-- The local "coefficient" function defined within the "c" function -- pattern matches on a bunch of integers, but doesn't handle the -- "otherwise" case for performance reasons. {-# OPTIONS_GHC -Wno-incomplete-patterns #-} {-# LANGUAGE BangPatterns #-} module Tetrahedron ( Tetrahedron(..), b0, -- Cube test b1, -- Cube test b2, -- Cube test b3, -- Cube test barycenter, c, polynomial, tetrahedron_properties, tetrahedron_tests, volume ) -- Cube test where import Data.Vector ( singleton, snoc ) import qualified Data.Vector as V ( sum ) import Test.Tasty ( TestTree, testGroup ) import Test.Tasty.HUnit ( Assertion, assertEqual, testCase ) import Test.Tasty.QuickCheck ( Arbitrary( arbitrary ), Gen, Property, (==>), testProperty ) import Comparisons ( (~=) ) import FunctionValues ( FunctionValues( front, back, left, right, top, down, front_left, front_right, front_down, front_top, back_left, back_right, back_down, back_top, left_down, left_top, right_down, right_top, front_left_down, front_left_top, front_right_down, front_right_top, interior ), empty_values ) import Misc ( factorial ) import Point ( Point(Point), scale ) import RealFunction ( RealFunction, cmult, fexp ) data Tetrahedron = Tetrahedron { function_values :: FunctionValues, v0 :: !Point, v1 :: !Point, v2 :: !Point, v3 :: !Point, precomputed_volume :: !Double } deriving (Eq) instance Arbitrary Tetrahedron where arbitrary = do rnd_v0 <- arbitrary :: Gen Point rnd_v1 <- arbitrary :: Gen Point rnd_v2 <- arbitrary :: Gen Point rnd_v3 <- arbitrary :: Gen Point rnd_fv <- arbitrary :: Gen FunctionValues -- We can't assign an incorrect precomputed volume, -- so we have to calculate the correct one here. let t' = Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 0 let vol = volume t' return (Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 vol) instance Show Tetrahedron where show t = "Tetrahedron:\n" ++ " function_values: " ++ (show (function_values t)) ++ "\n" ++ " v0: " ++ (show (v0 t)) ++ "\n" ++ " v1: " ++ (show (v1 t)) ++ "\n" ++ " v2: " ++ (show (v2 t)) ++ "\n" ++ " v3: " ++ (show (v3 t)) ++ "\n" -- | Find the barycenter of the given tetrahedron. -- We just average the four vertices. barycenter :: Tetrahedron -> Point barycenter (Tetrahedron _ v0' v1' v2' v3' _) = (v0' + v1' + v2' + v3') `scale` (1/4) {-# INLINE polynomial #-} polynomial :: Tetrahedron -> (RealFunction Point) polynomial t = V.sum $ singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `snoc` ((c t 0 0 1 2) `cmult` (beta t 0 0 1 2)) `snoc` ((c t 0 0 2 1) `cmult` (beta t 0 0 2 1)) `snoc` ((c t 0 0 3 0) `cmult` (beta t 0 0 3 0)) `snoc` ((c t 0 1 0 2) `cmult` (beta t 0 1 0 2)) `snoc` ((c t 0 1 1 1) `cmult` (beta t 0 1 1 1)) `snoc` ((c t 0 1 2 0) `cmult` (beta t 0 1 2 0)) `snoc` ((c t 0 2 0 1) `cmult` (beta t 0 2 0 1)) `snoc` ((c t 0 2 1 0) `cmult` (beta t 0 2 1 0)) `snoc` ((c t 0 3 0 