new file mode 100644 --- /dev/null +++ b/library/Geometry/ConvexHull3.hs @@ -0,0 +1,339 @@ +{-#LANGUAGE FlexibleInstances#-} + +module Geometry.ConvexHull3 + +( + -- * Data types + Point3D, + Vertex(..), + Edge(..), + Facet(..), + ConvexHull(..), + ConflictGraph, + + -- * Convex hull + convexHull3, + fromConvexHull, + + -- * Computations of geometrical objects + computeSegment, + computeTriangle, + computeTetrahedron, + fromVertices, + + -- * Operations with points + isBetween3D, + mergePoints, + fromFacet, + + + -- * Testing + isColinearIn3DFromList +) + +where + + + +import Data.List +import Data.Maybe +import qualified Data.Map.Strict as MS +import Data.Matrix +import Data.Function + +import Geometry.ConvexHull2 + + +type Point3D = (Int, Int, Int) + +getX,getY,getZ :: Point3D -> Int + +getX (x,_,_) = x +getY (_,y,_) = y +getZ (_,_,z) = z + + +newtype Vertex = Vertex {coordinates :: Point3D} deriving (Eq) + +-- | Edge as a pair of vertices. When constructing the initial tetrahedron, the order of the vertices must be counterclockwise. +newtype Edge = Edge {vertices :: (Vertex, Vertex)} deriving (Show, Eq) + +-- | Facet in this case is a 2-face. It is stored as a collection of edges. +newtype Facet = Facet {edges :: [Edge]} + +-- | The convex hull of a set is the collection of all the facets that conform that polyhedron. +newtype ConvexHull = ConvexHull {facets :: [Facet]} deriving (Eq) + +-- | The conflict graph is a data structure that stores for each point the list of facets that points views. And for every facet the list of points that facet views. +data ConflictGraph = ConflictGraph { + verticesF :: MS.Map Vertex [Facet], + facetsV :: MS.Map Facet [Vertex] + } deriving (Show) + + +instance Show ConvexHull where + show = show.sort.facets + +instance Eq Facet where + f1 == f2 = ((==) `on` (sort.fromFacet)) f1 f2 + +instance Show Facet where + show = show.fromFacet + +instance Ord Vertex where + compare (Vertex v1) (Vertex v2) = compare v1 v2 + +instance Show Vertex where + show (Vertex v) = show v + +instance Ord Edge where + compare (Edge e1) (Edge e2) = compare e1 e2 + +instance Ord Facet where + compare (Facet f1) (Facet f2) = compare f1 f2 + + +instance Num (Int, Int, Int) where + (x1,y1,z1) + (x2,y2,z2) = (x1+x2, y1+y2, z1+z2) + negate (x,y,z) = (-x,-y,-z) + +-- | Assume every point is different +convexHull3 :: [Point3D] -> Maybe ConvexHull +convexHull3 points + | length points < 4 = Just $ convexHull2In3 points + | isNothing tetraHedron = Just $ convexHull2In3 points + | otherwise = Just convexHull + where + convexHull = addPoints initialCH afterTetrahedron conflictGraph + conflictGraph = startConflictGraph initialCH afterTetrahedron + initialCH = (initializeCH . fromJust) tetraHedron + tetraHedron = computeTetrahedron points + afterTetrahedron = points \\ fromJust tetraHedron + + + +convexHull2In3 :: [Point3D] -> ConvexHull +convexHull2In3 points3 = ConvexHull [fromVertices $ map lift2To3 $ convexHull2 points2] + where + points2 = map project3To2 points3 +------------------------------------------------- + + +startConflictGraph :: ConvexHull -> [Point3D] -> ConflictGraph +startConflictGraph convexHull = matchFacetsPoints facetsCH + where + facetsCH = facets convexHull + +matchFacetsPoints :: [Facet] -> [Point3D] -> ConflictGraph +matchFacetsPoints faces points = foldr (\(f,p) conflictGraph -> insert' f (Vertex p) conflictGraph) (ConflictGraph MS.empty MS.empty) pairsFacetPoint + where + pairsFacetPoint = [(f,p) | f <- faces, p <- points, isInFrontOf f p] + insert' f p (ConflictGraph vs fs) + | p == Vertex (3,3,3) = ConflictGraph (MS.insertWith (++) p [f] vs) (MS.insertWith (++) f [p] fs) + | otherwise = ConflictGraph (MS.insertWith (++) p [f] vs) (MS.insertWith (++) f [p] fs) + +initializeCH :: [Point3D] -> ConvexHull -- ^ Counterclockwise +initializeCH [a,b,c,d] = ConvexHull $ map fromVertices [f1,f2,f3,f4] + where + middlePoint = (xs,ys,zs) + xs = ((/4).sum) $ map (toRational.getX) [a,b,c,d] + ys = ((/4).sum) $ map (toRational.getY) [a,b,c,d] + zs = ((/4).sum) $ map (toRational.getZ) [a,b,c,d] + isInFront [p1,p2,p3] p4 = let + completeVectors = map (\(x,y,z)-> map toRational [x,y,z,1]) [p1,p2,p3] + completePoint = (\(x,y,z) -> [x,y,z,1]) p4 + matrix = fromLists (completeVectors ++ [completePoint]) + res = detLU matrix + in res < 0 + + f1 + | isInFront [a,b,c] middlePoint = [a,c,b] + | otherwise = [a,b,c] + f2 + | isInFront [a,c,d] middlePoint = [a,d,c] + | otherwise = [a,c,d] + f3 + | isInFront [a,d,b] middlePoint = [a,b,d] + | otherwise = [a,d,b] + f4 + | isInFront [b,d,c] middlePoint = [b,c,d] + | otherwise = [b,d,c] + + + +addPoints :: ConvexHull -> [Point3D] -> ConflictGraph -> ConvexHull +addPoints convexHull [] _ = convexHull +addPoints convexHull (p:ps) conflictGraph + | inside p convexHull = addPoints convexHull ps conflictGraph + | otherwise = addPoints (ConvexHull nextFacets) ps newConflictGraph + where + facetsCH = facets convexHull + + conflictVertices = verticesF conflictGraph + + visibleFaces = conflictVertices MS.! Vertex p + horizon = findHorizon visibleFaces + newFacets = map (fromVertices . (++ [p]) . (\(a,b) -> map coordinates [a,b]) . vertices) horizon + + nextFacets = filter (/= Facet []) $ checkCoplanarity (facetsCH\\visibleFaces) newFacets + newConflictGraph = matchFacetsPoints nextFacets ps + + +checkCoplanarity :: [Facet] -> [Facet] -> [Facet] +checkCoplanarity facets1 facets2 = foldr (\(merged,fs) acc -> (acc \\ fs)++merged) (facets1++facets2) coplanarFaces + where + mergeCoplanar f1 f2 + | length ((pointsFromGinF `on` fromFacet) f1 f2) < 2 = ([],[]) + | areCoplanarFacets f1 f2 = ([fromVertices $ (mergePoints `on` fromFacet) f1 f2],[f1,f2]) + | otherwise = ([],[]) + + coplanarFaces = map (uncurry mergeCoplanar) [(f1,f2) | f1 <- facets1, f2 <- facets2] + pointsFromGinF f= filter (`elem` f) + + + +areCoplanarFacets :: Facet -> Facet -> Bool +areCoplanarFacets f1 f2 = isCoplanar (fromFacet f1) (nicePoint f2 f1) + where + nicePoint f g = + head $ ((\\) `on` fromFacet) f g + +findHorizon :: [Facet] -> [Edge] +findHorizon facets = dropTwins $ concatMap edges facets + where + dropTwins [] = [] + dropTwins edges@(Edge (v1,v2):es) = if Edge (v2,v1) `elem` edges then dropTwins (edges\\[Edge (v1,v2),Edge (v2,v1)]) + else Edge (v1,v2):dropTwins es + +-- | Generates a facet from its vertices. Points must be ordered counterclockwise +fromVertices :: [Point3D] -> Facet +fromVertices points@(p:ps) = Facet edges + where + edges = getEdges $ map Vertex (points++[p]) + getEdges [a] = [] + getEdges (x:y:z) = Edge (x,y):getEdges (y:z) + +fromConvexHull :: ConvexHull -> [Point3D] +fromConvexHull convexHull = sort.nub $ concatMap fromFacet (facets convexHull) + where + dropTwins [] = [] + dropTwins edges@(Edge (v1,v2):es) = if Edge (v2,v1) `elem` es then dropTwins (edges\\[Edge (v2,v1)]) + else Edge (v1,v2):dropTwins es + + +-- | Merges two lists of points ensuring that the counterclockwise orientation remains. +mergePoints :: [Point3D] -> [Point3D] -> [Point3D] +mergePoints p1@(p:ps) p2@(x:y:xs) + | elem x p1 && elem y p1 = if last p1 == x then checkColinearity $ x:init p1 ++ xs + else mergePoints (ps++[p]) p2 + | otherwise = mergePoints p1 ((y:xs) ++ [x]) + + +checkColinearity :: [Point3D] -> [Point3D] +checkColinearity points@(p1:p2:p3:_) = points \\ midPoints + where + midPoints = map (\(_,p,_) -> p) nicePoints + nicePoints = [(p1,p2,p3) | p1 <- points, p2 <- points, p3 <- points, isColinearIn3D [p1,p3] p2, isBetween3D [p1,p3] p2, p1 /= p2, p2/=p3, p1/=p3] + +isBetween3D :: [Point3D] -> Point3D -> Bool +isBetween3D [p1,p3] p2 + | (p2X-p1X)*(p3X-p2X) < 0 = False + | (p2Y-p1Y)*(p3Y-p2Y) < 0 = False + | (p2Z-p1Z)*(p3Z-p2Z) < 0 = False + | otherwise = True + where + [p1X, p2X, p3X] = map getX [p1,p2,p3] + [p1Y, p2Y, p3Y] = map getY [p1,p2,p3] + [p1Z, p2Z, p3Z] = map getZ [p1,p2,p3] + + +--------------------------------------------------------------------------------------------- +computeSegment :: [Point3D] -> Maybe [Point3D] +computeSegment [] = Nothing -- Not enough points or all points are equal +computeSegment (x:xs) = case find (/=x) xs of + Nothing -> Nothing + Just y -> Just [x,y] + +computeTriangle :: [Point3D] -> Maybe [Point3D] +computeTriangle [] = Nothing +computeTriangle points + | isNothing segment = Nothing + | isNothing nicePoint = Nothing + | otherwise = (++) <$> segment <*> fmap return nicePoint -- Applicative functors (<$>, <*>), simple functors (fmap) and monads (return) in action! + where + segment = computeSegment points + nicePoint = find (not . isColinearIn3D (fromJust segment)) points + +computeTetrahedron :: [Point3D] -> Maybe [Point3D] +computeTetrahedron [] = Nothing +computeTetrahedron points + | isNothing triangle = Nothing + | isNothing nicePoint = Nothing + | otherwise = (++) <$> triangle <*> fmap return nicePoint -- Applicative functors (<$>, <*>), simple functors (fmap) and monads (return) in action! + where + triangle = computeTriangle points + nicePoint = find (not . isCoplanar (fromJust triangle)) points + + +isColinearIn3D :: [Point3D] -> Point3D -> Bool -- ^ Not able to use determinant algorithm because the matrix is not square +isColinearIn3D [a,b] c = let + distanceXYZ (a,b,c) (d,e,f) = [a-d, b-e, c-f] + ab = map toRational $ distanceXYZ a b + ac = map toRational $ distanceXYZ a c + ratio = checkRatio ab ac + in case ratio of + Nothing -> False + Just ls -> all (== head ls) (tail ls) + +isColinearIn3DFromList :: [Point3D] -> Bool -- ^ Not able to use determinant algorithm because the matrix is not square +isColinearIn3DFromList [a,b,c] = isColinearIn3D [a,b] c + +checkRatio :: [Rational] -> [Rational] -> Maybe [Rational] +checkRatio l1 l2 + | l1 == [0,0,0] = Just [0,0,0] + | l2 == [0,0,0] = Just [0,0,0] + | 0 `elem` l1' && 0 `elem` l2' = Nothing + | 0 `elem` l1' = Just $ zipWith (/) l1' l2' + | otherwise = Just $ zipWith (/) l2' l1' + where + (l1', l2') = filterZeros l1 l2 + + +filterZeros :: [Rational] -> [Rational] -> ([Rational], [Rational]) +filterZeros [] [] = ([],[]) +filterZeros (x:xs) (y:ys) + | x == 0 && y == 0 = newXs + | otherwise = (x:fst newXs, y:snd newXs) + where + newXs = filterZeros xs ys + + +isCoplanar :: [Point3D] -> Point3D -> Bool +isCoplanar (a:b:c:s) d = let + matrix = fromLists (map (\(x,y,z)-> map toRational [x,y,z,1]) [a,b,c,d]) + res = detLU matrix + in res == 0 + +isInFrontOf :: Facet -> Point3D -> Bool +isInFrontOf facet d = let + (a:b:c:ps) = fromFacet facet + matrix = fromLists (map (\(x,y,z)-> map toRational [x,y,z,1]) [a,b,c,d]) + res = detLU matrix + in res < 0 + + + +fromFacet :: Facet -> [Point3D] +fromFacet facet = let + verticesPairs = map vertices $ edges facet + verticesFromEdges = map fst verticesPairs + in map coordinates verticesFromEdges + +inside :: Point3D -> ConvexHull -> Bool +inside p convexHull = all (not.flip isInFrontOf p) (facets convexHull) + +isCoplanarCH :: Point3D -> ConvexHull -> Bool +isCoplanarCH p convexHull = any (`isCoplanar` p) (map fromFacet $ facets convexHull) + + + diff --git a/library/Geometry/Polyhedral.hs b/library/Geometry/Polyhedral.hs new file mode 100644 --- /dev/null +++ b/library/Geometry/Polyhedral.hs @@ -0,0 +1,79 @@ +module Geometry.Polyhedral where + +import Geometry.ConvexHull3 +import Data.List +import Data.Maybe +import Data.Function + + +-- | Given a vertex of a convex hull, computes the facets adjacent to the vertex. +adjacentFacets :: Point3D -> ConvexHull -> [Facet] +adjacentFacets p c = sortFacets p $ filter (isPointInFacet p) (facets c) + + +-- | For each vertex of a polyhedron, the facets must be ordered clockwise since we need there normals to point inside the polyhedron + +isPointInFacet :: Point3D -> Facet -> Bool +isPointInFacet p f = p `elem` fromFacet f + + +-- | Checks if two facets are adjacent with respect to a vertex in counterclockwise orientation. +areFacetsAdjacent :: Point3D -> Facet -> Facet -> Bool +areFacetsAdjacent p f1 f2 + | isJust edge1 && isJust edge2 = True + | otherwise = False + where + edge1, edge2 :: Maybe Edge + edge1 = find (\e -> (coordinates.snd.vertices) e == p) (edges f1) + edge2 = find (\e -> (coordinates.fst.vertices) e == p) (edges f2) + + +-- | The facets are ordered counterclockwise with respect to a vertex. [Optimizable] + +sortFacets :: Point3D -> [Facet] -> [Facet] +sortFacets _ [a,b] = [a,b] +sortFacets p (f:g:fs) + | areFacetsAdjacent p f g = f:sortFacets p (g:fs) + | otherwise = sortFacets p (f:fs)++[g] + + +-- | Computes normal vector to a facet of the polyhedron with respect to a vertex. +normalVector :: Vertex -> Facet -> Point3D +normalVector v f = crossProduct vector1 vector2 + where + (v1,v2) = vertices.fromJust $ find (\e -> (fst.vertices) e == v) (edges f) + (v3,v4) = vertices.fromJust $ find (\e -> (snd.vertices) e == v) (edges f) + vector1 = ((-) `on` coordinates) v2 v1 + vector2 = ((-) `on` coordinates) v3 v4 + + + + +crossProduct :: Point3D -> Point3D -> Point3D +crossProduct (x1,y1,z1) (x2,y2,z2) = (y1*z2-y2*z1,-(x1*z2-x2*z1),x1*y2-x2*y1) + + +normalCone :: Vertex -> [Facet] -> [Point3D] +normalCone v facets@(f:fs) = computeNormalCone normals + where + normals = map (normalVector v) (facets++[f]) + computeNormalCone [v1,v2] = [crossProduct v1 v2] + computeNormalCone (v1:v2:vs) = (crossProduct v1 v2):computeNormalCone (v2:vs) + +normalFan :: ConvexHull -> [[Point3D]] +normalFan convexHull = normalFanPointsFacets (fromConvexHull convexHull) convexHull + +normalFanPointsFacets :: [Point3D] -> ConvexHull -> [[Point3D]] +normalFanPointsFacets [] _ = [] +normalFanPointsFacets (v:vs) convexHull = (normalCone (Vertex v) properFacets):normalFanPointsFacets vs convexHull + where + properFacets = adjacentFacets v convexHull + + + +isLowerFace :: Facet -> Bool +isLowerFace facet = trd' normal < 0 + where + trd' (_,_,a) = a + vertex = Vertex $ head $ fromFacet facet + normal = normalVector vertex facet diff --git a/library/Geometry/Polytope.hs b/library/Geometry/Polytope.hs new file mode 100644 --- /dev/null +++ b/library/Geometry/Polytope.hs @@ -0,0 +1,52 @@ +{-# LANGUAGE DataKinds, TypeFamilies, FlexibleContexts, FlexibleInstances, PolyKinds #-} +{-# LANGUAGE UndecidableInstances, MultiParamTypeClasses #-} + +module Geometry.Polytope where + +import Data.List + +import Geometry.ConvexHull3 +import Geometry.ConvexHull2 +import Geometry.Polyhedral +import Debug.Trace + +import Polynomial.Prelude +import Polynomial.Monomial +import qualified Data.Map.Strict as MS +import qualified Data.Sized as DS + +import Data.Maybe + + + +-- | Computes the projection R^3 -> R^2 of a convexhull. The input is a convexhull in R^3 and the output will regular subdivision of the convexhull in R^2 + +projectionToR2 :: ConvexHull -> [[Point2D]] +projectionToR2 convexHull = let lengthConvexHull = length $ facets convexHull in + if lengthConvexHull == 1 then + projected.triangles.facetsInPoints $ facets convexHull + else + projected.triangles.facetsInPoints $ lowerFaces + where + lowerFaces = filter isLowerFace $ facets convexHull + facetsInPoints = map fromFacet + triangles = filter (\points -> length points == 3) + projected = map (map project3To2) + + +delete' :: [Point2D] -> [[Point2D]] -> [[Point2D]] +delete' _ [] = [] +delete' point (p:ps) + | sort point == sort p = ps + | otherwise = p:delete' point ps + + +subdivision :: (Integral k) => Polynomial k ord n -> [[Point2D]] +subdivision poly = (projectionToR2.fromJust.convexHull3) points + where + terms = MS.toList $ getTerms poly + monExps = DS.toList . getMonomial + toPoints (mon, coef) = let [a,b] = monExps mon in (a, b, fromIntegral coef) + points = map toPoints terms + + diff --git a/library/Graphics/Drawings.hs b/library/Graphics/Drawings.hs new file mode 100644 --- /dev/null +++ b/library/Graphics/Drawings.hs @@ -0,0 +1,35 @@ +{-# LANGUAGE DataKinds, TypeFamilies, FlexibleContexts, FlexibleInstances, PolyKinds #-} +{-# LANGUAGE UndecidableInstances, MultiParamTypeClasses #-} + +module Graphics.Drawings where + +import Graphics.Gloss + +import Polynomial.Monomial +import Polynomial.Prelude +import Polynomial.Hypersurface +import Geometry.ConvexHull2 (Point2D) + +fromPoint2D :: Point2D -> Point +fromPoint2D (x,y) = (fromIntegral x, fromIntegral y) + + +tupleToList :: (Point2D, Point2D) -> [Point] +tupleToList (p1, p2) = [fromPoint2D p1, fromPoint2D p2] + +makeFig :: (IsMonomialOrder ord, Ord k, Integral k) => Polynomial k ord n -> IO () +makeFig poly = let + figure = color blue $ Pictures $ map (Line . tupleToList) (hypersurface poly) + scaled = scale 10.0 10.0 figure + in + display (InWindow "My Window" (400, 400) (10, 10)) white scaled + + +makeFigs :: (IsMonomialOrder ord, Ord k, Integral k) => [Polynomial k ord n] -> IO() +makeFigs polys = let + colors = [red, green, blue, yellow, cyan, magenta, rose, violet, azure, aquamarine, chartreuse, orange] + figures = map (Pictures . map (Line . tupleToList) . hypersurface) polys + putColors = zipWith color colors figures + scaled = scale 10.0 10.0 (Pictures putColors) + in + display (InWindow "Tropical curves" (400, 400) (10,10)) white scaled diff --git a/library/Polynomial/Hypersurface.hs b/library/Polynomial/Hypersurface.hs new file mode 100644 --- /dev/null +++ b/library/Polynomial/Hypersurface.hs @@ -0,0 +1,168 @@ +{-# LANGUAGE DataKinds, TypeFamilies, FlexibleContexts, FlexibleInstances, PolyKinds #-} +{-# LANGUAGE UndecidableInstances, MultiParamTypeClasses #-} + + +module Polynomial.Hypersurface where + +import Polynomial.Prelude +import Polynomial.Monomial +import Geometry.ConvexHull2 +import qualified Data.Map.Strict as MS +import qualified Data.Sized as DS +import Data.Maybe +import Debug.Trace +import Data.List +import Geometry.ConvexHull3 (Point3D) +import Geometry.Polytope + +-- | The tropical hypersurface of a polynomial f is the n-1 skeleton of the Newton polyotpe of f with a regular subdivision induced by a vector w in R^n. The hypersurface will be stored as a set of points. + + +type Polygon = [Point2D] +type Normals = [Point2D] + +-- | This function produces a key-value map of the terms of a polynomial with their corresponding coordinates for the Newton polytope +mapTermPoint :: (IsMonomialOrder ord, Ord k, Integral k) + => Polynomial k ord n -> MS.Map Point2D (Monomial ord n, k) +mapTermPoint poly = MS.fromList $ zip points terms + where + terms = MS.toList $ getTerms poly + monExps = DS.toList . getMonomial + toPoints (mon,coef) = let [a,b] = monExps mon in (a, b) + points = map toPoints terms + + +-- | Finds the vertex of a tropical line which corresponds to the intersection of the polyhedral fan of a triangle. The output is the result of the systems of equations given by: ax + by +c = 0; dx + ey f = 0; gx + hy + i = 0. + +computeIntersection :: (Integral k) => (Monomial ord n, k) -> (Monomial ord n, k) -> (Monomial ord n, k) -> Point2D +computeIntersection (mon1, c1) (mon2, c2) (mon3, c3) = (x, y) + where + [a,b] = DS.toList $ getMonomial mon1 + [d,e] = DS.toList $ getMonomial mon2 + [g,h] = DS.toList $ getMonomial mon3 + c = fromIntegral c1 + f = fromIntegral c2 + i = fromIntegral c3 + y = ((f-c)*(d-g) - (i-f)*(a-d)) `div` ((b-e)*(d-g)-(e-h)*(a-d)) + x = ((f-c)-(b-e)*y) `div` (a-d) + +findFanNVertex :: (Integral k) => MS.Map Point2D (Monomial ord n, k) -> Polygon -> (Point2D,Normals) +findFanNVertex mapPointMon points = (computeIntersection mon1 mon2 mon3, sort [inormal1,inormal2,inormal3]) + where + [pp1,pp2,pp3] = sort points -- This sorting is done to ensure that pp1 has non-zero degree on x. + mon1 = fromJust $ MS.lookup pp3 mapPointMon + mon2 = fromJust $ MS.lookup pp1 mapPointMon + mon3 = fromJust $ MS.lookup pp2 mapPointMon + inormal1 = innerNormal pp1 pp2 pp3 + inormal2 = innerNormal pp2 pp3 pp1 + inormal3 = innerNormal pp3 pp1 pp2 + +findPolygonNVertex :: (Integral k) => MS.Map Point2D (Monomial ord n, k) -> Polygon -> (Polygon, Point2D) +findPolygonNVertex mapPointMon points = ( [pp1,pp2,pp3], computeIntersection mon1 mon2 mon3) + where + [pp1,pp2,pp3] = sort points -- This sorting is done to ensure that pp1 has non-zero degree on x. + mon1 = fromJust $ MS.lookup pp3 mapPointMon + mon2 = fromJust $ MS.lookup pp1 mapPointMon + mon3 = fromJust $ MS.lookup pp2 mapPointMon + +innerNormal :: Point2D -> Point2D -> Point2D -> Point2D +innerNormal a@(x1,y1) b@(x2,y2) c@(x3,y3) + | dot > 0 = simplify nab + | dot < 0 = simplify (y1-y2,x2-x1) + where + ab = (x2-x1,y2-y1) + ac = (x3-x1,y3-y1) + nab = (y2-y1,x1-x2) -- (y,-x) + dot = (y2-y1)*(x3-x1) + (x1-x2)*(y3-y1) + simplify (0, q) = (0, div q (abs q)) + simplify (q, 0) = (div q (abs q), 0) + simplify (p, q) = let g = gcd p q in (div p g, div q g) + + + + + +innerNormals :: Point2D -> Point2D -> Point2D -> Normals +innerNormals a@(x1,y1) b@(x2,y2) c@(x3,y3) = map simplify [inner1, inner2, inner3] -- simplify is to normalize the normals + where + inner1 = innerNormal a b c + inner2 = innerNormal b c a + inner3 = innerNormal c a b + simplify (0, q) = (0, div q (abs q)) + simplify (q, 0) = (div q (abs q), 0) + simplify (p, q) = let g = gcd p q in (div p g, div q g) + + + +verticesNormals :: (IsMonomialOrder ord, Ord k, Integral k) => Polynomial k ord n -> MS.Map Point2D Normals +verticesNormals poly = MS.fromList $ map (findFanNVertex polyMap) triangles + where + polyMap = mapTermPoint poly + triangles = subdivision poly + + +---- For plotting + +neighborTriangles :: [Polygon] -> MS.Map Polygon [Polygon] -> MS.Map Polygon [Polygon] +neighborTriangles [] mapContainer = mapContainer +neighborTriangles (p:ps) mapContainer + | null ps && MS.null mapContainer = MS.fromList [(p, [])] -- The case that there is only one triangle. This will correspond to a tropical line + | otherwise = MS.map (map sort) $ neighborTriangles ps (foldr (lookAndInsert p) mapContainer ps) + where + lookAndInsert p1 p2 acc = if length (p1\\p2) == (length p1) - 2 then -- Checks is triangles share two vertices. + MS.insertWith (++) p2 [p1] $ MS.insertWith (++) p1 [p2] acc + else acc + + + +pointsWithTriangles :: (IsMonomialOrder ord, Ord k, Integral k) => Polynomial k ord n -> MS.Map Polygon Point2D +pointsWithTriangles poly = MS.fromList $ map (findPolygonNVertex polyMap) triangles + where + polyMap = mapTermPoint poly -- MS.Map Point2D (Monomial ord n, k) + triangles = subdivision poly -- [Polygon] + + +polygonCenter :: MS.Map Polygon Point2D -> Polygon -> Point2D +polygonCenter pointPolygonMap polygon = fromJust $ MS.lookup polygon pointPolygonMap + +convertMap :: MS.Map Polygon [Polygon] -> MS.Map Polygon Point2D -> MS.Map Point2D [Point2D] +convertMap map1 map2 = MS.fromList $ map fromPolygons $ MS.toList map1 + where + fromPolygons (polygon, polygons) = (polygonCenter map2 polygon, map (polygonCenter map2) polygons) + + +computeEdges :: MS.Map Point2D [Point2D] -> MS.Map Point2D Normals -> [(Point2D, Point2D)] +computeEdges map1 map2 = concatMap getEdges pointsWithNormals + where + listMap1 = MS.toList map1 + listMap2 = MS.toList map2 + getNormals point2D = (point2D, fromJust $ MS.lookup point2D map2) + attachNormals = map (\(e,l) -> (getNormals e, map getNormals l)) + + getEdges ((p,n), []) = map (\normal-> (p, p+ (10 >*< normal))) n + getEdges ((p,n), (p1,n1):ps) = let ((point, newNormals),edge) = analizeNormals (p,n) (p1,n1) in + edge:(getEdges ((point, newNormals), ps)) + + pointsWithNormals = attachNormals listMap1 + +isInverse :: Point2D -> Point2D -> Point2D -> Point2D ->Bool +isInverse (x1,y1) (nx1, ny1) (x2,y2) (nx2, ny2) + | x1 == x2 = nx1 == 0 && nx2 == 0 && ny1*ny2 < 0 + | y1 == y2 = ny1 == 0 && ny2 == 0 && nx1*nx2 < 0 + | nx1 == 0 = False + | div (y2-y1) (x2-x1) == div ny1 nx1 = True + | otherwise = False + + +analizeNormals :: (Point2D, Normals) -> (Point2D, Normals) -> ((Point2D,Normals),(Point2D, Point2D)) +analizeNormals (p1, n1) (p2, n2) = ((p1, newNormals), (p1,p2)) + where + isThereTwin p1 normal p2 normals = any (isInverse p1 normal p2) normals + newNormals = foldr (\normal acc -> if isThereTwin p1 normal p2 n2 then acc else normal:acc) [] n1 + +hypersurface :: (IsMonomialOrder ord, Ord k, Integral k) => Polynomial k ord n -> [(Point2D, Point2D)] +hypersurface poly = nub $ computeEdges (convertMap neighbors pointTriangles) pointNormals + where + pointNormals = verticesNormals poly + neighbors = neighborTriangles (map sort $ subdivision poly) MS.empty + pointTriangles = pointsWithTriangles poly diff --git a/library/Polynomial/Monomial.hs b/library/Polynomial/Monomial.hs new file mode 100644 --- /dev/null +++ b/library/Polynomial/Monomial.hs @@ -0,0 +1,123 @@ +{-# LANGUAGE DataKinds, FlexibleInstances, GADTs, PolyKinds, TemplateHaskell #-} +{-# LANGUAGE MultiParamTypeClasses, TypeFamilies #-} + +module Polynomial.Monomial +( + -- * Types + Monomial(..), + Mon, + SNat, + Lex, + Revlex, + + -- * Classes + IsMonomialOrder, + + -- * Functions + toMonomial + +) + +where + +import Data.Function +import Numeric.Algebra hiding ((+),(>)) +import Prelude hiding (lex) +import qualified Data.Sized as DS +import qualified Data.Sequence as Seq +import Data.Singletons.Prelude +import GHC.TypeLits +import Control.Lens (makeLenses, makeWrapped) + + + +type SNat (n :: Nat) = Sing n + +type Sized' n a = DS.Sized Seq.Seq n a +type Mon n = Sized' n Int + +-- | Monomial is defined as an array of exponents +newtype Monomial ord n = Monomial {getMonomial :: Mon n} deriving(Eq) + + +------------------------------------------ +makeLenses ''Monomial +makeWrapped ''Monomial + +showMonomial :: [Int] -> Int -> String +showMonomial [] _ = "" +showMonomial (x:xs) var + | x == 0 = showMonomial xs (var+1) + | x == 1 = "X_" ++ show var ++ showMonomial xs (var+1) + | otherwise = "X_" ++ show var ++ "^" ++ show x ++ showMonomial xs (var+1) + + +instance Show (Monomial ord n) where + show monomial = showMonomial (DS.toList $ getMonomial monomial) 0 +------------------------------------------ + +-- | Definition of what a monomial order must meet +class IsMonomialOrder (ord :: *) where + compareMonomial :: Monomial ord n -> Monomial ord n -> Ordering +----------------------------- + +data Lex = Lex -- ^ Just the datatype for Lex ordering +data Revlex = Revlex -- ^ Just the datatype for Revlex ordering + +lex :: Monomial ord n -> Monomial ord n -> Ordering +lex = lex' `on` (DS.toList . getMonomial) + +lex' :: [Int] -> [Int] -> Ordering +lex' [] [] = EQ +lex' [] _ = LT +lex' _ [] = GT +lex' (x:xs) (y:ys) + | (x == 0 && y == 0) || x==y = lex' xs ys + | x > y = GT + | otherwise = LT + + +revlex :: Monomial ord n -> Monomial ord n -> Ordering +revlex= revlex' `on` (DS.toList . getMonomial) + +revlex' :: [Int] -> [Int] -> Ordering +revlex' [] [] = EQ +revlex' [] _ = LT +revlex' _ [] = GT +revlex' x y + | (xr == 0 && yr == 0) || xr==yr = revlex' (reverse xrs) (reverse yrs) + | xr > yr = GT + | otherwise = LT + where + (xr:xrs) = reverse x + (yr:yrs) = reverse y + + +-- | convert NAry list into Monomial. +fromList :: SNat n -> [Int] -> Mon n +fromList len = DS.fromListWithDefault len 0 + +toMonomial :: (IsMonomialOrder ord, KnownNat n) => [Int] -> Monomial ord n +toMonomial a = Monomial $ fromList sing a + +instance IsMonomialOrder Lex where + compareMonomial = lex + +instance IsMonomialOrder Revlex where + compareMonomial = revlex + +instance (IsMonomialOrder ord) => Ord (Monomial ord n) where + compare = compareMonomial + +instance (IsMonomialOrder ord, KnownNat n) => Unital (Monomial ord n) where + one = toMonomial [] + +instance (IsMonomialOrder ord, KnownNat n) => Multiplicative (Monomial ord n) where + (*) = prodMon + +prodMon :: (IsMonomialOrder ord, KnownNat n) => Monomial ord n -> Monomial ord n -> Monomial ord n +prodMon mon1 mon2 + | mon1 == one = mon2 + | mon2 == one = mon1 + | otherwise = toMonomial $ (zipWith (+) `on` (DS.toList . getMonomial)) mon1 mon2 + diff --git a/library/Polynomial/Prelude.hs b/library/Polynomial/Prelude.hs new file mode 100644 --- /dev/null +++ b/library/Polynomial/Prelude.hs @@ -0,0 +1,165 @@ +{-# LANGUAGE DataKinds, TypeFamilies, FlexibleContexts, FlexibleInstances, PolyKinds #-} +{-# LANGUAGE UndecidableInstances, MultiParamTypeClasses #-} + + +--ConstrainedClassMethods, +module Polynomial.Prelude ( + -- * Types + Polynomial(..), + + -- * Classes + IsPolynomial(..), + IsOrderedPolynomial(..), + + -- * Functions + polytope2, + (!*) +) where + +import Control.Lens +import Prelude as P +import qualified Data.Map.Strict as MS +import Numeric.Algebra as NA +import qualified Numeric.Additive.Class as AD +import Debug.Trace +import Data.List +import qualified Data.Sized as DS +import GHC.TypeLits +import Data.Type.Ordinal.Builtin +import Data.Singletons.Prelude + + +import Polynomial.Monomial +import Arithmetic.Numbers +import Geometry.ConvexHull2 + + + + + +-- infix 7 !*! + + +class (DecidableZero r, Rig r, Commutative r, Eq r) => CoeffRig r + +-- | Synonym for instances. +instance (DecidableZero r, Rig r, Commutative r, Eq r) => CoeffRig r + + +-- | Polynomial requires just the type of the coefficient and the monomial ordering. +-- | Arity is given when defining variables with 'variable' function +newtype Polynomial k ord n = Polynomial { getTerms :: MS.Map (Monomial ord n) k} deriving(Eq) + +instance (Unital k, Show k, Eq k) => Show (Polynomial k Lex n) where + show = dropPlusSign . showTerms . reverse . MS.toList . getTerms + +instance (KnownNat n, Unital k, Show k, Eq k) => Show (Polynomial k Revlex n) where + show = dropPlusSign . showTerms . map reverseMon . reverse . MS.toList . getTerms + +reverseMon :: (KnownNat n, IsMonomialOrder ord) => (Monomial ord n, a) -> (Monomial ord n, a) +reverseMon (Monomial mon, a) = ((toMonomial . reverse . DS.toList) mon, a) + + +dropPlusSign :: String -> String +dropPlusSign [] = error "String too short in dropPlusSign function" +dropPlusSign [_] = error "String too short in dropPlusSign function" +dropPlusSign [_,_] = error "String too short in dropPlusSign function" +dropPlusSign s@(x:y:z:a) + | (x:y:[z]) == " + " = a + | otherwise = s + +showTerms :: (Unital k, Eq k, Show k) => [(Monomial ord n, k)] -> String +showTerms [] = "" +showTerms (t:ts) + | coeff == one = " + " ++ show mon ++ showTerms ts + | otherwise = " + " ++ show coeff ++ show mon ++ showTerms ts + where + coeff = snd t + mon = fst t + + +-- | Every polynomial must implement this class +class (CoeffRig (Coeff poly), KnownNat (Arity poly)) => IsPolynomial poly where + type Coeff poly :: * + type Arity poly :: Nat + + arity :: poly -> SNat (Arity poly) + + toPolynomial :: (Mon (Arity poly), Coeff poly) -> poly + + fromMonomial :: Mon (Arity poly) -> poly + fromMonomial mon = toPolynomial (mon, one) + + variable :: Ordinal (Arity poly) -> poly + variable idx = fromMonomial $ DS.replicate sing 0 & ix idx .