diff --git a/ChangeLog.md b/ChangeLog.md new file mode 100644 --- /dev/null +++ b/ChangeLog.md @@ -0,0 +1,5 @@ +# Revision history for spatial-rotations + +## 0.1.0.0 -- YYYY-mm-dd + +* First version. Released on an unsuspecting world. 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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But first, please read +. diff --git a/Math/Rotations/Class.hs b/Math/Rotations/Class.hs new file mode 100644 --- /dev/null +++ b/Math/Rotations/Class.hs @@ -0,0 +1,166 @@ +-- | +-- Module : Math.Rotations.Class +-- Copyright : (c) Justus Sagemüller 2018 +-- License : GPL v3 +-- +-- Maintainer : (@) jsagemue $ uni-koeln.de +-- Stability : experimental +-- Portability : portable +-- + +{-# LANGUAGE TypeFamilies #-} +{-# LANGUAGE FlexibleInstances #-} + +module Math.Rotations.Class ( Rotatable (..) + , xAxis, yAxis, zAxis + , (°) + -- * Utility + , rotateViaEulerAnglesYZ + , rotateℝ³AboutCenteredAxis + -- * Internals + , rotmatrixForAxis + , eulerAnglesZYZForMatrix + , rotmatrixForEulerAnglesZYZ + ) where + +import Math.Manifold.Core.Types +import Data.VectorSpace +import Linear.V3 (V3(V3)) + +class Rotatable m where + type AxisSpace m :: * + rotateAbout :: AxisSpace m -> S¹ -> m -> m + +instance (Rotatable m) => Rotatable (m -> Double) where + type AxisSpace (m -> Double) = AxisSpace m + rotateAbout ax (S¹Polar δφ) f = f . rotateAbout ax (S¹Polar $ -δφ) + +instance Rotatable S¹ where + type AxisSpace S¹ = ℝP⁰ + rotateAbout ℝPZero (S¹Polar δφ) (S¹Polar φ) + | φ' > pi = S¹Polar $ φ'-tau + | φ' < -pi = S¹Polar $ φ'+tau + | otherwise = S¹Polar φ' + where φ' = φ + δφ + +rotateViaEulerAnglesYZ + :: (S¹ -> m -> m) -- ^ Y-axis rotation method + -> (S¹ -> m -> m) -- ^ Z-axis rotation method + -> (ℝP² -> S¹ -> m -> m) -- ^ Suitable definition for 'rotateAbout' +rotateViaEulerAnglesYZ yRot zRot ax = rotAroundAxis . rotmatrixForAxis ax + where rotAroundAxis mat = case eulerAnglesZYZForMatrix mat of + [θz₀, θy, θz₁] -> zRot (S¹Polar θz₁) . yRot (S¹Polar θy) . zRot (S¹Polar θz₀) + +rotmatrixForAxis :: ℝP² -> S¹ -> [[ℝ]] +rotmatrixForAxis (HemisphereℝP²Polar θax φax) = rotAroundAxis + where rotAroundAxis (S¹Polar θ) = [[r₀₀,r₀₁,r₀₂] + ,[r₁₀,r₁₁,r₁₂] + ,[r₂₀,r₂₁,r₂₂]] + -- https://en.wikipedia.org/w/index.php?title=Rotation_formalisms_in_three_dimensions&oldid=823164970#Rotation_matrix_%E2%86%94_Euler_axis/angle + where r₀₀ = one_cosθ*e₀^2 + cosθ + r₀₁ = one_cosθ*e₀*e₁ - e₂*sinθ + r₀₂ = one_cosθ*e₀*e₂ + e₁*sinθ + r₁₀ = one_cosθ*e₁*e₀ + e₂*sinθ + r₁₁ = one_cosθ*e₁^2 + cosθ + r₁₂ = one_cosθ*e₁*e₂ - e₀*sinθ + r₂₀ = one_cosθ*e₂*e₀ - e₁*sinθ + r₂₁ = one_cosθ*e₂*e₁ + e₀*sinθ + r₂₂ = one_cosθ*e₂^2 + cosθ + cosθ = cos θ + sinθ = sin θ + one_cosθ = 1 - cos θ + + e₀ = cos φax * sin θax + e₁ = sin φax * sin θax + e₂ = cos θax + +rotmatrixForEulerAnglesZYZ :: [ℝ] -> [[ℝ]] +rotmatrixForEulerAnglesZYZ angles + = [[ cy*cz₀*cz₁-sz₀*sz₁, -cy*sz₀*cz₁-cz₀*sz₁, sy*cz₁ ] + ,[ cy*cz₀*sz₁+cz₁*sz₀, cz₀*cz₁-cy*sz₀*sz₁, sy*sz₁ ] + ,[ -sy*cz₀ , sy*sz₀ , cy ]] + where [cz₀,cy,cz₁] = cos<$>angles + [sz₀,sy,sz₁] = sin<$>angles + +eulerAnglesZYZForMatrix :: [[ℝ]] -> [ℝ] +eulerAnglesZYZForMatrix [[r₀₀,r₀₁,r₀₂] + ,[r₁₀,r₁₁,r₁₂] + ,[r₂₀,r₂₁,r₂₂]] + = [θz₀,θy,θz₁] + where + -- Rotation matrix for z₀-y-z₁ rotation, with cy := cos θy etc.: + -- + -- ⎛ r₀₀ r₀₁ r₀₂ ⎞ ⎛ cz₁ -sz₁ 0 ⎞ ⎛ cy 0 sy ⎞ ⎛ cz₀ -sz₀ 0 ⎞ + -- ⎜ r₁₀ r₁₁ r₁₂ ⎟ = ⎜ sz₁ cz₁ 0 ⎟ · ⎜ 0 1 0 ⎟ · ⎜ sz₀ cz₀ 0 ⎟ + -- ⎝ r₂₀ r₂₁ r₂₂ ⎠ ⎝ 0 0 1 ⎠ ⎝-sy 0 cy ⎠ ⎝ 0 0 1 ⎠ + -- + -- ⎛ cz₁ -sz₁ 0 ⎞ ⎛ cy·cz₀ -cy·sz₀ sy ⎞ + -- = ⎜ sz₁ cz₁ 0 ⎟ · ⎜ sz₀ cz₀ 0 ⎟ + -- ⎝ 0 0 1 ⎠ ⎝-sy·cz₀ sy·sz₀ cy ⎠ + -- + -- ⎛ cy·cz₀·cz₁−sz₀·sz₁ -cy·sz₀·cz₁−cz₀·sz₁ sy·cz₁ ⎞ + -- = ⎜ cy·cz₀·sz₁+cz₁·sz₀ cz₀·cz₁−cy·sz₀·sz₁ sy·sz₁ ⎟ + -- ⎝ -sy·cz₀ sy·sz₀ cy ⎠ + -- + -- Here, one can immediately read off + cy = r₂₂ + -- ...but the naïve choice + -- θy = acos cy = acos r₂₂ + -- is unstable. Better: + -- sqrt(r₂₀²+r₂₁²) = |sy|·(cz₁²+sz₁²) = sy + sy = sqrt $ r₂₀^2+r₂₁^2 + θy = atan2 sy cy + -- We can always choose + -- θz₀ = atan2 sz₀ cz₀ + -- = atan2 (sy·sz₀) (sy·cz₀) ∀sy‡0 + θz₀ = atan2 r₂₁ (-r₂₀) ; sz₀ = sin θz₀; cz₀ = cos θz₀ + -- ...noting however that this becomes underconstrained for small |sy|, so + -- the analogous θz₁ = atan2 r₂₁ (-r₂₀) should /not/ be used. Instead, put + -- in the y unit vector turned back by θz₀ (θy has no effect): + -- ⎛ r₀₀ r₀₁ r₀₂ ⎞ ⎛ cy·cz₀ -cy·sz₀ sy ⎞⁻¹ ⎛0⎞ ⎛ cz₁ -sz₁ 0 ⎞⎛0⎞ ⎛-sz₁⎞ + -- ⎜ r₁₀ r₁₁ r₁₂ ⎟ · ⎜ sz₀ cz₀ 0 ⎟ $ ⎜1⎟ = ⎜ sz₁ cz₁ 0 ⎟⎜1⎟ = ⎜ cz₁⎟ + -- ⎝ r₂₀ r₂₁ r₂₂ ⎠ ⎝-sy·cz₀ sy·sz₀ cy ⎠ ⎝0⎠ ⎝ 0 0 1 ⎠⎝0⎠ ⎝ 0 ⎠ + -- + -- Here we have, using orthogonality, + -- ⎛ cy·cz₀ -cy·sz₀ sy ⎞⁻¹⎛0⎞ ⎛ cy·cz₀ sz₀ -sy·cz₀ ⎞⎛0⎞ ⎛sz₀⎞ + -- ⎜ sz₀ cz₀ 0 ⎟ ⎜1⎟ = ⎜ -cy·sz₀ cz₀ sy·sz₀ ⎟⎜1⎟ = ⎜cz₀⎟ + -- ⎝-sy·cz₀ sy·sz₀ cy ⎠ ⎝0⎠ ⎝ sy 0 cy ⎠⎝0⎠ ⎝ 0 ⎠ + θz₁ = atan2 (-r₀₀*sz₀ - r₀₁*cz₀) + ( r₁₀*sz₀ + r₁₁*cz₀) + +instance Rotatable S² where + type AxisSpace S² = ℝP² + rotateAbout = rotateViaEulerAnglesYZ + (\(S¹Polar β) (S²Polar θ φ) + -> let x₀ = cos φ * sin θ + y = sin φ * sin θ + z₀ = cos θ + x₁ = x₀ * cos β + z₀ * sin β + z₁ = -x₀ * sin β + z₀ * cos β + rxy = sqrt $ x₁^2 + y^2 + in S²Polar (atan2 rxy z₁) (atan2 y x₁) ) + (\γ (S²Polar θ φ) -> case rotateAbout ℝPZero γ (S¹Polar φ) of + S¹Polar φ' -> S²Polar θ φ') + +xAxis, yAxis, zAxis :: ℝP² +xAxis = HemisphereℝP²Polar (pi/2) 0 +yAxis = HemisphereℝP²Polar (pi/2) (pi/2) +zAxis = HemisphereℝP²Polar 0 0 + +infix 5 ° + +-- | Rotate by an angle specified in degrees. +(°) :: Rotatable m => ℝ -> AxisSpace m -> m -> m +angle° axis = rotateAbout axis . S¹Polar $ angle * pi/180 + + +rotateℝ³AboutCenteredAxis :: ℝP² -> S¹ -> V3 ℝ -> V3 ℝ +rotateℝ³AboutCenteredAxis axis angle = case rotmatrixForAxis axis angle of + [ [r₀₀,r₀₁,r₀₂] + ,[r₁₀,r₁₁,r₁₂] + ,[r₂₀,r₂₁,r₂₂] ] -> \(V3 x y z) -> V3 (r₀₀*x + r₀₁*y + r₀₂*z) + (r₁₀*x + r₁₁*y + r₁₂*z) + (r₂₀*x + r₂₁*y + r₂₂*z) + +tau :: ℝ +tau = 2*pi 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/spatial-rotations.cabal b/spatial-rotations.cabal new file mode 100644 --- /dev/null +++ b/spatial-rotations.cabal @@ -0,0 +1,49 @@ +-- Initial spatial-rotations.cabal generated by cabal init. For further +-- documentation, see http://haskell.org/cabal/users-guide/ + +name: spatial-rotations +version: 0.1.0.0 +synopsis: Rotate about any suitable axis +-- description: +homepage: https://github.com/leftaroundabout/rotations +license: GPL-3 +license-file: LICENSE +author: Justus Sagemüller +maintainer: (@) jsagemue $ uni-koeln.de +-- copyright: +category: Math +build-type: Simple +extra-source-files: ChangeLog.md +cabal-version: >=1.10 + +library + exposed-modules: Math.Rotations.Class + -- other-modules: + -- other-extensions: + build-depends: base >=4.10 && <4.11 + , manifolds-core >=0.5 && <0.6 + , vector-space >=0.12 && <0.13 + , linear + -- hs-source-dirs: + default-language: Haskell2010 + +test-suite test + default-language: + Haskell2010 + type: + exitcode-stdio-1.0 + hs-source-dirs: + test/tasty + main-is: + test.hs + build-depends: + base >= 4 && < 5 + , tasty >= 0.7 + , tasty-hunit + , tasty-quickcheck + , manifolds + , pragmatic-show + , containers + , vector-space + , spatial-rotations + diff --git a/test/tasty/test.hs b/test/tasty/test.hs new file mode 100644 --- /dev/null +++ b/test/tasty/test.hs @@ -0,0 +1,183 @@ +-- | +-- Module : Main +-- Copyright : (c) Justus Sagemüller 2018 +-- License : GPL v3 +-- +-- Maintainer : (@) sagemueller $ geo.uni-koeln.de +-- Stability : experimental +-- Portability : portable +-- + +{-# LANGUAGE OverloadedLists, TypeFamilies, FlexibleContexts, UndecidableInstances #-} +{-# LANGUAGE TypeOperators #-} + +module Main where + +import qualified Text.Show.Pragmatic as SP +import Math.Rotations.Class + +import Data.Manifold + +import Test.Tasty +import Test.Tasty.HUnit +import qualified Test.Tasty.QuickCheck as QC +import Test.Tasty.QuickCheck ((==>)) + + +main = defaultMain tests + +tests :: TestTree +tests = testGroup "Tests" + [ testGroup "Rotation matrices" + [ testCase "180° x-axis" $ rotmatrixForAxis xAxis (rad π) + @?≈ [ [1,0,0], [0,-1,0], [0,0,-1] ] + , testCase " 90° x-axis" $ rotmatrixForAxis xAxis (rad π'₂) + @?≈ [ [1,0,0], [0,0,-1], [0,1,0] ] + , testCase "-90° x-axis" $ rotmatrixForAxis xAxis (rad $ -π'₂) + @?≈ [ [1,0,0], [0,0,1], [0,-1,0] ] + , testCase " 45° x-axis" $ rotmatrixForAxis xAxis (rad π'₄) + @?≈ [ [1,0,0], [0,sqrt 2/2,-sqrt 2/2], [0,sqrt 2/2,sqrt 2/2] ] + , testCase "180° y-axis" $ rotmatrixForAxis yAxis (rad π) + @?≈ [ [-1,0,0], [0,1,0], [0,0,-1] ] + , testCase " 90° y-axis" $ rotmatrixForAxis yAxis (rad π'₂) + @?≈ [ [0,0,1], [0,1,0], [-1,0,0] ] + , testCase "-90° y-axis" $ rotmatrixForAxis yAxis (rad $ -π'₂) + @?≈ [ [0,0,-1], [0,1,0], [1,0,0] ] + , testCase " 45° y-axis" $ rotmatrixForAxis yAxis (rad π'₄) + @?≈ [ [sqrt 2/2,0,sqrt 2/2], [0,1,0], [-sqrt 2/2,0,sqrt 2/2] ] + , testCase "180° z-axis" $ rotmatrixForAxis zAxis (rad π) + @?≈ [ [-1,0,0], [0,-1,0], [0,0,1] ] + , testCase " 90° z-axis" $ rotmatrixForAxis zAxis (rad π'₂) + @?≈ [ [0,-1,0], [1,0,0], [0,0,1] ] + , testCase "-90° z-axis" $ rotmatrixForAxis zAxis (rad $ -π'₂) + @?≈ [ [0,1,0], [-1,0,0], [0,0,1] ] + , testCase " 45° z-axis" $ rotmatrixForAxis zAxis (rad π'₄) + @?≈ [ [sqrt 2/2,-sqrt 2/2,0], [sqrt 2/2,sqrt 2/2,0], [0,0,1] ] + ] + , testGroup "Euler angles" + [ testCase "180° x-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis xAxis (rad π)) + @?≈ [ π³₄, π ,-π'₄] + , testCase " 90° x-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis xAxis (rad π'₂)) + @?≈ [ π'₂, π'₂,-π'₂] + , testCase "-90° x-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis xAxis (rad $ -π'₂)) + @?≈ [-π'₂, π'₂, π'₂] + , testCase " 45° x-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis xAxis (rad π'₄)) + @?≈ [ π'₂, π'₄,-π'₂] + , testCase "180° y-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis yAxis (rad π)) + @?≈ [ π'₄, π , π'₄] + , testCase " 90° y-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis yAxis (rad π'₂)) + @?