4 import Numeric.LinearAlgebra hiding (i, scale)
5 import Prelude hiding (LT)
6 import Test.QuickCheck (Arbitrary(..), Gen)
9 import Comparisons (nearly_ge)
11 import Misc (factorial)
14 import ThreeDimensional
16 data Tetrahedron = Tetrahedron { fv :: FunctionValues,
21 precomputed_volume :: Double }
25 instance Arbitrary Tetrahedron where
27 rnd_v0 <- arbitrary :: Gen Point
28 rnd_v1 <- arbitrary :: Gen Point
29 rnd_v2 <- arbitrary :: Gen Point
30 rnd_v3 <- arbitrary :: Gen Point
31 rnd_fv <- arbitrary :: Gen FunctionValues
32 -- We can't assign an incorrect precomputed volume,
33 -- so we have to calculate the correct one here.
34 let t' = Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 0
36 return (Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 vol)
39 instance Show Tetrahedron where
40 show t = "Tetrahedron:\n" ++
41 " fv: " ++ (show (fv t)) ++ "\n" ++
42 " v0: " ++ (show (v0 t)) ++ "\n" ++
43 " v1: " ++ (show (v1 t)) ++ "\n" ++
44 " v2: " ++ (show (v2 t)) ++ "\n" ++
45 " v3: " ++ (show (v3 t)) ++ "\n"
48 instance ThreeDimensional Tetrahedron where
49 center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
51 b0_unscaled `nearly_ge` 0 &&
52 b1_unscaled `nearly_ge` 0 &&
53 b2_unscaled `nearly_ge` 0 &&
54 b3_unscaled `nearly_ge` 0
56 -- Drop the useless division and volume calculation that we
57 -- would do if we used the regular b0,..b3 functions.
59 b0_unscaled = volume inner_tetrahedron
60 where inner_tetrahedron = t { v0 = p }
63 b1_unscaled = volume inner_tetrahedron
64 where inner_tetrahedron = t { v1 = p }
67 b2_unscaled = volume inner_tetrahedron
68 where inner_tetrahedron = t { v2 = p }
71 b3_unscaled = volume inner_tetrahedron
72 where inner_tetrahedron = t { v3 = p }
75 polynomial :: Tetrahedron -> (RealFunction Point)
77 sum [ (c t i j k l) `cmult` (beta t i j k l) | i <- [0..3],
84 -- | Returns the domain point of t with indices i,j,k,l.
85 -- Simply an alias for the domain_point function.
86 xi :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
89 -- | Returns the domain point of t with indices i,j,k,l.
90 domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
91 domain_point t i j k l
92 | i + j + k + l == 3 = weighted_sum `scale` (1/3)
93 | otherwise = error "domain point index out of bounds"
95 v0' = (v0 t) `scale` (fromIntegral i)
96 v1' = (v1 t) `scale` (fromIntegral j)
97 v2' = (v2 t) `scale` (fromIntegral k)
98 v3' = (v3 t) `scale` (fromIntegral l)
99 weighted_sum = v0' + v1' + v2' + v3'
102 -- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a
103 -- capital 'B' in the Sorokina/Zeilfelder paper.
104 beta :: Tetrahedron -> Int -> Int -> Int -> Int -> (RealFunction Point)
106 | (i + j + k + l == 3) =
107 coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
108 | otherwise = error "basis function index out of bounds"
110 denominator = (factorial i)*(factorial j)*(factorial k)*(factorial l)
111 coefficient = 6 / (fromIntegral denominator)
112 b0_term = (b0 t) `fexp` i
113 b1_term = (b1 t) `fexp` j
114 b2_term = (b2 t) `fexp` k
115 b3_term = (b3 t) `fexp` l
118 -- | The coefficient function. c t i j k l returns the coefficient
119 -- c_ijkl with respect to the tetrahedron t. The definition uses
120 -- pattern matching to mimic the definitions given in Sorokina and
121 -- Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
122 -- function will simply error.
