+-- 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
- )
+ volume ) -- Cube test
where
-import qualified Data.Vector as V (
- singleton,
- snoc,
- sum
- )
-
-import Test.Framework (Test, testGroup)
-import Test.Framework.Providers.HUnit (testCase)
-import Test.Framework.Providers.QuickCheck2 (testProperty)
-import Test.HUnit (Assertion, assertEqual)
-import Test.QuickCheck (Arbitrary(..), Gen, Property, (==>))
-
-import Comparisons ((~=), nearly_ge)
-import FunctionValues (FunctionValues(..), empty_values)
-import Misc (factorial)
-import Point (Point, scale)
-import RealFunction (RealFunction, cmult, fexp)
-import ThreeDimensional (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 { function_values :: FunctionValues,
- v0 :: Point,
- v1 :: Point,
- v2 :: Point,
- v3 :: Point,
- precomputed_volume :: Double
+ v0 :: !Point,
+ v1 :: !Point,
+ v2 :: !Point,
+ v3 :: !Point,
+ precomputed_volume :: !Double
}
deriving (Eq)
" 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 p0 =
- 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 = p0 }
-
- b1_unscaled :: Double
- b1_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v1 = p0 }
+-- | 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)
- b2_unscaled :: Double
- b2_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v2 = p0 }
-
- b3_unscaled :: Double
- b3_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v3 = p0 }
+{-# INLINE polynomial #-}
polynomial :: Tetrahedron -> (RealFunction Point)
polynomial t =
- V.sum $ V.singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `V.snoc`
- ((c t 0 0 1 2) `cmult` (beta t 0 0 1 2)) `V.snoc`
- ((c t 0 0 2 1) `cmult` (beta t 0 0 2 1)) `V.snoc`
- ((c t 0 0 3 0) `cmult` (beta t 0 0 3 0)) `V.snoc`
- ((c t 0 1 0 2) `cmult` (beta t 0 1 0 2)) `V.snoc`
- ((c t 0 1 1 1) `cmult` (beta t 0 1 1 1)) `V.snoc`
- ((c t 0 1 2 0) `cmult` (beta t 0 1 2 0)) `V.snoc`
- ((c t 0 2 0 1) `cmult` (beta t 0 2 0 1)) `V.snoc`
- ((c t 0 2 1 0) `cmult` (beta t 0 2 1 0)) `V.snoc`
- ((c t 0 3 0 0) `cmult` (beta t 0 3 0 0)) `V.snoc`
- ((c t 1 0 0 2) `cmult` (beta t 1 0 0 2)) `V.snoc`
- ((c t 1 0 1 1) `cmult` (beta t 1 0 1 1)) `V.snoc`
- ((c t 1 0 2 0) `cmult` (beta t 1 0 2 0)) `V.snoc`
- ((c t 1 1 0 1) `cmult` (beta t 1 1 0 1)) `V.snoc`
- ((c t 1 1 1 0) `cmult` (beta t 1 1 1 0)) `V.snoc`
- ((c t 1 2 0 0) `cmult` (beta t 1 2 0 0)) `V.snoc`
- ((c t 2 0 0 1) `cmult` (beta t 2 0 0 1)) `V.snoc`
- ((c t 2 0 1 0) `cmult` (beta t 2 0 1 0)) `V.snoc`
- ((c t 2 1 0 0) `cmult` (beta t 2 1 0 0)) `V.snoc`
+ 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))
--- | 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'
-
-- | 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 i j k l =
+c !t !i !j !k !l =
coefficient i j k l
where
fvs = function_values t
+ (1/96)*(lt + fl + ft + rt + bt + fr)
+ (1/96)*(fd + ld + bd + br + rd + bl)
- coefficient _ _ _ _ = error "coefficient index out of bounds"
-
-- | Compute the determinant of the 4x4 matrix,
det p0 p1 p2 p3 =
term5 + term6
where
- (x1, y1, z1) = p0
- (x2, y2, z2) = p1
- (x3, y3, z3) = p2
- (x4, y4, z4) = p3
+ 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
-- | Computed using the formula from Lai & Schumaker, Definition 15.4,
-- page 436.
+{-# INLINE volume #-}
volume :: Tetrahedron -> Double
-volume t
- | v0' == v1' = 0
- | v0' == v2' = 0
- | v0' == v3' = 0
- | v1' == v2' = 0
- | v1' == v3' = 0
- | v2' == v3' = 0
- | otherwise = (1/6)*(det v0' v1' v2' v3')
- where
- v0' = v0 t
- v1' = v1 t
- v2' = v2 t
- v3' = v3 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
--- Tests
-
-
-- | Check the volume of a particular tetrahedron (computed by hand)
--- and whether or not it contains a specific point chosen to be
--- outside of it. Its vertices are in clockwise order, so the volume
--- should be negative.
