snoc,
sum
)
-import Numeric.LinearAlgebra hiding (i, scale)
+
import Prelude hiding (LT)
import Test.Framework (Test, testGroup)
import Test.Framework.Providers.HUnit (testCase)
import ThreeDimensional
data Tetrahedron =
- Tetrahedron { fv :: FunctionValues,
+ Tetrahedron { function_values :: FunctionValues,
v0 :: Point,
v1 :: Point,
v2 :: Point,
instance Show Tetrahedron where
show t = "Tetrahedron:\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" ++
center (Tetrahedron _ v0' v1' v2' v3' _) =
(v0' + v1' + v2' + v3') `scale` (1/4)
- contains_point t p =
+ contains_point t p0 =
b0_unscaled `nearly_ge` 0 &&
b1_unscaled `nearly_ge` 0 &&
b2_unscaled `nearly_ge` 0 &&
-- would do if we used the regular b0,..b3 functions.
b0_unscaled :: Double
b0_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v0 = p }
+ where inner_tetrahedron = t { v0 = p0 }
b1_unscaled :: Double
b1_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v1 = p }
+ where inner_tetrahedron = t { v1 = p0 }
b2_unscaled :: Double
b2_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v2 = p }
+ where inner_tetrahedron = t { v2 = p0 }
b3_unscaled :: Double
b3_unscaled = volume inner_tetrahedron
- where inner_tetrahedron = t { v3 = p }
+ where inner_tetrahedron = t { v3 = p0 }
polynomial :: Tetrahedron -> (RealFunction Point)
-- 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) $
+c t 0 0 3 0 = eval (function_values t) $
(1/8) * (I + F + L + T + LT + FL + FT + FLT)
-c t 0 0 0 3 = eval (fv t) $
+c t 0 0 0 3 = eval (function_values t) $
(1/8) * (I + F + R + T + RT + FR + FT + FRT)
-c t 0 0 2 1 = eval (fv t) $
+c t 0 0 2 1 = eval (function_values t) $
(5/24)*(I + F + T + FT) +
(1/24)*(L + FL + LT + FLT)
-c t 0 0 1 2 = eval (fv t) $
+c t 0 0 1 2 = eval (function_values t) $
(5/24)*(I + F + T + FT) +
(1/24)*(R + FR + RT + FRT)
-c t 0 1 2 0 = eval (fv t) $
+c t 0 1 2 0 = eval (function_values t) $
(5/24)*(I + F) +
(1/8)*(L + T + FL + FT) +
(1/24)*(LT + FLT)
-c t 0 1 0 2 = eval (fv t) $
+c t 0 1 0 2 = eval (function_values t) $
(5/24)*(I + F) +
(1/8)*(R + T + FR + FT) +
(1/24)*(RT + FRT)
-c t 0 1 1 1 = eval (fv t) $
+c t 0 1 1 1 = eval (function_values 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) $
+c t 0 2 1 0 = eval (function_values 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) $
+c t 0 2 0 1 = eval (function_values 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) $
+c t 0 3 0 0 = eval (function_values 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) $
+c t 1 0 2 0 = eval (function_values t) $
(1/4)*I +
(1/6)*(F + L + T) +
(1/12)*(LT + FL + FT)
-c t 1 0 0 2 = eval (fv t) $
+c t 1 0 0 2 = eval (function_values t) $
(1/4)*I +
(1/6)*(F + R + T) +
(1/12)*(RT + FR + FT)
-c t 1 0 1 1 = eval (fv t) $
+c t 1 0 1 1 = eval (function_values 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) $
+c t 1 1 1 0 = eval (function_values t) $
(1/3)*I +
(5/24)*F +
(1/8)*(L + T) +
(1/48)*(D + R + LT) +
(1/96)*(FD + LD + RT + FR)
-c t 1 1 0 1 = eval (fv t) $
+c t 1 1 0 1 = eval (function_values t) $
(1/3)*I +
(5/24)*F +
(1/8)*(R + T) +
(1/48)*(D + L + RT) +
(1/96)*(FD + LT + RD + FL)
-c t 1 2 0 0 = eval (fv t) $
+c t 1 2 0 0 = eval (function_values 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) $
+c t 2 0 1 0 = eval (function_values 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) $
+c t 2 0 0 1 = eval (function_values 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) $
+c t 2 1 0 0 = eval (function_values t) $
(3/8)*I +
(1/12)*(T + R + L + D) +
(1/64)*(FT + FR + FL + FD) +
(1/96)*(RT + LD + LT + RD) +
(1/192)*(BT + BR + BL + BD)
-c t 3 0 0 0 = eval (fv t) $
+c t 3 0 0 0 = eval (function_values t) $
(3/8)*I +
(1/12)*(T + F + L + R + D + B) +
(1/96)*(LT + FL + FT + RT + BT + FR) +
--- | 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
+det :: Point -> Point -> Point -> Point -> Double
+det p0 p1 p2 p3 =
+-- Both of these results are just copy/pasted from Sage. One of them
+-- might be more numerically stable, faster, or both.
+--
+-- x1*y2*z4 - x1*y2*z3 + x1*y3*z2 - x1*y3*z4 - x1*y4*z2 + x1*y4*z3 +
+-- x2*y1*z3 - x2*y1*z4 - x2*y3*z1 + x2*y3*z4 +
+-- x2*y4*z1 - x2*y4*z3 - x3*y1*z2 + x3*y1*z4 + x3*y2*z1 - x3*y2*z4 - x3*y4*z1 +
+-- x3*y4*z2 + x4*y1*z2 - x4*y1*z3 - x4*y2*z1 + x4*y2*z3 + x4*y3*z1 - x4*y3*z2
+ -((x2 - x3)*y1 - (x1 - x3)*y2 + (x1 - x2)*y3)*z4 + ((x2 - x4)*y1 - (x1 - x4)*y2 + (x1 - x2)*y4)*z3 + ((x3 - x4)*y2 - (x2 - x4)*y3 + (x2 - x3)*y4)*z1 - ((x3 - x4)*y1 - (x1 - x4)*y3 + (x1 - x3)*y4)*z2
+ where
+ (x1, y1, z1) = p0
+ (x2, y2, z2) = p1
+ (x3, y3, z3) = p2
+ (x4, y4, z4) = p3
+
-- | 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.
v1 = p1,
v2 = p2,
v3 = p3,
- fv = empty_values,
+ function_values = empty_values,
precomputed_volume = 0 }
volume1 :: Assertion
v1 = p1,
v2 = p2,
v3 = p3,
- fv = empty_values,
+ function_values = empty_values,
precomputed_volume = 0 }
volume1 :: Assertion
v1 = p1,
v2 = p2,
v3 = p3,
- fv = empty_values,
+ function_values = empty_values,
precomputed_volume = 0 }
contained = contains_point t exterior_point
v1 = p1,
v2 = p2,
v3 = p3,
- fv = empty_values,
+ function_values = empty_values,
precomputed_volume = 0 }
contained = contains_point t exterior_point
v1 = p1,
v2 = p2,
v3 = p3,
- fv = empty_values,
+ function_values = empty_values,
precomputed_volume = 0 }
contained = contains_point t exterior_point
v1 = p1,
v2 = p2,
v3 = p3,
- fv = empty_values,
+ function_values = empty_values,
precomputed_volume = 0 }
contained = contains_point t exterior_point
projects / spline3.git / blobdiff