src/Tetrahedron.hs

   1 module Tetrahedron
   2 where
   3
   4 import qualified Data.Vector as V (
   5   singleton,
   6   snoc,
   7   sum
   8   )
   9 import Numeric.LinearAlgebra hiding (i, scale)
  10 import Prelude hiding (LT)
  11 import Test.QuickCheck (Arbitrary(..), Gen)
  12
  13 import Cardinal
  14 import Comparisons (nearly_ge)
  15 import FunctionValues
  16 import Misc (factorial)
  17 import Point
  18 import RealFunction
  19 import ThreeDimensional
  20
  21 data Tetrahedron =
  22   Tetrahedron { fv :: FunctionValues,
  23                 v0 :: Point,
  24                 v1 :: Point,
  25                 v2 :: Point,
  26                 v3 :: Point,
  27                 precomputed_volume :: Double
  28               }
  29     deriving (Eq)
  30
  31
  32 instance Arbitrary Tetrahedron where
  33     arbitrary = do
  34       rnd_v0 <- arbitrary :: Gen Point
  35       rnd_v1 <- arbitrary :: Gen Point
  36       rnd_v2 <- arbitrary :: Gen Point
  37       rnd_v3 <- arbitrary :: Gen Point
  38       rnd_fv <- arbitrary :: Gen FunctionValues
  39
  40       -- We can't assign an incorrect precomputed volume,
  41       -- so we have to calculate the correct one here.
  42       let t' = Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 0
  43       let vol = volume t'
  44       return (Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 vol)
  45
  46
  47 instance Show Tetrahedron where
  48     show t = "Tetrahedron:\n" ++
  49              "  fv: " ++ (show (fv t)) ++ "\n" ++
  50              "  v0: " ++ (show (v0 t)) ++ "\n" ++
  51              "  v1: " ++ (show (v1 t)) ++ "\n" ++
  52              "  v2: " ++ (show (v2 t)) ++ "\n" ++
  53              "  v3: " ++ (show (v3 t)) ++ "\n"
  54
  55
  56 instance ThreeDimensional Tetrahedron where
  57     center (Tetrahedron _ v0' v1' v2' v3' _) =
  58         (v0' + v1' + v2' + v3') `scale` (1/4)
  59
  60     contains_point t p =
  61       b0_unscaled `nearly_ge` 0 &&
  62       b1_unscaled `nearly_ge` 0 &&
  63       b2_unscaled `nearly_ge` 0 &&
  64       b3_unscaled `nearly_ge` 0
  65       where
  66         -- Drop the useless division and volume calculation that we
  67         -- would do if we used the regular b0,..b3 functions.
  68         b0_unscaled :: Double
  69         b0_unscaled = volume inner_tetrahedron
  70           where inner_tetrahedron = t { v0 = p }
  71
  72         b1_unscaled :: Double
  73         b1_unscaled = volume inner_tetrahedron
  74           where inner_tetrahedron = t { v1 = p }
  75
  76         b2_unscaled :: Double
  77         b2_unscaled = volume inner_tetrahedron
  78           where inner_tetrahedron = t { v2 = p }
  79
  80         b3_unscaled :: Double
  81         b3_unscaled = volume inner_tetrahedron
  82           where inner_tetrahedron = t { v3 = p }
  83
  84
  85 polynomial :: Tetrahedron -> (RealFunction Point)
  86 polynomial t =
  87     V.sum $ V.singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `V.snoc`
  88             ((c t 0 0 1 2) `cmult` (beta t 0 0 1 2)) `V.snoc`
  89             ((c t 0 0 2 1) `cmult` (beta t 0 0 2 1)) `V.snoc`
  90             ((c t 0 0 3 0) `cmult` (beta t 0 0 3 0)) `V.snoc`
  91             ((c t 0 1 0 2) `cmult` (beta t 0 1 0 2)) `V.snoc`
  92             ((c t 0 1 1 1) `cmult` (beta t 0 1 1 1)) `V.snoc`
  93             ((c t 0 1 2 0) `cmult` (beta t 0 1 2 0)) `V.snoc`
  94             ((c t 0 2 0 1) `cmult` (beta t 0 2 0 1)) `V.snoc`
  95             ((c t 0 2 1 0) `cmult` (beta t 0 2 1 0)) `V.snoc`
  96             ((c t 0 3 0 0) `cmult` (beta t 0 3 0 0)) `V.snoc`
  97             ((c t 1 0 0 2) `cmult` (beta t 1 0 0 2)) `V.snoc`
  98             ((c t 1 0 1 1) `cmult` (beta t 1 0 1 1)) `V.snoc`
  99             ((c t 1 0 2 0) `cmult` (beta t 1 0 2 0)) `V.snoc`
 100             ((c t 1 1 0 1) `cmult` (beta t 1 1 0 1)) `V.snoc`
 101             ((c t 1 1 1 0) `cmult` (beta t 1 1 1 0)) `V.snoc`
 102             ((c t 1 2 0 0) `cmult` (beta t 1 2 0 0)) `V.snoc`
 103             ((c t 2 0 0 1) `cmult` (beta t 2 0 0 1)) `V.snoc`
 104             ((c t 2 0 1 0) `cmult` (beta t 2 0 1 0)) `V.snoc`
 105             ((c t 2 1 0 0) `cmult` (beta t 2 1 0 0)) `V.snoc`
 106             ((c t 3 0 0 0) `cmult` (beta t 3 0 0 0))
 107
 108
 109 -- | Returns the domain point of t with indices i,j,k,l.
