b1, -- Cube test
b2, -- Cube test
b3, -- Cube test
+ barycenter,
c,
+ contains_point,
polynomial,
tetrahedron_properties,
tetrahedron_tests,
import Misc (factorial)
import Point (Point(..), scale)
import RealFunction (RealFunction, cmult, fexp)
-import ThreeDimensional (ThreeDimensional(..))
data Tetrahedron =
Tetrahedron { function_values :: FunctionValues,
" v3: " ++ (show (v3 t)) ++ "\n"
-instance ThreeDimensional Tetrahedron where
- 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
+-- | 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)
+
+-- | A point is internal to a tetrahedron if all of its barycentric
+-- coordinates with respect to that tetrahedron are non-negative.
+contains_point :: Tetrahedron -> Point -> Bool
+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
- -- 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 }
+ inner_tetrahedron = t { v0 = p0 }
+
+ b1_unscaled :: Double
+ b1_unscaled = volume inner_tetrahedron
+ where inner_tetrahedron = t { v1 = 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 }
- 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 }
- b3_unscaled :: Double
- b3_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v3 = p0 }
{-# INLINE polynomial #-}
polynomial :: Tetrahedron -> (RealFunction Point)
-- | 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)
-- | 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 =
coefficient i j k l
+ (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,
-- 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 #-}
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"
+ 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)