-tetrahedron1_geometry_tests :: Test.Framework.Test
+-- 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,
- testCase "doesn't contain point1" doesnt_contain_point1]
+ [ testCase "volume1" volume1 ]
where
- p0 = (0, -0.5, 0)
- p1 = (0, 0.5, 0)
- p2 = (2, 0, 0)
- p3 = (1, 0, 1)
+ 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,
where
vol = volume t
- doesnt_contain_point1 :: Assertion
- doesnt_contain_point1 =
- assertEqual "doesn't contain an exterior point" False contained
- where
- exterior_point = (5, 2, -9.0212)
- contained = contains_point t exterior_point
-
-- | Check the volume of a particular tetrahedron (computed by hand)
--- and whether or not it contains a specific point chosen to be
--- inside of it. Its vertices are in counter-clockwise order, so the
--- volume should be positive.
-tetrahedron2_geometry_tests :: Test.Framework.Test
+-- 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,
- testCase "contains point1" contains_point1]
+ [ testCase "volume1" volume1 ]
where
- p0 = (0, -0.5, 0)
- p1 = (2, 0, 0)
- p2 = (0, 0.5, 0)
- p3 = (1, 0, 1)
+ 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,
where
vol = volume t
- contains_point1 :: Assertion
- contains_point1 = assertEqual "contains an inner point" True contained
- where
- inner_point = (1, 0, 0.5)
- contained = contains_point t inner_point
-
-
--- | Ensure that tetrahedra do not contain a particular point chosen to
--- be outside of them.
-containment_tests :: Test.Framework.Test
-containment_tests =
- testGroup "containment tests"
- [ testCase "doesn't contain point2" doesnt_contain_point2,
- testCase "doesn't contain point3" doesnt_contain_point3,
- testCase "doesn't contain point4" doesnt_contain_point4,
- testCase "doesn't contain point5" doesnt_contain_point5]
- where
- p2 = (0.5, 0.5, 1)
- p3 = (0.5, 0.5, 0.5)
- exterior_point = (0, 0, 0)
-
- doesnt_contain_point2 :: Assertion
- doesnt_contain_point2 =
- assertEqual "doesn't contain an exterior point" False contained
- where
- p0 = (0, 1, 1)
- p1 = (1, 1, 1)
- t = Tetrahedron { v0 = p0,
- v1 = p1,
- v2 = p2,
- v3 = p3,
- function_values = empty_values,
- precomputed_volume = 0 }
- contained = contains_point t exterior_point
-
-
- doesnt_contain_point3 :: Assertion
- doesnt_contain_point3 =
- assertEqual "doesn't contain an exterior point" False contained
- where
- p0 = (1, 1, 1)
- p1 = (1, 0, 1)
- t = Tetrahedron { v0 = p0,
- v1 = p1,
- v2 = p2,
- v3 = p3,
- function_values = empty_values,
- precomputed_volume = 0 }
- contained = contains_point t exterior_point
-
-
- doesnt_contain_point4 :: Assertion
- doesnt_contain_point4 =
- assertEqual "doesn't contain an exterior point" False contained
- where
- p0 = (1, 0, 1)
- p1 = (0, 0, 1)
- t = Tetrahedron { v0 = p0,
- v1 = p1,
- v2 = p2,
- v3 = p3,
- function_values = empty_values,
- precomputed_volume = 0 }
- contained = contains_point t exterior_point
-
-
- doesnt_contain_point5 :: Assertion
- doesnt_contain_point5 =
- assertEqual "doesn't contain an exterior point" False contained
- where
- p0 = (0, 0, 1)
- p1 = (0, 1, 1)
- t = Tetrahedron { v0 = p0,
- v1 = p1,
- v2 = p2,
- v3 = p3,
- function_values = empty_values,
- precomputed_volume = 0 }
- contained = contains_point t exterior_point
-- | The barycentric coordinate of v0 with respect to itself should
(volume t) > 0 ==> (b3 t) (v2 t) ~= 0
--- | Used for convenience in the next few tests; not a test itself.
-p :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
-p t i j k l = (polynomial t) (xi 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))
-
-
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
-tetrahedron_tests :: Test.Framework.Test
+tetrahedron_tests :: TestTree
tetrahedron_tests =
- testGroup "Tetrahedron Tests" [
+ testGroup "Tetrahedron tests" [
tetrahedron1_geometry_tests,
- tetrahedron2_geometry_tests,
- containment_tests ]
+ tetrahedron2_geometry_tests ]
-p78_24_properties :: Test.Framework.Test
+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]
+ 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 :: Test.Framework.Test
+tetrahedron_properties :: TestTree
tetrahedron_properties =
- testGroup "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 "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" $
+ testProperty "swapping_vertices_doesnt_affect_coefficients1"
prop_swapping_vertices_doesnt_affect_coefficients1,
- testProperty "swapping_vertices_doesnt_affect_coefficients2" $
+ testProperty "swapping_vertices_doesnt_affect_coefficients2"
prop_swapping_vertices_doesnt_affect_coefficients2,
- testProperty "swapping_vertices_doesnt_affect_coefficients3" $
+ testProperty "swapping_vertices_doesnt_affect_coefficients3"
prop_swapping_vertices_doesnt_affect_coefficients3,
- testProperty "swapping_vertices_doesnt_affect_coefficients4" $
+ testProperty "swapping_vertices_doesnt_affect_coefficients4"
prop_swapping_vertices_doesnt_affect_coefficients4 ]
projects / spline3.git / blobdiff