0) `cmult` (beta t 0 3 0 0)) `snoc` ((c t 1 0 0 2) `cmult` (beta t 1 0 0 2)) `snoc` ((c t 1 0 1 1) `cmult` (beta t 1 0 1 1)) `snoc` ((c t 1 0 2 0) `cmult` (beta t 1 0 2 0)) `snoc` ((c t 1 1 0 1) `cmult` (beta t 1 1 0 1)) `snoc` ((c t 1 1 1 0) `cmult` (beta t 1 1 1 0)) `snoc` ((c t 1 2 0 0) `cmult` (beta t 1 2 0 0)) `snoc` ((c t 2 0 0 1) `cmult` (beta t 2 0 0 1)) `snoc` ((c t 2 0 1 0) `cmult` (beta t 2 0 1 0)) `snoc` ((c t 2 1 0 0) `cmult` (beta t 2 1 0 0)) `snoc` ((c t 3 0 0 0) `cmult` (beta t 3 0 0 0)) -- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a -- capital 'B' in the Sorokina/Zeilfelder paper. beta :: Tetrahedron -> Int -> Int -> Int -> Int -> (RealFunction Point) beta t i j k l = coefficient `cmult` (b0_term * b1_term * b2_term * b3_term) where denominator = (factorial i)*(factorial j)*(factorial k)*(factorial l) coefficient = (6 / (fromIntegral denominator)) :: Double b0_term = (b0 t) `fexp` i b1_term = (b1 t) `fexp` j b2_term = (b2 t) `fexp` k b3_term = (b3 t) `fexp` l -- | The coefficient function. c t i j k l returns the coefficient -- c_ijkl with respect to the tetrahedron t. The definition uses -- pattern matching to mimic the definitions given in Sorokina and -- Zeilfelder, pp. 84-86. If incorrect indices are supplied, the world -- will end. This is for performance reasons. c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double c !t !i !j !k !l = coefficient i j k l where fvs = function_values t f = front fvs b = back fvs r = right fvs l' = left fvs t' = top fvs d = down fvs fl = front_left fvs fr = front_right fvs fd = front_down fvs ft = front_top fvs bl = back_left fvs br = back_right fvs bd = back_down fvs bt = back_top fvs ld = left_down fvs lt = left_top fvs rd = right_down fvs rt = right_top fvs fld = front_left_down fvs flt = front_left_top fvs frd = front_right_down fvs frt = front_right_top fvs i' = interior fvs coefficient :: Int -> Int -> Int -> Int -> Double coefficient 0 0 3 0 = (1/8) * (i' + f + l' + t' + lt + fl + ft + flt) coefficient 0 0 0 3 = (1/8) * (i' + f + r + t' + rt + fr + ft + frt) coefficient 0 0 2 1 = (5/24)*(i' + f + t' + ft) + (1/24)*(l' + fl + lt + flt) coefficient 0 0 1 2 = (5/24)*(i' + f + t' + ft) + (1/24)*(r + fr + rt + frt) coefficient 0 1 2 0 = (5/24)*(i' + f) + (1/8)*(l' + t' + fl + ft) + (1/24)*(lt + flt) coefficient 0 1 0 2 = (5/24)*(i' + f) + (1/8)*(r + t' + fr + ft) + (1/24)*(rt + frt) coefficient 0 1 1 1 = (13/48)*(i' + f) + (7/48)*(t' + ft) + (1/32)*(l' + r + fl + fr) + (1/96)*(lt + rt + flt + frt) coefficient 0 2 1 0 = (13/48)*(i' + f) + (17/192)*(l' + t' + fl + ft) + (1/96)*(lt + flt) + (1/64)*(r + d + fr + fd) + (1/192)*(rt + ld + frt + fld) coefficient 0 2 0 1 = (13/48)*(i' + f) + (17/192)*(r + t' + fr + ft) + (1/96)*(rt + frt) + (1/64)*(l' + d + fl + fd) + (1/192)*(rd + lt + flt + frd) coefficient 0 3 0 0 = (13/48)*(i' + f) + (5/96)*(l' + r + t' + d + fl + fr + ft + fd) + (1/192)*(rt + rd + lt + ld + frt + frd + flt + fld) coefficient 1 0 2 0 = (1/4)*i' + (1/6)*(f + l' + t') + (1/12)*(lt + fl + ft) coefficient 1 0 0 2 = (1/4)*i' + (1/6)*(f + r + t') + (1/12)*(rt + fr + ft) coefficient 1 0 1 1 = (1/3)*i' + (5/24)*(f + t') + (1/12)*ft + (1/24)*(l' + r) + (1/48)*(lt + rt + fl + fr) coefficient 1 1 1 0 = (1/3)*i' + (5/24)*f + (1/8)*(l' + t') + (5/96)*(fl + ft) + (1/48)*(d + r + lt) + (1/96)*(fd + ld + rt + fr) coefficient 1 1 0 1 = (1/3)*i' + (5/24)*f + (1/8)*(r + t') + (5/96)*(fr + ft) + (1/48)*(d + l' + rt) + (1/96)*(fd + lt + rd + fl) coefficient 1 2 0 0 = (1/3)*i' + (5/24)*f + (7/96)*(l' + r + t' + d) + (1/32)*(fl + fr + ft + fd) + (1/96)*(rt + rd + lt + ld) coefficient 2 0 1 0 = (3/8)*i' + (7/48)*(f + t' + l') + (1/48)*(r + d + b + lt + fl + ft) + (1/96)*(rt + bt + fr + fd + ld + bl) coefficient 2 0 0 1 = (3/8)*i' + (7/48)*(f + t' + r) + (1/48)*(l' + d + b + rt + fr + ft) + (1/96)*(lt + bt + fl + fd + rd + br) coefficient 2 1 0 0 = (3/8)*i' + (1/12)*(t' + r + l' + d) + (1/64)*(ft + fr + fl + fd) + (7/48)*f + (1/48)*b + (1/96)*(rt + ld + lt + rd) + (1/192)*(bt + br + bl + bd) coefficient 3 0 0 0 = (3/8)*i' + (1/12)*(t' + f + l' + r + d + b) + (1/96)*(lt + fl + ft + rt + bt + fr) + (1/96)*(fd + ld + bd + br + rd + bl) -- | Compute the determinant of the 4x4 matrix, -- -- [1] -- [x] -- [y] -- [z] -- -- where [1] = [1, 1, 1, 1], -- [x] = [x1,x2,x3,x4], -- -- et cetera. -- -- The termX nonsense is an attempt to prevent Double overflow. -- which has been observed to happen with large coordinates. -- det :: Point -> Point -> Point -> Point -> Double det p0 p1 p2 p3 = term5 + term6 where Point x1 y1 z1 = p0 Point x2 y2 z2 = p1 Point x3 y3 z3 = p2 Point x4 y4 z4 = p3 term1 = ((x2 - x4)*y1 - (x1 - x4)*y2 + (x1 - x2)*y4)*z3 term2 = ((x2 - x3)*y1 - (x1 - x3)*y2 + (x1 - x2)*y3)*z4 term3 = ((x3 - x4)*y2 - (x2 - x4)*y3 + (x2 - x3)*y4)*z1 term4 = ((x3 - x4)*y1 - (x1 - x4)*y3 + (x1 - x3)*y4)*z2 term5 = term1 - term2 term6 = term3 - term4 -- | Computed using the formula from Lai & Schumaker, Definition 15.4, -- page 436. {-# INLINE volume #-} volume :: Tetrahedron -> Double volume (Tetrahedron _ v0' v1' v2' v3' _) = (1/6)*(det v0' v1' v2' v3') -- | The barycentric coordinates of a point with respect to v0. {-# INLINE b0 #-} b0 :: Tetrahedron -> (RealFunction Point) b0 t point = (volume inner_tetrahedron) / (precomputed_volume t) where inner_tetrahedron = t { v0 = point } -- | The barycentric coordinates of a point with respect to v1. {-# INLINE b1 #-} b1 :: Tetrahedron -> (RealFunction Point) b1 t point = (volume inner_tetrahedron) / (precomputed_volume t) where inner_tetrahedron = t { v1 = point } -- | The barycentric coordinates of a point with respect to v2. {-# INLINE b2 #-} b2 :: Tetrahedron -> (RealFunction Point) b2 t point = (volume inner_tetrahedron) / (precomputed_volume