≈ [ 0 , π'₂, 0 ] + , testCase "-90° y-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis yAxis (rad $ -π'₂)) + @?≈ [-π , π'₂, π ] + , testCase " 45° y-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis yAxis (rad π'₄)) + @?≈ [ 0 , π'₄, 0 ] + , testCase "180° z-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis zAxis (rad π)) + @?≈ [ π , 0 , 0 ] + , testCase " 90° z-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis zAxis (rad π'₂)) + @?≈ [ π , 0 ,-π'₂] + , testCase "-90° z-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis zAxis (rad $ -π'₂)) + @?≈ [ π , 0 , π'₂] + , testCase " 45° z-axis" $ eulerAnglesZYZForMatrix (rotmatrixForAxis zAxis (rad π'₄)) + @?≈ [ π , 0 ,-π³₄] + , QC.testProperty "Arbitrary rotation-matrix reconstruction" + $ \ax α -> let mat = rotmatrixForAxis ax α + in rotmatrixForEulerAnglesZYZ (eulerAnglesZYZForMatrix mat) ≈ mat + ] + , testGroup "Concrete 180° rotations" + [ testCase "x around x" $ (180° xAxis) (π'₂ ⦺ 0 ) @?≈ π'₂ ⦺ 0 + , testCase "..full rot" $ (180° xAxis) (π'₂ ⦺ 0 )@?≈ π'₂ ⦺ 0 + , testCase "x around y" $ (180° yAxis) (π'₂ ⦺ 0 ) @?≈ π'₂ ⦺ π + , testCase "..full rot" $ (180° yAxis) (π'₂ ⦺ π )@?≈ π'₂ ⦺ 0 + , testCase "x around z" $ (180° zAxis) (π'₂ ⦺ 0 ) @?≈ π'₂ ⦺ π + , testCase "..full rot" $ (180° zAxis) (π'₂ ⦺ π )@?≈ π'₂ ⦺ 0 + , testCase "y around x" $ (180° xAxis) (π'₂ ⦺ π'₂) @?≈ π'₂ ⦺ -π'₂ + , testCase "..full rot" $ (180° xAxis) (π'₂ ⦺ -π'₂)@?≈ π'₂ ⦺ π'₂ + , testCase "y around y" $ (180° yAxis) (π'₂ ⦺ π'₂) @?≈ π'₂ ⦺ π'₂ + , testCase "..full rot" $ (180° yAxis) (π'₂ ⦺ π'₂)@?≈ π'₂ ⦺ π'₂ + , testCase "y around z" $ (180° zAxis) (π'₂ ⦺ π'₂) @?≈ π'₂ ⦺ -π'₂ + , testCase "..full rot" $ (180° zAxis) (π'₂ ⦺ -π'₂)@?≈ π'₂ ⦺ π'₂ + , testCase "z around x" $ (180° xAxis) (0 ⦺ 0 ) @?≈ π ⦺ 0 + , testCase "..full rot" $ (180° xAxis) (π ⦺ 0 )@?≈ 0 ⦺ 0 + , testCase "z around y" $ (180° yAxis) (0 ⦺ 0 ) @?≈ π ⦺ 0 + , testCase "..full rot" $ (180° yAxis) (π ⦺ 0 )@?≈ 0 ⦺ 0 + , testCase "z around z" $ (180° zAxis) (0 ⦺ 0 ) @?≈ 0 ⦺ 0 + , testCase "..full rot" $ (180° zAxis) (0 ⦺ 0 )@?≈ 0 ⦺ 0 + ] + , testGroup "Concrete 90° rotations" + [ testCase "x around x" $ (90° xAxis) (π'₂ ⦺ 0 ) @?≈ π'₂ ⦺ 0 + , testCase "x around y" $ (90° yAxis) (π'₂ ⦺ 0 ) @?≈ π ⦺ 0 + , testCase "x around z" $ (90° zAxis) (π'₂ ⦺ 0 ) @?≈ π'₂ ⦺ π'₂ + , testCase "y around x" $ (90° xAxis) (π'₂ ⦺ π'₂) @?≈ 0 ⦺ 0 + , testCase "y around y" $ (90° yAxis) (π'₂ ⦺ π'₂) @?≈ π'₂ ⦺ π'₂ + , testCase "y around z" $ (90° zAxis) (π'₂ ⦺ π'₂) @?