~ 1 + + -- (!*!) :: Coeff poly -> poly -> poly + -- (!*!) = (!*) + + +class (IsMonomialOrder (MonOrder poly), IsPolynomial poly) => IsOrderedPolynomial poly where + type MonOrder poly :: * + + terms :: poly -> MS.Map (Monomial (MonOrder poly) (Arity poly)) (Coeff poly) + + leadingTerm :: poly -> (Coeff poly, Monomial (MonOrder poly) (Arity poly)) + leadingTerm = (,) <$> leadingCoeff <*> leadingMonomial + leadingMonomial :: poly -> Monomial (MonOrder poly) (Arity poly) + leadingMonomial = snd . leadingTerm + + leadingCoeff :: poly -> Coeff poly + leadingCoeff = fst . leadingTerm + +instance (KnownNat n, IsMonomialOrder ord, CoeffRig k) => IsPolynomial (Polynomial k ord n) where + type Coeff (Polynomial k ord n) = k + type Arity (Polynomial k ord n) = n + + arity = DS.sLength . getMonomial . leadingMonomial + toPolynomial (mon, coeff) = Polynomial $ MS.singleton (Monomial mon) coeff + + +instance (KnownNat n, CoeffRig k, IsMonomialOrder ord) => IsOrderedPolynomial (Polynomial k ord n) where + type MonOrder (Polynomial k ord n) = ord + terms = getTerms + leadingTerm (Polynomial d) = case MS.lookupMax d of + Just (mon, coeff) -> (coeff, mon) + Nothing -> (one, one) + +instance (IsMonomialOrder ord, KnownNat n) => Num (Polynomial (Tropical Integer) ord n) where + (+) (Polynomial terms1) (Polynomial terms2) = Polynomial $ MS.unionWith (P.+) terms1 terms2 + (*) (Polynomial terms1) (Polynomial terms2) = Polynomial $ MS.fromListWith (P.+) [ prodTerm t1 t2 | t1 <- MS.toList terms1, t2 <- MS.toList terms2] + fromInteger x = Polynomial $ MS.singleton one (P.fromInteger x) + negate poly = Polynomial $ MS.map P.negate $ terms poly + +instance (IsMonomialOrder ord, KnownNat n) => Fractional (Polynomial (Tropical Integer) ord n) where + recip (Polynomial terms1) = Polynomial $ (MS.fromList . map negateTerm . MS.toList) terms1 -- Pseudo recip, only work with monic polynomials + where + negateTerm (mon, coef) = let + listExps = DS.toList $ getMonomial mon + negated = map P.negate listExps + in (toMonomial negated, coef) + + + +instance (Num k, IsMonomialOrder ord) => AD.Additive (Polynomial k ord n) where + (+) (Polynomial terms1) (Polynomial terms2) = Polynomial $ MS.unionWith (P.+) terms1 terms2 + + +instance (AD.Additive (Polynomial k ord n), Semiring k, Num k) => LeftModule k (Polynomial k ord n) where + num .* poly = num !* poly + +instance (AD.Additive (Polynomial k ord n), Semiring k, Num k) => RightModule k (Polynomial k ord n) where + poly *. num = num !* poly + + +-- | Aux functions + +prodTerm :: (KnownNat n, Num k, IsMonomialOrder ord) => (Monomial ord n, k) -> (Monomial ord n, k) -> (Monomial ord n, k) +prodTerm (mon1, coeff1) (mon2, coeff2) = (mon1 NA.* mon2, coeff1 P.* coeff2) + +(!*) :: (Num k) => k -> Polynomial k ord n -> Polynomial k ord n +num !* poly = Polynomial $ MS.map (P.* num) (getTerms poly) + + +polytope2 :: IsOrderedPolynomial poly => poly -> [Point2D] +polytope2 = convexHull2 . map ( (\[a,b] -> (a,b)) . DS.toList . getMonomial . fst) . MS.toList . terms diff --git a/package.yaml b/package.yaml new file mode 100644 --- /dev/null +++ b/package.yaml @@ -0,0 +1,75 @@ +# This YAML file describes your package. Stack will automatically generate a +# Cabal file when you run `stack build`. See the hpack website for help with +# this file: . +name: tropical-geometry +version: '0.0.0' +github: "ferynando7/tropical-geometry" +license: GPL-3 +author: "Fernando Zhapa" +maintainer: "Fernando Zhapa" +# synopsis: A new Haskeleton package. +# description: Tropical is a new Haskeleton package. +# category: Other + +extra-source-files: +- CHANGELOG.md +#- LICENSE.md +- package.yaml +- README.md +- stack.yaml + +ghc-options: -Wall + +library: + dependencies: + - base + - containers + - numeric-prelude + - semiring-simple + - algebra + - singletons + - lens + - matrix + - gloss + source-dirs: library + + +executables: + tropical-geometry: + source-dirs: executable + main: Main.hs + dependencies: + - base + - Tropical + ghc-options: + - -rtsopts + - -threaded + - -with-rtsopts=-N + +benchmarks: + tropical-geometry-benchmarks: + source-dirs: benchmark + main: Main.hs + dependencies: + - base + - Tropical + - criterion + ghc-options: + - -rtsopts + - -threaded + - -with-rtsopts=-N + +tests: + tropical-geometry-test-suite: + source-dirs: test-suite + main: Main.hs + dependencies: + - base + - Tropical + - tasty + - tasty-hspec + - tasty-hunit + ghc-options: + - -rtsopts + - -threaded + - -with-rtsopts=-N diff --git a/stack.yaml b/stack.yaml new file mode 100644 --- /dev/null +++ b/stack.yaml @@ -0,0 +1,73 @@ +# This file was automatically generated by 'stack init' +# +# Some commonly used options have been documented as comments in this file. +# For advanced use and comprehensive documentation of the format, please see: +# https://docs.haskellstack.org/en/stable/yaml_configuration/ + +# Resolver to choose a 'specific' stackage snapshot or a compiler version. +# A snapshot resolver dictates the compiler version and the set of packages +# to be used for project dependencies. For example: +# +# resolver: lts-3.5 +# resolver: nightly-2015-09-21 +# resolver: ghc-7.10.2 +# resolver: ghcjs-0.1.0_ghc-7.10.2 +# +# The location of a snapshot can be provided as a file or url. Stack assumes +# a snapshot provided as a file might change, whereas a url resource does not. +# +# resolver: ./custom-snapshot.yaml +# resolver: https://example.com/snapshots/2018-01-01.yaml +resolver: lts-12.26 +#lts-11.17 + +# User packages to be built. +# Various formats can be used as shown in the example below. +# +# packages: +# - some-directory +# - https://example.com/foo/bar/baz-0.0.2.tar.gz +# - location: +# git: https://github.com/commercialhaskell/stack.git +# commit: e7b331f14bcffb8367cd58fbfc8b40ec7642100a +# - location: https://github.com/commercialhaskell/stack/commit/e7b331f14bcffb8367cd58fbfc8b40ec7642100a +# subdirs: +# - auto-update +# - wai +packages: +- . +# Dependency packages to be pulled from upstream that are not in the resolver +# using the same syntax as the packages field. +# (e.g., acme-missiles-0.3) +extra-deps: +- sized-0.3.0.0 +- equational-reasoning-0.6.0.0 +- ghc-typelits-presburger-0.2.0.5 +- type-natural-0.8.2.0 +- hlint-test-0.1.0.0 +- algebra-4.3.1 + +# Override default flag values for local packages and extra-deps +# flags: {} + + +# Extra package databases containing global packages +# extra-package-dbs: [] + +# Control whether we use the GHC we find on the path +# system-ghc: true +# +# Require a specific version of stack, using