123 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
124 c t 0 0 3 0 = eval (fv t) $
125 (1/8) * (I + F + L + T + LT + FL + FT + FLT)
127 c t 0 0 0 3 = eval (fv t) $
128 (1/8) * (I + F + R + T + RT + FR + FT + FRT)
130 c t 0 0 2 1 = eval (fv t) $
131 (5/24)*(I + F + T + FT) +
132 (1/24)*(L + FL + LT + FLT)
134 c t 0 0 1 2 = eval (fv t) $
135 (5/24)*(I + F + T + FT) +
136 (1/24)*(R + FR + RT + FRT)
138 c t 0 1 2 0 = eval (fv t) $
140 (1/8)*(L + T + FL + FT) +
143 c t 0 1 0 2 = eval (fv t) $
145 (1/8)*(R + T + FR + FT) +
148 c t 0 1 1 1 = eval (fv t) $
151 (1/32)*(L + R + FL + FR) +
152 (1/96)*(LT + RT + FLT + FRT)
154 c t 0 2 1 0 = eval (fv t) $
156 (17/192)*(L + T + FL + FT) +
158 (1/64)*(R + D + FR + FD) +
159 (1/192)*(RT + LD + FRT + FLD)
161 c t 0 2 0 1 = eval (fv t) $
163 (17/192)*(R + T + FR + FT) +
165 (1/64)*(L + D + FL + FD) +
166 (1/192)*(RD + LT + FLT + FRD)
168 c t 0 3 0 0 = eval (fv t) $
170 (5/96)*(L + R + T + D + FL + FR + FT + FD) +
171 (1/192)*(RT + RD + LT + LD + FRT + FRD + FLT + FLD)
173 c t 1 0 2 0 = eval (fv t) $
176 (1/12)*(LT + FL + FT)
178 c t 1 0 0 2 = eval (fv t) $
181 (1/12)*(RT + FR + FT)
183 c t 1 0 1 1 = eval (fv t) $
188 (1/48)*(LT + RT + FL + FR)
190 c t 1 1 1 0 = eval (fv t) $
195 (1/48)*(D + R + LT) +
196 (1/96)*(FD + LD + RT + FR)
198 c t 1 1 0 1 = eval (fv t) $
203 (1/48)*(D + L + RT) +
204 (1/96)*(FD + LT + RD + FL)
206 c t 1 2 0 0 = eval (fv t) $
209 (7/96)*(L + R + T + D) +
210 (1/32)*(FL + FR + FT + FD) +
211 (1/96)*(RT + RD + LT + LD)
213 c t 2 0 1 0 = eval (fv t) $
216 (1/48)*(R + D + B + LT + FL + FT) +
217 (1/96)*(RT + BT + FR + FD + LD + BL)
219 c t 2 0 0 1 = eval (fv t) $
222 (1/48)*(L + D + B + RT + FR + FT) +
223 (1/96)*(LT + BT + FL + FD + RD + BR)
225 c t 2 1 0 0 = eval (fv t) $
227 (1/12)*(T + R + L + D) +
228 (1/64)*(FT + FR + FL + FD) +
231 (1/96)*(RT + LD + LT + RD) +
232 (1/192)*(BT + BR + BL + BD)
234 c t 3 0 0 0 = eval (fv t) $
236 (1/12)*(T + F + L + R + D + B) +
237 (1/96)*(LT + FL + FT + RT + BT + FR) +
238 (1/96)*(FD + LD + BD + BR + RD + BL)
240 c _ _ _ _ _ = error "coefficient index out of bounds"
244 -- | The matrix used in the tetrahedron volume calculation as given in
245 -- Lai & Schumaker, Definition 15.4, page 436.
246 vol_matrix :: Tetrahedron -> Matrix Double
247 vol_matrix t = (4><4)
258 -- | Computed using the formula from Lai & Schumaker, Definition 15.4,
260 volume :: Tetrahedron -> Double
262 | (v0 t) == (v1 t) = 0
263 | (v0 t) == (v2 t) = 0
264 | (v0 t) == (v3 t) = 0
265 | (v1 t) == (v2 t) = 0
266 | (v1 t) == (v3 t) = 0
267 | (v2 t) == (v3 t) = 0
268 | otherwise = (1/6)*(det (vol_matrix t))
271 -- | The barycentric coordinates of a point with respect to v0.
272 b0 :: Tetrahedron -> (RealFunction Point)
273 b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
275 inner_tetrahedron = t { v0 = point }
278 -- | The barycentric coordinates of a point with respect to v1.
279 b1 :: Tetrahedron -> (RealFunction Point)
280 b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
282 inner_tetrahedron = t { v1 = point }
285 -- | The barycentric coordinates of a point with respect to v2.
286 b2 :: Tetrahedron -> (RealFunction Point)
287 b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
289 inner_tetrahedron = t { v2 = point }
292 -- | The barycentric coordinates of a point with respect to v3.
293 b3 :: Tetrahedron -> (RealFunction Point)
294 b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
296 inner_tetrahedron = t { v3 = point }