-module Tetrahedron
+{-# LANGUAGE BangPatterns #-}
+module Tetrahedron (
+ Tetrahedron(..),
+ b0, -- Cube test
+ b1, -- Cube test
+ b2, -- Cube test
+ b3, -- Cube test
+ c,
+ polynomial,
+ tetrahedron_properties,
+ tetrahedron_tests,
+ volume -- Cube test
+ )
where
-import Numeric.LinearAlgebra hiding (i, scale)
+import qualified Data.Vector as V (
+ singleton,
+ snoc,
+ sum
+ )
-import Cube (back,
- Cube(datum),
- down,
- front,
- Gridded,
- left,
- right,
- top)
+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
-import RealFunction
-import ThreeDimensional
+import Point (Point, scale)
+import RealFunction (RealFunction, cmult, fexp)
+import ThreeDimensional (ThreeDimensional(..))
+
+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)
-data Tetrahedron = Tetrahedron { cube :: Cube,
- v0 :: Point,
- v1 :: Point,
- v2 :: Point,
- v3 :: Point }
- deriving (Eq)
instance Show Tetrahedron where
- show t = "Tetrahedron (Cube: " ++ (show (cube t)) ++ ") " ++
- "(v0: " ++ (show (v0 t)) ++ ") (v1: " ++ (show (v1 t)) ++
- ") (v2: " ++ (show (v2 t)) ++ ") (v3: " ++ (show (v3 t)) ++
- ")\n\n"
-
-instance Gridded Tetrahedron where
- back t = back (cube t)
- down t = down (cube t)
- front t = front (cube t)
- left t = left (cube t)
- right t = right (cube t)
- top t = top (cube t)
+ 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"
instance ThreeDimensional Tetrahedron where
- center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
- contains_point t p =
- (b0 t p) >= 0 && (b1 t p) >= 0 && (b2 t p) >= 0 && (b3 t p) >= 0
+ center (Tetrahedron _ v0' v1' v2' v3' _) =
+ (v0' + v1' + v2' + v3') `scale` (1/4)
+
+ -- contains_point is only used in tests.
+ 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 }
+
+ 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 }
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 $ 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`
+ ((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
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
+-- function will simply error.
c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
-c x 0 0 3 0 = datum $ (1/8) * (i + f + l + t + lt + fl + ft + flt)
- where
- f = front x
- flt = front (left (top x))
- fl = front (left x)
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- t = top x
-
-
-c x 0 0 0 3 = datum $ (1/8) * (i + f + r + t + rt + fr + ft + frt)
- where
- f = front x
- fr = front (right x)
- frt = front (right (top x))
- ft = front (top x)
- i = cube x
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 0 0 2 1 = datum $ (5/24)*(i + f + t + ft) + (1/24)*(l + fl + lt + flt)
- where
- f = front x
- flt = front (left (top x))
- fl = front (left x)
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- t = top x
-
-
-c x 0 0 1 2 = datum $ (5/24)*(i + f + t + ft) + (1/24)*(r + fr + rt + frt)
- where
- f = front x
- frt = front (right (top x))
- fr = front (right x)
- ft = front (top x)
- i = cube x
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 0 1 2 0 = datum $
- (5/24)*(i + f) +
- (1/8)*(l + t + fl + ft) +
- (1/24)*(lt + flt)
- where
- f = front x
- flt = front (left (top x))
- fl = front (left x)
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- t = top x
-
-
-c x 0 1 0 2 = datum $
- (5/24)*(i + f) +
- (1/8)*(r + t + fr + ft) +
- (1/24)*(rt + frt)
- where
- f = front x
- fr = front (right x)
- frt = front (right (top x))
- ft = front (top x)
- i = cube x
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 0 1 1 1 = datum $
- (13/48)*(i + f) +
- (7/48)*(t + ft) +
- (1/32)*(l + r + fl + fr) +
- (1/96)*(lt + rt + flt + frt)
- where
- f = front x
- flt = front (left (top x))
- fl = front (left x)
- fr = front (right x)
- frt = front (right (top x))
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 0 2 1 0 = datum $