version ranges +# require-stack-version: -any # Default +# require-stack-version: ">=1.7" +# +# Override the architecture used by stack, especially useful on Windows +# arch: i386 +# arch: x86_64 +# +# Extra directories used by stack for building +# extra-include-dirs: [/path/to/dir] +# extra-lib-dirs: [/path/to/dir] +# +# Allow a newer minor version of GHC than the snapshot specifies +# compiler-check: newer-minor diff --git a/test-suite/Main.hs b/test-suite/Main.hs new file mode 100644 --- /dev/null +++ b/test-suite/Main.hs @@ -0,0 +1,37 @@ +-- Tasty makes it easy to test your code. It is a test framework that can +-- combine many different types of tests into one suite. See its website for +-- help: . + +--{-# OPTIONS_GHC -F -pgmF hlint-test #-} + +import Test.Tasty + +import TPolynomial.TMonomial +import TPolynomial.TPrelude +import TArithmetic.TNumbers +import TArithmetic.TMatrix +import TGeometry.TConvexHull2 +import TGeometry.TConvexHull3 +import TGeometry.TPolyhedral +import TGeometry.TPolytope +import TPolynomial.THypersurface + +main :: IO () +main = defaultMain allTests + + + +allTests :: TestTree +allTests = testGroup "Tasty tests" [ + testGroup "List of tests:" [ + testsNumbers, + testsMatrices, + testsMonomial, + testsPrelude, + testsConvexHull2, + testsConvexHull3, + testsPolyhedral, + testsPolytope, + testsHypersurface] + ] + diff --git a/test-suite/TArithmetic/TMatrix.hs b/test-suite/TArithmetic/TMatrix.hs new file mode 100644 --- /dev/null +++ b/test-suite/TArithmetic/TMatrix.hs @@ -0,0 +1,97 @@ +module TArithmetic.TMatrix (testsMatrices) where + + import Test.Tasty + import Test.Tasty.HUnit as HU + import Arithmetic.Numbers + import Arithmetic.Matrix + import Data.List (transpose) + + + a = TMatrix + [ + [1, 2, 3], + [2, Inf, 5], + [4, 8, -2] + ] :: TMatrix (Tropical Integer) + + b = TMatrix + [ + [6, Inf, -1], + [0, -9, 8], + [Inf, 2, 0] + ] :: TMatrix (Tropical Integer) + + c = TMatrix + [ + [0, 6, 2], + [1, 0, 1] + ] :: TMatrix (Tropical Integer) + + d = TMatrix + [ + [4, 1], + [2, 2], + [0, Inf] + ] :: TMatrix (Tropical Integer) + +-- | Results + + aPb = TMatrix + [ + [1, 2, -1], + [0, -9, 5], + [4, 2, -2] + ] :: TMatrix (Tropical Integer) + + cPd' = TMatrix + [ + [0, 2, 0], + [1, 0, 1] + ] :: TMatrix (Tropical Integer) + + aTb = TMatrix + [ + [2, -7, 0], + [8, 7, 1], + [8, -1, -2] + ] :: TMatrix (Tropical Integer) + + cTa = TMatrix + [ + [1, 2, 0], + [2, 3, -1] + ] :: TMatrix (Tropical Integer) + + cTd = TMatrix + [ + [2, 1], + [1, 2] + ] :: TMatrix (Tropical Integer) + + dTc = TMatrix + [ + [2, 1, 2], + [2, 2, 3], + [0, 6, 2] + ] :: TMatrix (Tropical Integer) + + + + testSumMatrix :: TestTree + testSumMatrix = HU.testCase "Sum of matrices" $ do + a+b @?= aPb + c + transp d @?= cPd' + + testProdMatrix :: TestTree + testProdMatrix = HU.testCase "Product of matrices" $ do + a*b @?= aTb + c*a @?= cTa + c*d @?= cTd + d*c @?= dTc + + testsMatrices :: TestTree + testsMatrices = testGroup "Test for tropical matrices" [testSumMatrix, testProdMatrix] + + + + diff --git a/test-suite/TArithmetic/TNumbers.hs b/test-suite/TArithmetic/TNumbers.hs new file mode 100644 --- /dev/null +++ b/test-suite/TArithmetic/TNumbers.hs @@ -0,0 +1,27 @@ +module TArithmetic.TNumbers (testsNumbers) where + + import Test.Tasty + import Test.Tasty.HUnit as HU + import Arithmetic.Numbers + + + + testSumTropicals :: TestTree + testSumTropicals = HU.testCase "Sum tropical numbers" $ do + Tropical 3 + Tropical 3 @?= (Tropical 3 :: Tropical Integer) + Inf + Tropical 3 @?= (Tropical 3 :: Tropical Integer) + Tropical (-2) + Tropical 3 @?= (Tropical (-2) :: Tropical Integer) + + testProdTropicals :: TestTree + testProdTropicals = HU.testCase "Multiply tropical numbers" $ do + Tropical 3 * Tropical 3 @?= (Tropical 6 :: Tropical Integer) + Inf * Tropical 3 @?= (Inf :: Tropical Integer) + Tropical (-2) * Tropical 3 @?= (Tropical 1 :: Tropical Integer) + + + testsNumbers :: TestTree + testsNumbers = testGroup "Test for tropical numbers" [testSumTropicals, testProdTropicals] + + + + diff --git a/test-suite/TGeometry/TConvexHull2.hs b/test-suite/TGeometry/TConvexHull2.hs new file mode 100644 --- /dev/null +++ b/test-suite/TGeometry/TConvexHull2.hs @@ -0,0 +1,18 @@ +module TGeometry.TConvexHull2 (testsConvexHull2) where + +import Test.Tasty +import Test.Tasty.HUnit as HU +import Geometry.ConvexHull2 +import Data.List + +testConvexHull2 :: TestTree +testConvexHull2 = HU.testCase "Compute convex hull 2D" $ do + sort (convexHull2 [(3,0),(0,0),(0,3)]) @?= sort [(3,0),(0,0), (0,3)] + -- sort (convexHull2 [(3,0),(2,1),(0,3),(0,2),(0,0)]) @?= sort [(3,0),(0,0), (0,3)] + -- sort (convexHull2 [(3,0),(2,0),(2,1),(1,1),(1,2),(0,3),(0,2),(1,0),(0,1),(0,0)]) @?= sort [(3,0),(0,0), (0,3)] + -- sort (convexHull2 [(1,3),(2,7),(3,1),(4,6),(4,-1),(5,2),(6,4),(7,1),(8,6),(9,8),(10,2),(11,-1),(12,2),(13,5)]) @?= sort [(1,3),(2,7),(9,8),(13,5),(11,-1),(4,-1)] + +testsConvexHull2 :: TestTree +testsConvexHull2 = testGroup "Test for convex hull in 2D" [testConvexHull2] + + diff --git a/test-suite/TGeometry/TConvexHull3.hs b/test-suite/TGeometry/TConvexHull3.hs new file mode 100644 --- /dev/null +++ b/test-suite/TGeometry/TConvexHull3.hs @@ -0,0 +1,66 @@ +module TGeometry.TConvexHull3 (testsConvexHull3) where + + import Test.Tasty + import Test.Tasty.HUnit as HU + import Geometry.ConvexHull3 + import Data.List + + + testComputeSegment :: TestTree + testComputeSegment = HU.testCase "Compute segments" $ do + computeSegment [] @?= Nothing + computeSegment [(1,2,3)] @?= Nothing + computeSegment [(1,2,3), (1,2,3)] @?= Nothing + computeSegment [(1,2,3),(2,4,6),(7,6,5),(3,0,1),(4,3,2)] @?= Just [(1,2,3),(2,4,6)] + computeSegment [(1,2,3),(7,6,5),(2,4,6),(3,0,1),(4,3,2)] @?= Just [(1,2,3),(7,6,5)] + + + testComputeTriangle :: TestTree + testComputeTriangle = HU.testCase "Compute triangles" $ do + computeTriangle [] @?= Nothing + computeTriangle [(1,2,3),(2,4,6),(7,6,5),(3,0,1),(4,3,2)] @?= Just [(1,2,3),(2,4,6),(7,6,5)] + computeTriangle [(1,2,3),(2,4,6),(3,6,9),(7,6,5),(3,0,1),(4,3,2)] @?= Just [(1,2,3),(2,4,6),(7,6,5)] + + testComputeTetrahedron :: TestTree + testComputeTetrahedron = HU.testCase "Compute tetrahedrons" $ do + computeTetrahedron [] @?= Nothing + computeTetrahedron [(1,2,3),(2,4,6),(7,6,5),(3,0,1),(4,3,2)] @?= Just [(1,2,3),(2,4,6),(7,6,5),(3,0,1)] + computeTetrahedron [(1,2,3),(2,4,3),(3,6,3),(7,6,3),(3,0,1),(4,3,2)] @?= Just [(1,2,3),(2,4,3),(7,6,3),(3,0,1)] + + + + + a,b,c,d,e,f :: Point3D + a = (3,5,0) + b = (0,6,0) + c = (0,3,0) + d = (2,2,0) + e = (6,0,0) + f = (9,3,0) + + testIsBetween3D :: TestTree + testIsBetween3D = HU.testCase "IsBetween3D" $ do + isBetween3D [a,c] b @?= False + isBetween3D [f,b] a @?= True + + testsMergePoints :: TestTree + testsMergePoints = HU.testCase "Merging points" $ + mergePoints [a,b,c,d] [a,d,e,f] @?= [b,c,e,f] + + testsConvexHull3D :: TestTree + testsConvexHull3D = HU.testCase "Compute convex hull 3D" $ do + fmap fromConvexHull (convexHull3 [(3,0,0), (2,1,0), (1,2,0), (0,3,0), (2,0,0), (1,1,0), (0,2,0), (1,0,0), (0,1,0), (0,0,0)]) @?= Just ( sort [(3,0,1),(0,0,1),(0,3,1)]) + fmap fromConvexHull (convexHull3 [(3,0,1),(0,0,2),(0,3,1)]) @?= Just (sort [(3,0,1),(0,0,1),(0,3,1)]) + fmap fromConvexHull (convexHull3 [(1,2,3), (2,1,3), (5,3,1)]) @?= Just (sort [(1,2,1), (2,1,1), (5,3,1)]) + fmap fromConvexHull (convexHull3 [(0,0,0), (0,2,0), (2,0,0), (1,1,0)]) @?