4 import Numeric.LinearAlgebra hiding (i, scale)
5 import Prelude hiding (LT)
9 import Misc (factorial)
12 import ThreeDimensional
14 data Tetrahedron = Tetrahedron { fv :: FunctionValues,
21 instance Show Tetrahedron where
22 show t = "Tetrahedron:\n" ++
23 " fv: " ++ (show (fv t)) ++ "\n" ++
24 " v0: " ++ (show (v0 t)) ++ "\n" ++
25 " v1: " ++ (show (v1 t)) ++ "\n" ++
26 " v2: " ++ (show (v2 t)) ++ "\n" ++
27 " v3: " ++ (show (v3 t)) ++ "\n"
30 instance ThreeDimensional Tetrahedron where
31 center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
33 (b0 t p) >= 0 && (b1 t p) >= 0 && (b2 t p) >= 0 && (b3 t p) >= 0
36 polynomial :: Tetrahedron -> (RealFunction Point)
38 sum [ (c t i j k l) `cmult` (beta t i j k l) | i <- [0..3],
45 -- | Returns the domain point of t with indices i,j,k,l.
46 -- Simply an alias for the domain_point function.
47 xi :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
50 -- | Returns the domain point of t with indices i,j,k,l.
51 domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
52 domain_point t i j k l
53 | i + j + k + l == 3 = weighted_sum `scale` (1/3)
54 | otherwise = error "domain point index out of bounds"
56 v0' = (v0 t) `scale` (fromIntegral i)
57 v1' = (v1 t) `scale` (fromIntegral j)
58 v2' = (v2 t) `scale` (fromIntegral k)
59 v3' = (v3 t) `scale` (fromIntegral l)
60 weighted_sum = v0' + v1' + v2' + v3'
63 -- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a
64 -- capital 'B' in the Sorokina/Zeilfelder paper.
65 beta :: Tetrahedron -> Int -> Int -> Int -> Int -> (RealFunction Point)
67 | (i + j + k + l == 3) =
68 coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
69 | otherwise = error "basis function index out of bounds"
71 denominator = (factorial i)*(factorial j)*(factorial k)*(factorial l)
72 coefficient = 6 / (fromIntegral denominator)
73 b0_term = (b0 t) `fexp` i
74 b1_term = (b1 t) `fexp` j
75 b2_term = (b2 t) `fexp` k
76 b3_term = (b3 t) `fexp` l
79 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
80 c t 0 0 3 0 = eval (fv t) $
81 (1/8) * (I + F + L + T + LT + FL + FT + FLT)
83 c t 0 0 0 3 = eval (fv t) $
84 (1/8) * (I + F + R + T + RT + FR + FT + FRT)
86 c t 0 0 2 1 = eval (fv t) $
87 (5/24)*(I + F + T + FT) +
88 (1/24)*(L + FL + LT + FLT)
90 c t 0 0 1 2 = eval (fv t) $
91 (5/24)*(I + F + T + FT) +
92 (1/24)*(R + FR + RT + FRT)
94 c t 0 1 2 0 = eval (fv t) $
96 (1/8)*(L + T + FL + FT) +
99 c t 0 1 0 2 = eval (fv t) $
101 (1/8)*(R + T + FR + FT) +
104 c t 0 1 1 1 = eval (fv t) $
107 (1/32)*(L + R + FL + FR) +
108 (1/96)*(LT + RT + FLT + FRT)
110 c t 0 2 1 0 = eval (fv t) $
112 (17/192)*(L + T + FL + FT) +
114 (1/64)*(R + D + FR + FD) +
115 (1/192)*(RT + LD + FRT + FLD)
117 c t 0 2 0 1 = eval (fv t) $
119 (17/192)*(R + T + FR + FT) +
121 (1/64)*(L + D + FL + FD) +
122 (1/192)*(RD + LT + FLT + FRD)
124 c t 0 3 0 0 = eval (fv t) $
126 (5/96)*(L + R + T + D + FL + FR + FT + FD) +
127 (1/192)*(RT + RD + LT + LD + FRT + FRD + FLT + FLD)
129 c t 1 0 2 0 = eval (fv t) $
132 (1/12)*(LT + FL + FT)
134 c t 1 0 0 2 = eval (fv t) $
137 (1/12)*(RT + FR + FT)
139 c t 1 0 1 1 = eval (fv t) $
144 (1/48)*(LT + RT + FL + FR)
146 c t 1 1 1 0 = eval (fv t) $
151 (1/48)*(D + R + LT) +
152 (1/96)*(FD + LD + RT + FR)
154 c t 1 1 0 1 = eval (fv t) $
159 (1/48)*(D + L + RT) +
160 (1/96)*(FD + LT + RD + FL)
162 c t 1 2 0 0 = eval (fv t) $
165 (7/96)*(L + R + T + D) +
166 (1/32)*(FL + FR + FT + FD) +
167 (1/96)*(RT + RD + LT + LD)
169 c t 2 0 1 0 = eval (fv t) $
172 (1/48)*(R + D + B + LT + FL + FT) +
173 (1/96)*(RT + BT + FR + FD + LD + BL)
175 c t 2 0 0 1 = eval (fv t) $
178 (1/48)*(L + D + B + RT + FR + FT) +
179 (1/96)*(LT + BT + FL + FD + RD + BR)
181 c t 2 1 0 0 = eval (fv t) $
183 (1/12)*(T + R + L + D) +
184 (1/64)*(FT + FR + FL + FD) +
187 (1/96)*(RT + LD + LT + RD) +
188 (1/192)*(BT + BR + BL + BD)
190 c t 3 0 0 0 = eval (fv t) $
192 (1/12)*(T + F + L + R + D + B) +
193 (1/96)*(LT + FL + FT + RT + BT + FR) +
194 (1/96)*(FD + LD + BD + BR + RD + BL)
196 c _ _ _ _ _ = error "coefficient index out of bounds"
200 vol_matrix :: Tetrahedron -> Matrix Double
201 vol_matrix t = (4><4) $
220 -- Computed using the formula from Lai & Schumaker, Definition 15.4,
222 volume :: Tetrahedron -> Double
224 | (v0 t) == (v1 t) = 0
225 | (v0 t) == (v2 t) = 0
226 | (v0 t) == (v3 t) = 0
227 | (v1 t) == (v2 t) = 0
228 | (v1 t) == (v3 t) = 0
229 | (v2 t) == (v3 t) = 0
230 | otherwise = (1/6)*(det (vol_matrix t))
233 b0 :: Tetrahedron -> (RealFunction Point)
234 b0 t point = (volume inner_tetrahedron) / (volume t)
236 inner_tetrahedron = t { v0 = point }
238 b1 :: Tetrahedron -> (RealFunction Point)
239 b1 t point = (volume inner_tetrahedron) / (volume t)
241 inner_tetrahedron = t { v1 = point }
243 b2 :: Tetrahedron -> (RealFunction Point)
244 b2 t point = (volume inner_tetrahedron) / (volume t)
246 inner_tetrahedron = t { v2 = point }
248 b3 :: Tetrahedron -> (RealFunction Point)
249 b3 t point = (volume inner_tetrahedron) / (volume t)
251 inner_tetrahedron = t { v3 = point }