- (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)
- where
- d = down x
- f = front x
- fd = front (down x)
- fld = front (left (down x))
- flt = front (left (top x))
- fl = front (left x)
- fr = front (right x)
- frt = front (right (top x))
- ft = front (top x)
- i = cube x
- l = left x
- ld = left (down x)
- lt = left (top x)
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 0 2 0 1 = datum $
- (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)
- where
- d = down x
- f = front x
- fd = front (down x)
- flt = front (left (top x))
- fl = front (left x)
- frd = front (right (down x))
- fr = front (right x)
- frt = front (right (top x))
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- r = right x
- rd = right (down x)
- rt = right (top x)
- t = top x
-
-
-c x 0 3 0 0 = datum $
- (13/48)*(i + f) +
- (5/96)*(l + r + t + d + fl + fr + ft + fd) +
- (1/192)*(rt + rd + lt + ld + frt + frd + flt + fld)
- where
- d = down x
- f = front x
- fd = front (down x)
- fld = front (left (down x))
- flt = front (left (top x))
- fl = front (left x)
- frd = front (right (down x))
- fr = front (right x)
- frt = front (right (top x))
- ft = front (top x)
- i = cube x
- l = left x
- ld = left (down x)
- lt = left (top x)
- r = right x
- rd = right (down x)
- rt = right (top x)
- t = top x
-
-c x 1 0 2 0 = datum $ (1/4)*i + (1/6)*(f + l + t) + (1/12)*(lt + fl + ft)
- where
- f = front x
- fl = front (left x)
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- t = top x
-
-
-c x 1 0 0 2 = datum $ (1/4)*i + (1/6)*(f + r + t) + (1/12)*(rt + fr + ft)
- where
- f = front x
- fr = front (right x)
- ft = front (top x)
- i = cube x
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 1 0 1 1 = datum $
- (1/3)*i +
- (5/24)*(f + t) +
- (1/12)*ft +
- (1/24)*(l + r) +
- (1/48)*(lt + rt + fl + fr)
- where
- f = front x
- fl = front (left x)
- fr = front (right x)
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 1 1 1 0 = datum $
- (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)
- where
- d = down x
- f = front x
- fd = front (down x)
- fl = front (left x)
- fr = front (right x)
- ft = front (top x)
- i = cube x
- l = left x
- ld = left (down x)
- lt = left (top x)
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 1 1 0 1 = datum $
- (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)
- where
- d = down x
- f = front x
- fd = front (down x)
- fl = front (left x)
- fr = front (right x)
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- r = right x
- rd = right (down x)
- rt = right (top x)
- t = top x
-
-
-
-c x 1 2 0 0 = datum $
- (1/3)*i +
- (5/24)*f +
- (7/96)*(l + r + t + d) +
- (1/32)*(fl + fr + ft + fd) +
- (1/96)*(rt + rd + lt + ld)
- where
- d = down x
- f = front x
- fd = front (down x)
- fl = front (left x)
- fr = front (right x)
- ft = front (top x)
- i = cube x
- l = left x
- ld = left (down x)
- lt = left (top x)
- r = right x
- rd = right (down x)
- rt = right (top x)
- t = top x
-
-
-c x 2 0 1 0 = datum $
- (3/8)*i +
- (7/48)*(f + t + l) +
- (1/48)*(r + d + b + lt + fl + ft) +
- (1/96)*(rt + bt + fr + fd + ld + bl)
- where
- b = back x
- bl = back (left x)
- bt = back (top x)
- d = down x
- f = front x
- fd = front (down x)
- fl = front (left x)
- fr = front (right x)
- ft = front (top x)
- i = cube x
- l = left x
- ld = left (down x)
- lt = left (top x)
- r = right x
- rt = right (top x)
- t = top x
-
-
-c x 2 0 0 1 = datum $
- (3/8)*i +
- (7/48)*(f + t + r) +
- (1/48)*(l + d + b + rt + fr + ft) +
- (1/96)*(lt + bt + fl + fd + rd + br)
- where
- b = back x
- br = back (right x)
- bt = back (top x)
- d = down x
- f = front x
- fd = front (down x)
- fl = front (left x)
- fr = front (right x)
- ft = front (top x)
- i = cube x
- l = left x
- lt = left (top x)
- r = right x