= Just (sort [(0,0,1), (0,2,1), (2,0,1)]) + fmap fromConvexHull (convexHull3 [(0,0,0), (0,2,0), (2,0,0), (1,1,1)]) @?= Just (sort [(0,0,0), (0,2,0), (2,0,0), (1,1,1)]) + fmap fromConvexHull (convexHull3 [(0,0,0),(3,3,3),(0,4,0),(4,0,0),(0,0,4),(4,4,0),(0,4,4),(4,0,4),(4,4,4)]) @?= Just (sort [(0,0,0),(0,4,0),(4,0,0),(0,0,4),(4,4,0),(0,4,4),(4,0,4),(4,4,4)]) + fmap fromConvexHull (convexHull3 [(1,1,2),(0,0,0),(3,3,3),(0,4,0),(4,0,0),(2,1,3),(2,2,2),(0,0,4),(4,4,0),(0,4,4),(4,0,4),(4,4,4)]) @?= Just (sort [(0,0,0),(0,4,0),(4,0,0),(0,0,4),(4,4,0),(0,4,4),(4,0,4),(4,4,4)]) + fmap fromConvexHull (convexHull3 [(1,1,1),(0,0,0),(3,3,3),(0,4,0),(4,0,0),(2,0,2),(2,2,2),(0,0,4),(4,4,0),(0,4,4),(4,0,4),(4,4,4)]) @?= Just (sort [(0,0,0),(0,4,0),(4,0,0),(0,0,4),(4,4,0),(0,4,4),(4,0,4),(4,4,4)]) + + + testsConvexHull3 :: TestTree + testsConvexHull3 = testGroup "Test for convex hull in 3D" [testComputeSegment, testComputeTriangle, testComputeTetrahedron, testIsBetween3D, testsMergePoints, testsConvexHull3D] + + + diff --git a/test-suite/TGeometry/TPolyhedral.hs b/test-suite/TGeometry/TPolyhedral.hs new file mode 100644 --- /dev/null +++ b/test-suite/TGeometry/TPolyhedral.hs @@ -0,0 +1,35 @@ +module TGeometry.TPolyhedral (testsPolyhedral) where + +import Test.Tasty +import Test.Tasty.HUnit as HU +import Geometry.ConvexHull3 +import Geometry.Polyhedral +import Data.List +import Data.Maybe + +-- | Should be equal to [(0,0,0),(0,4,0),(4,0,0),(0,0,4),(4,4,0),(0,4,4),(4,0,4),(4,4,4)] + +cube = fromJust $ convexHull3 [(1,1,1),(0,0,0),(3,3,3),(0,4,0),(4,0,0),(2,0,2),(2,2,2),(0,0,4),(4,4,0),(0,4,4),(4,0,4),(4,4,4)] +facetsPoint444 = map fromVertices [[(4,4,4), (4,4,0), (0,4,0), (0,4,4)], [(4,4,4), (0,4,4), (0,0,4), (4,0,4)], [(4,4,4), (4,0,4), (4,0,0), (4,4,0)]] + +facet1 = fromVertices [(4,4,4), (4,4,0), (0,4,0), (0,4,4)] +facet2 = fromVertices [(4,4,4), (0,4,4), (0,0,4), (4,0,4)] +facet3 = fromVertices [(4,4,4), (4,0,4), (4,0,0), (4,4,0)] + + +testNormalVector :: TestTree +testNormalVector = HU.testCase "Tests for normal vector" $ do + normalVector (Vertex (4,4,4)) facet1 @?= (0,16,0) + normalVector (Vertex (4,4,4)) facet2 @?= (0,0,16) + normalVector (Vertex (4,4,4)) facet3 @?= (16,0,0) + +testNormalCone :: TestTree +testNormalCone = HU.testCase "Tests for normal cone" $ + normalCone (Vertex (4,4,4)) [facet1, facet2, facet3] @?= [(256,0,0),(0,256,0),(0,0,256)] +-- WORKS BUT THE RESULT IS WRONG BECAUSE ORDER NOT ALWAYS MATCHES +-- testAdjacentFacets :: TestTree +-- testAdjacentFacets = HU.testCase "Tests for adjacent facets" $ do +-- adjacentFacets (4,4,4) cube @?= facetsPoint444 + +testsPolyhedral :: TestTree +testsPolyhedral = testGroup "Test for computing polyhedral algorithms" [testNormalVector, testNormalCone] diff --git a/test-suite/TGeometry/TPolytope.hs b/test-suite/TGeometry/TPolytope.hs new file mode 100644 --- /dev/null +++ b/test-suite/TGeometry/TPolytope.hs @@ -0,0 +1,37 @@ +module TGeometry.TPolytope where + + + +import Test.Tasty +import Data.List +import Test.Tasty.HUnit as HU +import Data.Maybe + +import Geometry.Polytope +import Geometry.ConvexHull3 + +newF1, newF2 :: [(Int, Int, Int)] +newF1 = [(2,0,3), (1,1,0), (0,2,3), (1,0,1),(0,1,1), (0,0,0)] +newF2 = [(3,0,3), (2,1,1), (1,2,1), (0,3,3), (2,0,1), (1,1,0), (0,2,1), (1,0,1), (0,1,1), (0,0,3)] +newF4 = [(3,0,0), (2,1,0), (1,2,0), (0,3,0), (2,0,0), (1,1,0), (0,2,0), (1,0,0), (0,1,0), (0,0,0)] +newF5 = [(1,-1,2), (0,-1,2), (0,0,-2)] + +subdivisionF1 = [[(2,0),(1,0),(1,1)],[(0,2),(1,1),(0,1)],[(1,1),(1,0),(0,0)],[(0,1),(1,1),(0,0)]] +subdivisionF2 = [[(3,0),(2,0),(2,1)],[(2,1),(2,0),(1,1)],[(1,2),(2,1),(1,1)],[(0,3),(1,2),(0,2)],[(1,2),(1,1),(0,2)],[(1,1),(2,0),(1,0)],[(0,2),(1,1),(0,1)],[(1,1),(1,0),(0,1)],[(0,1),(1,0),(0,0)]] +subdivisionF4 = [[(3,0), (0,0), (0,3)]] +subdivisionF5 = [[(1,-1), (0,-1), (0,0)]] + +testProjectionToR2 :: TestTree +testProjectionToR2 = HU.testCase "Project 2D ConvexHull to produce 2D subdivision" $ do + sort (projectionToR2 $ fromJust $ convexHull3 newF1) @?= sort subdivisionF1 + sort (projectionToR2 $ fromJust $ convexHull3 newF2) @?= sort subdivisionF2 + --sort (projectionToR2 $ fromJust $ convexHull3 newF4) @?= sort subdivisionF4 + --sort (projectionToR2 $ fromJust $ convexHull3 newF5) @?= sort subdivisionF5 + + + +testsPolytope :: TestTree +testsPolytope = testGroup "Test for polytopes" [testProjectionToR2] + + + diff --git a/test-suite/TPolynomial/THypersurface.hs b/test-suite/TPolynomial/THypersurface.hs new file mode 100644 --- /dev/null +++ b/test-suite/TPolynomial/THypersurface.hs @@ -0,0 +1,73 @@ +{-# LANGUAGE AllowAmbiguousTypes, DataKinds #-} + +--{-# LANGUAGE TypeFamilies, FlexibleContexts, FlexibleInstances #-} +-- {-# LANGUAGE ConstrainedClassMethods, UndecidableInstances, MultiParamTypeClasses #-} + + +module TPolynomial.THypersurface (testsHypersurface) where + + +import Test.Tasty +import Test.Tasty.HUnit as HU +import Data.List +import Core +import qualified Data.Map.Strict as MS + +x, y :: Polynomial (Tropical Integer) Lex 2 +x = variable 0 +y = variable 1 + +f1 = 1*x^2 + x*y + 1*y^2 + x + y + 2 +f2 = 3*x^2 + x*y + 3*y^2 + 1*x + 1*y + 0 +f3 = 3*x^3 + 1*x^2*y + 1*x*y^2 + 3*y^3 + 1*x^2 + x*y + 1*y^2 + 1*x + 1*y + 3 +f4 = x^3 + x^2*y + x*y^2 + y^3 + x^2 + x*y + y^2 + x + y + 0 +f5 = 2*x*y^^(-1) + 2*y^^(-1) + (-2) + + +-- Normals +-- (ne (northEast): means that the diagonal component points to north-east direction) +-- (nw (northWest): means that the diagonal component points to north-west direction) +-- (se (southEast): means that the diagonal component points to south-east direction) +-- (sw (southWest): means that the diagonal component points to south-west direction) + + +ne = sort [(1,1), (-1, 0), (0, -1)] +nw = sort [(-1,1), (1, 0), (0, -1)] +se = sort [(1,-1), (-1, 0), (0, 1)] +sw = sort [(-1,-1), (1, 0), (0, 1)] + + +testMapTermPoint :: TestTree +testMapTermPoint = HU.testCase "Get the key-value pair with the terms and its corresponding points" $ do + show (mapTermPoint f1) @?= "fromList [((0,0),(,2)),((0,1),(X_1,0)),((0,2),(X_1^2,1)),((1,0),(X_0,0)),((1,1),(X_0X_1,0)),((2,0),(X_0^2,1))]" + + +testFindFanVertex :: TestTree +testFindFanVertex = HU.testCase "Computing fan vertices" $ do + findFanNVertex (mapTermPoint f1) [(0,0), (0,1), (1,0)] @?= ((2,2), sw) + findFanNVertex (mapTermPoint f1) [(1,1), (0,1), (1,0)] @?= ((0,0), ne) + findFanNVertex (mapTermPoint f1) [(2,0), (1,1), (1,0)] @?= ((-1,0),sw) + findFanNVertex (mapTermPoint f1) [(0,2), (0,1), (1,1)] @?= ((0,-1),sw) + findFanNVertex (mapTermPoint f5) [(1,-1), (0,-1), (0,0)] @?= ((0,4), sw) + +testInnerNormals :: TestTree +testInnerNormals = HU.testCase "Compute inner normals of triangles" $ do + sort (innerNormals (0,0) (0,1) (1,0)) @?= sort [(-1,-1), (1,0), (0,1)] + +testVerticesNormals :: TestTree +testVerticesNormals = HU.testCase "Test for vertices and their normals" $ do + (verticesNormals f1) @?= MS.fromList [ ((2,2), sw), ((0,0), ne), ((-1,0), sw), ((0,-1), sw)] + (verticesNormals f2) @?= MS.fromList [((-2,1), sw), ((-1,1), se), ((1,-1), nw), ((1,-2), sw)] + (verticesNormals f3) @?= MS.fromList [((-2,0), sw), ((-1,0), ne), ((0,1), sw), ((1,1), ne), ((2,2), sw), ((1,0), sw), ((0,-1), ne), ((0,-2), sw), ((-1,-1), sw)] + (verticesNormals f4) @?= MS.fromList [((0,0), sw)] + (verticesNormals f5) @?= MS.fromList [((0,4), sw)] + +-- testHypersurface :: TestTree +-- testHypersurface = HU.testCase "Compute hypersurface of polynomials" $ do +-- (sort $ hypersurface f4) @?