 110 --   Simply an alias for the domain_point function.
 111 xi :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
 112 xi = domain_point
 113
 114 -- | Returns the domain point of t with indices i,j,k,l.
 115 domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
 116 domain_point t i j k l
 117    | i + j + k + l == 3 = weighted_sum `scale` (1/3)
 118    | otherwise = error "domain point index out of bounds"
 119    where
 120      v0' = (v0 t) `scale` (fromIntegral i)
 121      v1' = (v1 t) `scale` (fromIntegral j)
 122      v2' = (v2 t) `scale` (fromIntegral k)
 123      v3' = (v3 t) `scale` (fromIntegral l)
 124      weighted_sum = v0' + v1' + v2' + v3'
 125
 126
 127 -- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a
 128 --   capital 'B' in the Sorokina/Zeilfelder paper.
 129 beta :: Tetrahedron -> Int -> Int -> Int -> Int -> (RealFunction Point)
 130 beta t i j k l
 131   | (i + j + k + l == 3) =
 132       coefficient `cmult` (b0_term * b1_term * b2_term * b3_term)
 133   | otherwise = error "basis function index out of bounds"
 134   where
 135     denominator = (factorial i)*(factorial j)*(factorial k)*(factorial l)
 136     coefficient = 6 / (fromIntegral denominator)
 137     b0_term = (b0 t) `fexp` i
 138     b1_term = (b1 t) `fexp` j
 139     b2_term = (b2 t) `fexp` k
 140     b3_term = (b3 t) `fexp` l
 141
 142
 143 -- | The coefficient function. c t i j k l returns the coefficient
 144 --   c_ijkl with respect to the tetrahedron t. The definition uses
 145 --   pattern matching to mimic the definitions given in Sorokina and
 146 --   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
 147 --   function will simply error.
 148 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
 149 c t 0 0 3 0 = eval (fv t) $
 150               (1/8) * (I + F + L + T + LT + FL + FT + FLT)
 151
 152 c t 0 0 0 3 = eval (fv t) $
 153               (1/8) * (I + F + R + T + RT + FR + FT + FRT)
 154
 155 c t 0 0 2 1 = eval (fv t) $
 156               (5/24)*(I + F + T + FT) +
 157               (1/24)*(L + FL + LT + FLT)
 158
 159 c t 0 0 1 2 = eval (fv t) $
 160               (5/24)*(I + F + T + FT) +
 161               (1/24)*(R + FR + RT + FRT)
 162
 163 c t 0 1 2 0 = eval (fv t) $
 164               (5/24)*(I + F) +
 165               (1/8)*(L + T + FL + FT) +
 166               (1/24)*(LT + FLT)
 167
 168 c t 0 1 0 2 = eval (fv t) $
 169               (5/24)*(I + F) +
 170               (1/8)*(R + T + FR + FT) +
 171               (1/24)*(RT + FRT)
 172
 173 c t 0 1 1 1 = eval (fv t) $
 174           (13/48)*(I + F) +
 175           (7/48)*(T + FT) +
 176           (1/32)*(L + R + FL + FR) +
 177           (1/96)*(LT + RT + FLT + FRT)
 178
 179 c t 0 2 1 0 = eval (fv t) $
 180           (13/48)*(I + F) +
 181           (17/192)*(L + T + FL + FT) +
 182           (1/96)*(LT + FLT) +
 183           (1/64)*(R + D + FR + FD) +
 184           (1/192)*(RT + LD + FRT + FLD)
 185
 186 c t 0 2 0 1 = eval (fv t) $
 187           (13/48)*(I + F) +
 188           (17/192)*(R + T + FR + FT) +
 189           (1/96)*(RT + FRT) +
 190           (1/64)*(L + D + FL + FD) +
 191           (1/192)*(RD + LT + FLT + FRD)
 192
 193 c t 0 3 0 0 = eval (fv t) $
 194           (13/48)*(I + F) +
 195           (5/96)*(L + R + T + D + FL + FR + FT + FD) +
 196           (1/192)*(RT + RD + LT + LD + FRT + FRD + FLT + FLD)
 197
 198 c t 1 0 2 0 = eval (fv t) $
 199               (1/4)*I +
 200               (1/6)*(F + L + T) +
 201               (1/12)*(LT + FL + FT)
 202
 203 c t 1 0 0 2 = eval (fv t) $
 204               (1/4)*I +
 205               (1/6)*(F + R + T) +
 206               (1/12)*(RT + FR + FT)
 207
 208 c t 1 0 1 1 = eval (fv t) $
 209           (1/3)*I +
 210           (5/24)*(F + T) +