≈ π'₂ ⦺ π + , testCase "z around x" $ (90° xAxis) (0 ⦺ 0 ) @?≈ π'₂ ⦺ (-π'₂) + , testCase "z around y" $ (90° yAxis) (0 ⦺ 0 ) @?≈ π'₂ ⦺ 0 + , testCase "z around z" $ (90° zAxis) (0 ⦺ 0 ) @?≈ 0 ⦺ π'₂ + + , testCase "around y, over x" $ (90° yAxis) (π'₄ ⦺ 0 ) @?≈ π³₄ ⦺ 0 + , testCase "around z, over x" $ (90° zAxis) (π'₂ ⦺ -π'₄) @?≈ π'₂ ⦺ π'₄ + , testCase "around x, over y" $ (90° xAxis) (π³₄ ⦺ π'₂) @?≈ π'₄ ⦺ π'₂ + , testCase "around z, over y" $ (90° zAxis) (π'₂ ⦺ π'₄) @?≈ π'₂ ⦺ π³₄ + , testCase "around x, over z" $ (90° xAxis) (π'₄ ⦺ π'₂) @?≈ π'₄ ⦺ -π'₂ + , testCase "around y, over z" $ (90° yAxis) (π'₄ ⦺ π ) @?≈ π'₄ ⦺ 0 + ] + , testGroup "Concrete 45° rotations" + [ testCase "x around x" $ (45° xAxis) (π'₂ ⦺ 0 ) @?≈ π'₂ ⦺ 0 + , testCase "x around y" $ (45° yAxis) (π'₂ ⦺ 0 ) @?≈ π³₄ ⦺ 0 + , testCase "x around z" $ (45° zAxis) (π'₂ ⦺ 0 ) @?≈ π'₂ ⦺ π'₄ + , testCase "y around x" $ (45° xAxis) (π'₂ ⦺ π'₂) @?≈ π'₄ ⦺ π'₂ + , testCase "y around y" $ (45° yAxis) (π'₂ ⦺ π'₂) @?≈ π'₂ ⦺ π'₂ + , testCase "y around z" $ (45° zAxis) (π'₂ ⦺ π'₂) @?≈ π'₂ ⦺ π³₄ + , testCase "z around x" $ (45° xAxis) (0 ⦺ 0 ) @?≈ π'₄ ⦺ -π'₂ + , testCase "z around y" $ (45° yAxis) (0 ⦺ 0 ) @?≈ π'₄ ⦺ 0 + , testCase "z around z" $ (45° zAxis) (0 ⦺ 0 ) @?≈ 0 ⦺ π'₂ + ] + , testGroup "Reversability" + [ QC.testProperty "Arbitrary axis, angles and points" + $ \ax ψ p -> rotateAboutThenUndo ax ψ p ≈ p + ] + ] + where π = pi + π'₂ = pi/2 + π'₄ = pi/4 + π³₄ = 3*pi/4 + + + +infix 4 ≈ +class AEq e where + (≈) :: e -> e -> Bool + +instance AEq Double where + x ≈ y = abs (x-y) < 1e-9 +instance AEq S¹ where + S¹Polar φ ≈ S¹Polar ϕ + | φ > pi/2, ϕ < -pi/2 = S¹Polar (φ - 2*pi) ≈ S¹Polar ϕ + | ϕ > pi/2, φ < -pi/2 = S¹Polar φ ≈ S¹Polar (ϕ - 2*pi) + | otherwise = φ ≈ ϕ +instance AEq S² where + S²Polar θ φ ≈ S²Polar ϑ ϕ + | φ > pi/2, ϕ < -pi/2 = S²Polar θ (φ - 2*pi) ≈ S²Polar ϑ ϕ + | ϕ > pi/2, φ < -pi/2 = S²Polar θ φ ≈ S²Polar ϑ (ϕ - 2*pi) + | otherwise = θ ≈ ϑ && abs (φ - ϕ) * sin θ < 1e-9 +instance AEq a => AEq [a] where + [] ≈ [] = True + x:xs ≈ y:ys = x ≈ y && xs ≈ ys + _ ≈ _ = False + + +infix 1 @?≈ +(@?≈) :: (AEq e, Show e) => e -> e -> Assertion +a@?≈b + | a≈b = return () + | otherwise = assertFailure $ "Expected "++show b++", but got "++show a + +rad :: Double -> S¹ +rad = S¹Polar + +infix 5 ⦺ +(⦺) :: Double -> Double -> S² +(⦺) = S²Polar + +rotateAboutThenUndo :: ℝP² -> S¹ -> S² -> S² +rotateAboutThenUndo ax ψ@(S¹Polar w) p = rotateAbout ax (S¹Polar $ -w) $ rotateAbout ax ψ p