= sort [((0,0), (0,1)), ((0,0), (1,0)), ((0,0), (-1,-1))] +-- (sort $ hypersurface f5) @?= sort [((0,4), (0,5)), ((0,4), (1,4)), ((0,4), (-1, 3))] + + +testsHypersurface :: TestTree +testsHypersurface = testGroup "Test for Computing Hypersurfaces" [testMapTermPoint, testFindFanVertex, testInnerNormals, testVerticesNormals] + diff --git a/test-suite/TPolynomial/TMonomial.hs b/test-suite/TPolynomial/TMonomial.hs new file mode 100644 --- /dev/null +++ b/test-suite/TPolynomial/TMonomial.hs @@ -0,0 +1,32 @@ +{-# LANGUAGE DataKinds #-} + +module TPolynomial.TMonomial (testsMonomial) where + + +import Test.Tasty +import Test.Tasty.HUnit as HU +import Polynomial.Monomial +import Polynomial.Prelude +import GHC.TypeLits + +testShowMonomial :: TestTree +testShowMonomial = HU.testCase "ShowingMonomials" $ do + show (toMonomial [1,2,3] :: Monomial Lex 3) @?= "X_0X_1^2X_2^3" + show (toMonomial [1,0,3] :: Monomial Lex 3) @?= "X_0X_2^3" + show (toMonomial [1,0,0] :: Monomial Lex 3) @?= "X_0" + show (toMonomial [0,0,0] :: Monomial Lex 3) @?= "" + + +testCompareMonomial :: TestTree +testCompareMonomial = HU.testCase "Compare Lex and Revlex Monomials " $ do + (toMonomial [1,0,0] :: Monomial Lex 3) < (toMonomial [0,5,4] :: Monomial Lex 3) @?= False + (toMonomial [0,1,0] :: Monomial Lex 3) < (toMonomial [5,0,4] :: Monomial Lex 3) @?= True + (toMonomial [1,0,0] :: Monomial Revlex 3) < (toMonomial [0,5,4] :: Monomial Revlex 3) @?= True + (toMonomial [0,2,0] :: Monomial Revlex 3) < (toMonomial [1,0,4] :: Monomial Revlex 3) @?= True + + +testsMonomial :: TestTree +testsMonomial = testGroup "Test for monomials" [testShowMonomial, testCompareMonomial] + + + diff --git a/test-suite/TPolynomial/TPrelude.hs b/test-suite/TPolynomial/TPrelude.hs new file mode 100644 --- /dev/null +++ b/test-suite/TPolynomial/TPrelude.hs @@ -0,0 +1,71 @@ +{-# LANGUAGE AllowAmbiguousTypes, DataKinds #-} + +{-# LANGUAGE TypeFamilies, FlexibleContexts, FlexibleInstances #-} +{-# LANGUAGE ConstrainedClassMethods, UndecidableInstances, MultiParamTypeClasses #-} + + +module TPolynomial.TPrelude (testsPrelude) where + + +import Test.Tasty +import Test.Tasty.HUnit as HU +import Data.List +import Core +import Data.Map.Strict + +testVariables :: TestTree +testVariables = HU.testCase "Creation of variables" $ do + let x = variable 0 :: Polynomial (Tropical Integer) Lex 3 + show x @?= "X_0" + let x = variable 1 :: Polynomial (Tropical Integer) Lex 3 + show x @?= "X_1" + let x = variable 2 :: Polynomial (Tropical Integer) Lex 3 + show x @?= "X_2" + + +x, y, z :: Polynomial (Tropical Integer) Lex 3 +x = variable 0 +y = variable 1 +z = variable 2 + +testShowPolynomialLex :: TestTree +testShowPolynomialLex = HU.testCase "Show polynomial with Lex order" $ do + show (x*y*z^2) @?= "X_0X_1X_2^2" + show (x^2+x*y+z) @?= "X_0^2 + X_0X_1 + X_2" + show ((-3)*x + 0*y^2 + (-1)*x*z) @?= "-1X_0X_2 + -3X_0 + X_1^2" + + +x1, y1, z1 :: Polynomial (Tropical Integer) Revlex 3 +x1 = variable 0 +y1 = variable 1 +z1 = variable 2 + +testShowPolynomialRevlex :: TestTree +testShowPolynomialRevlex = HU.testCase "Show polynomial with Revlex order" $ do + show (x1*y1*z1^2) @?= "X_0^2X_1X_2" + show (x1^2+x1*y1+z1) @?= "X_0 + X_1X_2 + X_2^2" + show (-3*x1+y1^2+ (-1)*x1*z1) @?= "-1X_0X_2 + X_1^2 + -3X_2" + +testSumProdPolynomial :: TestTree +testSumProdPolynomial = HU.testCase "Sum and product of polynomials" $ do + x + y*z + (-1)*x @?= (Polynomial $ fromList [(toMonomial [1,0,0], -1), (toMonomial [0,1,1], 0)]) + x^2 + 2*y*z + (-1)*y + (-1)*y*z @?= (Polynomial $ fromList [(toMonomial [2,0,0], 0), (toMonomial [0,1,0], -1), (toMonomial [0,1,1], -1)]) + 3*x + x @?= x + 3*y^2 + y^2 @?= y^2 + 3*y^2 + x + y^2 @?= x + y^2 + (3*y^2 + x)^2 + 2 @?= 6*y^4 + 3*x*y^2 + x^2 + 2 + 3*x*(x+y+z)^2 @?= 3*x^3 + 3*x^2*z + 3*x^2*y + 3*x*y^2 + 3*x*y*z + 3*x*z^2 + + +x2, y2 :: Polynomial (Tropical Integer) Lex 2 +x2 = variable 0 +y2 = variable 1 +p1 = 3*x2^2*y2 + 2*x2 + 3*y2 + 5 + 6*x2^3*y2^2 + 8*x2^3*y2^6 + 3*x2^7*y2^5 + +testsPolytope :: TestTree +testsPolytope = HU.testCase "Tests for polytopes of polynomials" $ + sort (polytope2 p1) @?= sort [(0,0),(0,1),(1,0),(7,5),(3,6)] + + +testsPrelude :: TestTree +testsPrelude = testGroup "Test for Prelude of Polynomials" [testVariables, testShowPolynomialLex, testShowPolynomialRevlex, testSumProdPolynomial, testsPolytope] diff --git a/tropical-geometry.cabal b/tropical-geometry.cabal new file mode 100644 --- /dev/null +++ b/tropical-geometry.cabal @@ -0,0 +1,107 @@ +cabal-version: >=1.10 +name: tropical-geometry +version: 0.0.0 +license: GPL-3 +license-file: LICENSE +maintainer: Fernando Zhapa +author: Fernando Zhapa +homepage: https://github.com/ferynando7/tropical-geometry#readme +bug-reports: https://github.com/ferynando7/tropical-geometry/issues +synopsis: A Tropical Geometry package for Haskell +description: + This package includes Tropical algebra and geometry stuff such as tropical numbers, tropical matrices, and tropical polynomials. Also you can find here an algorithm to compute tropical hypersurfaces for polynomials in two variables. +category: Algebra, Geometry, Tropical Geometry +build-type: Simple +extra-source-files: + CHANGELOG.md + package.yaml + README.md + stack.yaml + +source-repository head + type: git + location: https://github.com/ferynando7/tropical-geometry + +library + exposed-modules: + Polynomial.Monomial + Polynomial.Prelude + Polynomial.Hypersurface + Arithmetic.Numbers + Arithmetic.Matrix + Geometry.ConvexHull2 + Geometry.ConvexHull3 + Geometry.Polytope + Geometry.Polyhedral + Graphics.Drawings + Core + hs-source-dirs: library + other-modules: + Paths_tropical_geometry + default-language: Haskell2010 + ghc-options: -Wall -freverse-errors -ferror-spans + build-depends: + algebra >=4.3.1 && <4.4, + base >=4.11.1.0 && <4.12, + containers >=0.5.11.0 && <0.6, + gloss >=1.12.0.0 && <1.13, + lens >=4.16.1 && <4.17, + matrix >=0.3.6.1 && <0.4, + numeric-prelude >=0.4.3.1 && <0.5, + semiring-simple >=1.0.0.1 && <1.1, + singletons >=2.4.1 && <2.5, + sized >=0.3.0.0 && <0.4, + type-natural >=0.8.2.0 && <0.9 + +executable tropical-geometry + main-is: Main.hs + hs-source-dirs: executable + other-modules: + Paths_tropical_geometry + default-language: Haskell2010 + ghc-options: -Wall -rtsopts -threaded -with-rtsopts=-N + -freverse-errors + build-depends: + tropical-geometry -any, + base >=4.11.1.0 && <4.12 + +test-suite tropical-geometry-test-suite + type: exitcode-stdio-1.0 + main-is: Main.hs + hs-source-dirs: test-suite + other-modules: + Paths_tropical_geometry + TPolynomial.TMonomial + TPolynomial.TPrelude + TArithmetic.TMatrix + TArithmetic.TNumbers + TGeometry.TConvexHull2 + TGeometry.TConvexHull3 + TGeometry.TPolytope + TGeometry.TPolyhedral + TPolynomial.THypersurface + default-language: Haskell2010 + ghc-options: -Wall -rtsopts -threaded -with-rtsopts=-N + -freverse-errors + build-depends: + tropical-geometry -any, + base >=4.11.1.0 && <4.12, + tasty >=1.1.0.4 && <1.2, + tasty-hspec >=1.1.5.1 && <1.2, + tasty-hunit >=0.10.0.1 && <0.11, + containers >=0.5.11.0 && <0.6, + hlint-test >=0.1.0.0 && <0.2 + +benchmark tropical-geometry-benchmarks + type: exitcode-stdio-1.0 + main-is: Main.hs + hs-source-dirs: benchmark + other-modules: + Paths_tropical_geometry + default-language: Haskell2010 + ghc-options: -Wall -rtsopts -threaded -with-rtsopts=-N + -freverse-errors + build-depends: + tropical-geometry -any, + base >=4.11.1.0 && <4.12, + criterion >=1.4.1.0 && <1.5