-module Tetrahedron
+-- 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 Numeric.LinearAlgebra hiding (i, scale)
-import Prelude hiding (LT)
-import Test.QuickCheck (Arbitrary(..), Gen, choose)
-
-import Cardinal
-import Comparisons (nearly_ge)
-import FunctionValues
-import Misc (factorial)
-import Point
-import RealFunction
-import ThreeDimensional
+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 { fv :: FunctionValues,
- v0 :: Point,
- v1 :: Point,
- v2 :: Point,
- v3 :: Point,
- precomputed_volume :: Double,
-
- -- | Between 0 and 23; used to quickly determine which
- -- tetrahedron I am in the parent 'Cube' without
- -- having to compare them all.
- number :: Int
+ Tetrahedron { function_values :: FunctionValues,
+ v0 :: !Point,
+ v1 :: !Point,
+ v2 :: !Point,
+ v3 :: !Point,
+ precomputed_volume :: !Double
}
deriving (Eq)
rnd_v2 <- arbitrary :: Gen Point
rnd_v3 <- arbitrary :: Gen Point
rnd_fv <- arbitrary :: Gen FunctionValues
- rnd_no <- choose (0,23)
-- 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 rnd_no
+ 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 rnd_no)
+ return (Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 vol)
instance Show Tetrahedron where
show t = "Tetrahedron:\n" ++
- " no: " ++ (show (number t)) ++ "\n" ++
- " fv: " ++ (show (fv t)) ++ "\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"
-instance ThreeDimensional Tetrahedron where
- center (Tetrahedron _ v0' v1' v2' v3' _ _) =
- (v0' + v1' + v2' + v3') `scale` (1/4)
-
- contains_point t p =
- b0_unscaled `nearly_ge` 0 &&
- b1_unscaled `nearly_ge` 0 &&
- b2_unscaled `nearly_ge` 0 &&
- b3_unscaled `nearly_ge` 0
- where
- -- Drop the useless division and volume calculation that we
- -- would do if we used the regular b0,..b3 functions.
- b0_unscaled :: Double
- b0_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v0 = p }
-
- b1_unscaled :: Double
- b1_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v1 = p }
-
- b2_unscaled :: Double
- b2_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v2 = p }
+-- | 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)
- b3_unscaled :: Double
- b3_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v3 = p }
+{-# INLINE polynomial #-}
polynomial :: Tetrahedron -> (RealFunction Point)
polynomial t =
- sum [ (c t i j k l) `cmult` (beta t i j k l) | i <- [0..3],
- j <- [0..3],
- k <- [0..3],
- l <- [0..3],
- i + j + k + l == 3]
-
-
--- | Returns the domain point of t with indices i,j,k,l.
--- Simply an alias for the domain_point function.
-xi :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
-xi = domain_point
-
--- | 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
- | i + j + k + l == 3 = weighted_sum `scale` (1/3)
- | otherwise = error "domain point index out of bounds"
- 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'
+ 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
- | (i + j + k + l == 3) =
- coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
- | otherwise = error "basis function index out of bounds"
+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)
+ coefficient = (6 / (fromIntegral denominator)) :: Double
b0_term = (b0 t) `fexp` i
b1_term = (b1 t) `fexp` j
b2_term = (b2 t) `fexp` k
-- | 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
--- function will simply error.
+-- 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 0 0 3 0 = eval (fv t) $
- (1/8) * (I + F + L + T + LT + FL + FT + FLT)
-
-c t 0 0 0 3 = eval (fv t) $
- (1/8) * (I + F + R + T + RT + FR + FT + FRT)
-
-c t 0 0 2 1 = eval (fv t) $
- (5/24)*(I + F + T + FT) +
- (1/24)*(L + FL + LT + FLT)
-
-c t 0 0 1 2 = eval (fv t) $
- (5/24)*(I + F + T + FT) +
- (1/24)*(R + FR + RT + FRT)
-
-c t 0 1 2 0 = eval (fv t) $
- (5/24)*(I + F) +
- (1/8)*(L + T + FL + FT) +
- (1/24)*(LT + FLT)
-
-c t 0 1 0 2 = eval (fv t) $
- (5/24)*(I + F) +
- (1/8)*(R + T + FR + FT) +
- (1/24)*(RT + FRT)
-
-c t 0 1 1 1 = eval (fv t) $
- (13/48)*(I + F) +
- (7/48)*(T + FT) +
- (1/32)*(L + R + FL + FR) +
- (1/96)*(LT + RT + FLT + FRT)
-
-c t 0 2 1 0 = eval (fv t) $
- (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)
-
-c t 0 2 0 1 = eval (fv t) $
- (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)
-
-c t 0 3 0 0 = eval (fv t) $
- (13/48)*(I + F) +
- (5/96)*(L + R + T + D + FL + FR + FT + FD) +
- (1/192)*(RT + RD + LT + LD + FRT + FRD + FLT + FLD)
-
-c t 1 0 2 0 = eval (fv t) $
- (1/4)*I +
- (1/6)*(F + L + T) +
- (1/12)*(LT + FL + FT)
-
-c t 1 0 0 2 = eval (fv t) $
- (1/4)*I +
- (1/6)*(F + R + T) +
- (1/12)*(RT + FR + FT)
-
-c t 1 0 1 1 = eval (fv t) $
- (1/3)*I +
- (5/24)*(F + T) +
- (1/12)*FT +
- (1/24)*(L + R) +
- (1/48)*(LT + RT + FL + FR)
-
-c t 1 1 1 0 = eval (fv t) $
- (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)
-
-c t 1 1 0 1 = eval (fv t) $
- (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)
-
-c t 1 2 0 0 = eval (fv t) $
- (1/3)*I +
- (5/24)*F +
- (7/96)*(L + R + T + D) +
- (1/32)*(FL + FR + FT + FD) +
- (1/96)*(RT + RD + LT + LD)
-
-c t 2 0 1 0 = eval (fv t) $
- (3/8)*I +
- (7/48)*(F + T + L) +
- (1/48)*(R + D + B + LT + FL + FT) +
- (1/96)*(RT + BT + FR + FD + LD + BL)
-
-c t 2 0 0 1 = eval (fv t) $
- (3/8)*I +
- (7/48)*(F + T + R) +
- (1/48)*(L + D + B + RT + FR + FT) +
- (1/96)*(LT + BT + FL + FD + RD + BR)
-
-c t 2 1 0 0 = eval (fv t) $
- (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)
-
-c t 3 0 0 0 = eval (fv t) $
- (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)
-
-c _ _ _ _ _ = error "coefficient index out of bounds"
-
-
-
--- | The matrix used in the tetrahedron volume calculation as given in
--- Lai & Schumaker, Definition 15.4, page 436.
-vol_matrix :: Tetrahedron -> Matrix Double
-vol_matrix t = (4><4)
- [1, 1, 1, 1,
- x1, x2, x3, x4,
- y1, y2, y3, y4,
- z1, z2, z3, z4 ]
- where
- (x1, y1, z1) = v0 t
- (x2, y2, z2) = v1 t
- (x3, y3, z3) = v2 t
- (x4, y4, z4) = v3 t
+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 t
- | (v0 t) == (v1 t) = 0
- | (v0 t) == (v2 t) = 0
- | (v0 t) == (v3 t) = 0
- | (v1 t) == (v2 t) = 0
- | (v1 t) == (v3 t) = 0
- | (v2 t) == (v3 t) = 0
- | otherwise = (1/6)*(det (vol_matrix t))
-
+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
-- | 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
-- | 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
-- | 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 ]
projects / spline3.git / blobdiff