t) where inner_tetrahedron = t { v2 = point } -- | The barycentric coordinates of a point with respect to v3. {-# INLINE b3 #-} b3 :: Tetrahedron -> (RealFunction Point) b3 t point = (volume inner_tetrahedron) / (precomputed_volume t) where inner_tetrahedron = t { v3 = point } -- | Check the volume of a particular tetrahedron (computed by hand) -- Its vertices are in clockwise order, so the volume should be -- negative. tetrahedron1_geometry_tests :: TestTree tetrahedron1_geometry_tests = testGroup "tetrahedron1 geometry" [ testCase "volume1" volume1 ] where p0 = Point 0 (-0.5) 0 p1 = Point 0 0.5 0 p2 = Point 2 0 0 p3 = Point 1 0 1 t = Tetrahedron { v0 = p0, v1 = p1, v2 = p2, v3 = p3, function_values = empty_values, precomputed_volume = 0 } volume1 :: Assertion volume1 = assertEqual "volume is correct" True (vol ~= (-1/3)) where vol = volume t -- | Check the volume of a particular tetrahedron (computed by hand) -- Its vertices are in counter-clockwise order, so the volume should -- be positive. tetrahedron2_geometry_tests :: TestTree tetrahedron2_geometry_tests = testGroup "tetrahedron2 geometry" [ testCase "volume1" volume1 ] where p0 = Point 0 (-0.5) 0 p1 = Point 2 0 0 p2 = Point 0 0.5 0 p3 = Point 1 0 1 t = Tetrahedron { v0 = p0, v1 = p1, v2 = p2, v3 = p3, function_values = empty_values, precomputed_volume = 0 } volume1 :: Assertion volume1 = assertEqual "volume1 is correct" True (vol ~= (1/3)) where vol = volume t -- | The barycentric coordinate of v0 with respect to itself should -- be one. prop_b0_v0_always_unity :: Tetrahedron -> Property prop_b0_v0_always_unity t = (volume t) > 0 ==> (b0 t) (v0 t) ~= 1.0 -- | The barycentric coordinate of v1 with respect to v0 should -- be zero. prop_b0_v1_always_zero :: Tetrahedron -> Property prop_b0_v1_always_zero t = (volume t) > 0 ==> (b0 t) (v1 t) ~= 0 -- | The barycentric coordinate of v2 with respect to v0 should -- be zero. prop_b0_v2_always_zero :: Tetrahedron -> Property prop_b0_v2_always_zero t = (volume t) > 0 ==> (b0 t) (v2 t) ~= 0 -- | The barycentric coordinate of v3 with respect to v0 should -- be zero. prop_b0_v3_always_zero :: Tetrahedron -> Property prop_b0_v3_always_zero t = (volume t) > 0 ==> (b0 t) (v3 t) ~= 0 -- | The barycentric coordinate of v1 with respect to itself should -- be one. prop_b1_v1_always_unity :: Tetrahedron -> Property prop_b1_v1_always_unity t = (volume t) > 0 ==> (b1 t) (v1 t) ~= 1.0 -- | The barycentric coordinate of v0 with respect to v1 should -- be zero. prop_b1_v0_always_zero :: Tetrahedron -> Property prop_b1_v0_always_zero t = (volume t) > 0 ==> (b1 t) (v0 t) ~= 0 -- | The barycentric coordinate of v2 with respect to v1 should -- be zero. prop_b1_v2_always_zero :: Tetrahedron -> Property prop_b1_v2_always_zero t = (volume t) > 0 ==> (b1 t) (v2 t) ~= 0 -- | The barycentric coordinate of v3 with respect to v1 should -- be zero. prop_b1_v3_always_zero :: Tetrahedron -> Property prop_b1_v3_always_zero t = (volume t) > 0 ==> (b1 t) (v3 t) ~= 0 -- | The barycentric coordinate of v2 with respect to itself should -- be one. prop_b2_v2_always_unity :: Tetrahedron -> Property prop_b2_v2_always_unity t = (volume t) > 0 ==> (b2 t) (v2 t) ~= 1.0 -- | The barycentric coordinate of v0 with respect to v2 should -- be zero. prop_b2_v0_always_zero :: Tetrahedron -> Property prop_b2_v0_always_zero t = (volume t) > 0 ==> (b2 t) (v0 t) ~= 0 -- | The barycentric coordinate of v1 with respect to v2 should -- be zero. prop_b2_v1_always_zero :: Tetrahedron -> Property prop_b2_v1_always_zero t = (volume t) > 0 ==> (b2 t) (v1 t) ~= 0 -- | The barycentric coordinate of v3 with respect to v2 should -- be zero. prop_b2_v3_always_zero :: Tetrahedron -> Property prop_b2_v3_always_zero t = (volume t) > 0 ==> (b2 t) (v3 t) ~= 0 -- | The barycentric coordinate of v3 with respect to itself should -- be one. prop_b3_v3_always_unity :: Tetrahedron -> Property prop_b3_v3_always_unity t = (volume t) > 0 ==> (b3 t) (v3 t) ~= 1.0 -- | The barycentric coordinate of v0 with respect to v3 should -- be zero. prop_b3_v0_always_zero :: Tetrahedron -> Property prop_b3_v0_always_zero t = (volume t) > 0 ==> (b3 t) (v0 t) ~= 0 -- | The barycentric coordinate of v1 with respect to v3 should -- be zero. prop_b3_v1_always_zero :: Tetrahedron -> Property prop_b3_v1_always_zero t = (volume t) > 0 ==> (b3 t) (v1 t) ~= 0 -- | The barycentric coordinate of v2 with respect to v3 should -- be zero. prop_b3_v2_always_zero :: Tetrahedron -> Property prop_b3_v2_always_zero t = (volume t) > 0 ==> (b3 t) (v2 t) ~= 0 prop_swapping_vertices_doesnt_affect_coefficients1 :: Tetrahedron -> Bool prop_swapping_vertices_doesnt_affect_coefficients1 t = c t 0 0 1 2 == c t' 0 0 1 2 where t' = t { v0 = (v1 t), v1 = (v0 t) } prop_swapping_vertices_doesnt_affect_coefficients2 :: Tetrahedron -> Bool prop_swapping_vertices_doesnt_affect_coefficients2 t = c t 0 1 1 1 == c t' 0 1 1 1 where t' = t { v2 = (v3 t), v3 = (v2 t) } prop_swapping_vertices_doesnt_affect_coefficients3 :: Tetrahedron -> Bool prop_swapping_vertices_doesnt_affect_coefficients3 t = c t 2 1 0 0 == c t' 2 1 0 0 where t' = t { v2 = (v3 t), v3 = (v2 t) } prop_swapping_vertices_doesnt_affect_coefficients4 :: Tetrahedron -> Bool prop_swapping_vertices_doesnt_affect_coefficients4 t = c t 2 0 0 1 == c t' 2 0 0 1 where t' = t { v0 = (v3 t), v3 = (v0 t) } tetrahedron_tests :: TestTree tetrahedron_tests = testGroup "Tetrahedron tests" [ tetrahedron1_geometry_tests, tetrahedron2_geometry_tests ] p78_24_properties :: TestTree p78_24_properties = testGroup "p. 78, Section (2.4) properties" [ testProperty "c3000 identity" prop_c3000_identity, testProperty "c2100 identity" prop_c2100_identity, testProperty "c1110 identity" prop_c1110_identity] where -- | Returns