- rd = right (down x)
- rt = right (top x)
- t = top x
-
-
-
-c x 2 1 0 0 = datum $
- (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)
- where
- b = back x
- bd = back (down x)
- bl = back (left x)
- br = back (right x)
- bt = back (top x)
- d = down x
- f = front x
- fd = front (down x)
- fl = front (left x)
- fr = front (right x)
- ft = front (top x)
- i = cube x
- l = left x
- ld = left (down x)
- lt = left (top x)
- r = right x
- rd = right (down x)
- rt = right (top x)
- t = top x
-
-
-c x 3 0 0 0 = datum $
- (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)
- where
- b = back x
- bd = back (down x)
- bl = back (left x)
- br = back (right x)
- bt = back (top x)
- d = down x
- f = front x
- fd = front (down x)
- fl = front (left x)
- fr = front (right x)
- ft = front (top x)
- i = cube x
- l = left x
- ld = left (down x)
- lt = left (top x)
- r = right x
- rd = right (down x)
- rt = right (top x)
- t = top x
-
-
-c _ _ _ _ _ = error "coefficient index out of bounds"
-
-
-
-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 = x_coord (v0 t)
- x2 = x_coord (v1 t)
- x3 = x_coord (v2 t)
- x4 = x_coord (v3 t)
- y1 = y_coord (v0 t)
- y2 = y_coord (v1 t)
- y3 = y_coord (v2 t)
- y4 = y_coord (v3 t)
- z1 = z_coord (v0 t)
- z2 = z_coord (v1 t)
- z3 = z_coord (v2 t)
- z4 = z_coord (v3 t)
-
--- Computed using the formula from Lai & Schumaker, Definition 15.4,
--- page 436.
+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)
+
+ coefficient _ _ _ _ = error "coefficient index out of bounds"
+
+
+
+-- | 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
+ (x1, y1, z1) = p0
+ (x2, y2, z2) = p1
+ (x3, y3, z3) = p2
+ (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.
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))
+ | 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
+-- | The barycentric coordinates of a point with respect to v0.
b0 :: Tetrahedron -> (RealFunction Point)
-b0 t point = (volume inner_tetrahedron) / (volume t)
+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.
b1 :: Tetrahedron -> (RealFunction Point)
-b1 t point = (volume inner_tetrahedron) / (volume t)
+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.
b2 :: Tetrahedron -> (RealFunction Point)
-b2 t point = (volume inner_tetrahedron) / (volume t)
+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.
b3 :: Tetrahedron -> (RealFunction Point)
-b3 t point = (volume inner_tetrahedron) / (volume t)
+b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
where
inner_tetrahedron = t { v3 = point }
+
+
+
+
+-- 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
+tetrahedron1_geometry_tests =
+ testGroup "tetrahedron1 geometry"
+ [ testCase "volume1" volume1,
+ testCase "doesn't contain point1" doesnt_contain_point1]
+ where
+ p0 = (0, -0.5, 0)
+ p1 = (0, 0.5, 0)
+ p2 = (2, 0, 0)
+ p3 = (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
+
+ 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
+tetrahedron2_geometry_tests =
+ testGroup "tetrahedron2 geometry"
+ [ testCase "volume1" volume1,
+ testCase "contains point1" contains_point1]
+ where
+ p0 = (0, -0.5, 0)
+ p1 = (2, 0, 0)
+ p2 = (0, 0.5, 0)
+ p3 = (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
+
+ 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
+-- 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 :: Test.Framework.Test
+tetrahedron_tests =
+ testGroup "Tetrahedron Tests" [
+ tetrahedron1_geometry_tests,
+ tetrahedron2_geometry_tests,
+ containment_tests ]
+
+
+
+p78_24_properties :: Test.Framework.Test
+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
+ | 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'
+
+
+ -- | 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 =
+ 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