module Tetrahedron
where
+import qualified Data.Vector as V (
+ singleton,
+ snoc,
+ sum
+ )
import Numeric.LinearAlgebra hiding (i, scale)
import Prelude hiding (LT)
+import Test.QuickCheck (Arbitrary(..), Gen)
import Cardinal
+import Comparisons (nearly_ge)
import FunctionValues
import Misc (factorial)
import Point
import RealFunction
import ThreeDimensional
-data Tetrahedron = Tetrahedron { fv :: FunctionValues,
- v0 :: Point,
- v1 :: Point,
- v2 :: Point,
- v3 :: Point }
- deriving (Eq)
+data Tetrahedron =
+ Tetrahedron { fv :: 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" ++
instance ThreeDimensional Tetrahedron where
- center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
+ center (Tetrahedron _ v0' v1' v2' v3' _) =
+ (v0' + v1' + v2' + v3') `scale` (1/4)
+
contains_point t p =
- (b0 t p) >= 0 && (b1 t p) >= 0 && (b2 t p) >= 0 && (b3 t p) >= 0
+ 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 }
+
+ b3_unscaled :: Double
+ b3_unscaled = volume inner_tetrahedron
+ where inner_tetrahedron = t { v3 = p }
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]
+ 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))
-- | Returns the domain point of t with indices i,j,k,l.
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 t 0 0 3 0 = eval (fv t) $
(1/8) * (I + F + L + T + LT + FL + FT + FLT)
+-- | 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,
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)
+ (x1, y1, z1) = v0 t
+ (x2, y2, z2) = v1 t
+ (x3, y3, z3) = v2 t
+ (x4, y4, z4) = v3 t
-- | Computed using the formula from Lai & Schumaker, Definition 15.4,
-- page 436.
-- | 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 }
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