the domain point of t with indices i,j,k,l. domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point domain_point t i j k l = weighted_sum `scale` (1/3) where v0' = (v0 t) `scale` (fromIntegral i) v1' = (v1 t) `scale` (fromIntegral j) v2' = (v2 t) `scale` (fromIntegral k) v3' = (v3 t) `scale` (fromIntegral l) weighted_sum = v0' + v1' + v2' + v3' -- | Used for convenience in the next few tests. p :: Tetrahedron -> Int -> Int -> Int -> Int -> Double p t i j k l = (polynomial t) (domain_point t i j k l) -- | Given in Sorokina and Zeilfelder, p. 78. prop_c3000_identity :: Tetrahedron -> Property prop_c3000_identity t = (volume t) > 0 ==> c t 3 0 0 0 ~= p t 3 0 0 0 -- | Given in Sorokina and Zeilfelder, p. 78. prop_c2100_identity :: Tetrahedron -> Property prop_c2100_identity t = (volume t) > 0 ==> c t 2 1 0 0 ~= (term1 - term2 + term3 - term4) where term1 = (1/3)*(p t 0 3 0 0) term2 = (5/6)*(p t 3 0 0 0) term3 = 3*(p t 2 1 0 0) term4 = (3/2)*(p t 1 2 0 0) -- | Given in Sorokina and Zeilfelder, p. 78. prop_c1110_identity :: Tetrahedron -> Property prop_c1110_identity t = (volume t) > 0 ==> c t 1 1 1 0 ~= (term1 + term2 - term3 - term4) where term1 = (1/3)*((p t 3 0 0 0) + (p t 0 3 0 0) + (p t 0 0 3 0)) term2 = (9/2)*(p t 1 1 1 0) term3 = (3/4)*((p t 2 1 0 0) + (p t 1 2 0 0) + (p t 2 0 1 0)) term4 = (3/4)*((p t 1 0 2 0) + (p t 0 2 1 0) + (p t 0 1 2 0)) tetrahedron_properties :: TestTree tetrahedron_properties = testGroup "Tetrahedron properties" [ p78_24_properties, testProperty "b0_v0_always_unity" prop_b0_v0_always_unity, testProperty "b0_v1_always_zero" prop_b0_v1_always_zero, testProperty "b0_v2_always_zero" prop_b0_v2_always_zero, testProperty "b0_v3_always_zero" prop_b0_v3_always_zero, testProperty "b1_v1_always_unity" prop_b1_v1_always_unity, testProperty "b1_v0_always_zero" prop_b1_v0_always_zero, testProperty "b1_v2_always_zero" prop_b1_v2_always_zero, testProperty "b1_v3_always_zero" prop_b1_v3_always_zero, testProperty "b2_v2_always_unity" prop_b2_v2_always_unity, testProperty "b2_v0_always_zero" prop_b2_v0_always_zero, testProperty "b2_v1_always_zero" prop_b2_v1_always_zero, testProperty "b2_v3_always_zero" prop_b2_v3_always_zero, testProperty "b3_v3_always_unity" prop_b3_v3_always_unity, testProperty "b3_v0_always_zero" prop_b3_v0_always_zero, testProperty "b3_v1_always_zero" prop_b3_v1_always_zero, testProperty "b3_v2_always_zero" prop_b3_v2_always_zero, testProperty "swapping_vertices_doesnt_affect_coefficients1" prop_swapping_vertices_doesnt_affect_coefficients1, testProperty "swapping_vertices_doesnt_affect_coefficients2" prop_swapping_vertices_doesnt_affect_coefficients2, testProperty "swapping_vertices_doesnt_affect_coefficients3" prop_swapping_vertices_doesnt_affect_coefficients3, testProperty "swapping_vertices_doesnt_affect_coefficients4" prop_swapping_vertices_doesnt_affect_coefficients4 ]