 211           (1/12)*FT +
 212           (1/24)*(L + R) +
 213           (1/48)*(LT + RT + FL + FR)
 214
 215 c t 1 1 1 0 = eval (fv t) $
 216           (1/3)*I +
 217           (5/24)*F +
 218           (1/8)*(L + T) +
 219           (5/96)*(FL + FT) +
 220           (1/48)*(D + R + LT) +
 221           (1/96)*(FD + LD + RT + FR)
 222
 223 c t 1 1 0 1 = eval (fv t) $
 224           (1/3)*I +
 225           (5/24)*F +
 226           (1/8)*(R + T) +
 227           (5/96)*(FR + FT) +
 228           (1/48)*(D + L + RT) +
 229           (1/96)*(FD + LT + RD + FL)
 230
 231 c t 1 2 0 0 = eval (fv t) $
 232           (1/3)*I +
 233           (5/24)*F +
 234           (7/96)*(L + R + T + D) +
 235           (1/32)*(FL + FR + FT + FD) +
 236           (1/96)*(RT + RD + LT + LD)
 237
 238 c t 2 0 1 0 = eval (fv t) $
 239           (3/8)*I +
 240           (7/48)*(F + T + L) +
 241           (1/48)*(R + D + B + LT + FL + FT) +
 242           (1/96)*(RT + BT + FR + FD + LD + BL)
 243
 244 c t 2 0 0 1 = eval (fv t) $
 245           (3/8)*I +
 246           (7/48)*(F + T + R) +
 247           (1/48)*(L + D + B + RT + FR + FT) +
 248           (1/96)*(LT + BT + FL + FD + RD + BR)
 249
 250 c t 2 1 0 0 = eval (fv t) $
 251           (3/8)*I +
 252           (1/12)*(T + R + L + D) +
 253           (1/64)*(FT + FR + FL + FD) +
 254           (7/48)*F +
 255           (1/48)*B +
 256           (1/96)*(RT + LD + LT + RD) +
 257           (1/192)*(BT + BR + BL + BD)
 258
 259 c t 3 0 0 0 = eval (fv t) $
 260           (3/8)*I +
 261           (1/12)*(T + F + L + R + D + B) +
 262           (1/96)*(LT + FL + FT + RT + BT + FR) +
 263           (1/96)*(FD + LD + BD + BR + RD + BL)
 264
 265 c _ _ _ _ _ = error "coefficient index out of bounds"
 266
 267
 268
 269 -- | The matrix used in the tetrahedron volume calculation as given in
 270 --   Lai & Schumaker, Definition 15.4, page 436.
 271 vol_matrix :: Tetrahedron -> Matrix Double
 272 vol_matrix t = (4><4)
 273                [1,  1,  1,  1,
 274                 x1, x2, x3, x4,
 275                 y1, y2, y3, y4,
 276                 z1, z2, z3, z4 ]
 277     where
 278       (x1, y1, z1) = v0 t
 279       (x2, y2, z2) = v1 t
 280       (x3, y3, z3) = v2 t
 281       (x4, y4, z4) = v3 t
 282
 283 -- | Computed using the formula from Lai & Schumaker, Definition 15.4,
 284 --   page 436.
 285 volume :: Tetrahedron -> Double
 286 volume t
 287        | (v0 t) == (v1 t) = 0
 288        | (v0 t) == (v2 t) = 0
 289        | (v0 t) == (v3 t) = 0
 290        | (v1 t) == (v2 t) = 0
 291        | (v1 t) == (v3 t) = 0
 292        | (v2 t) == (v3 t) = 0
 293        | otherwise = (1/6)*(det (vol_matrix t))
 294
 295
 296 -- | The barycentric coordinates of a point with respect to v0.
 297 b0 :: Tetrahedron -> (RealFunction Point)
 298 b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 299              where
 300                inner_tetrahedron = t { v0 = point }
 301
 302
 303 -- | The barycentric coordinates of a point with respect to v1.
 304 b1 :: Tetrahedron -> (RealFunction Point)
 305 b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 306              where
 307                inner_tetrahedron = t { v1 = point }
 308
 309
 310 -- | The barycentric coordinates of a point with respect to v2.
 311 b2 :: Tetrahedron -> (RealFunction Point)
 312 b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 313              where
 314                inner_tetrahedron = t { v2 = point }
 315
 316
 317 -- | The barycentric coordinates of a point with respect to v3.
 318 b3 :: Tetrahedron -> (RealFunction Point)
 319 b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
 320              where
 321                inner_tetrahedron = t { v3 = point }