src/Tetrahedron.hs

   1 module Tetrahedron
   2 where
   3
   4 import Numeric.LinearAlgebra hiding (i, scale)
   5 import Prelude hiding (LT)
   6
   7 import Cardinal
   8 import FunctionValues
   9 import Misc (factorial)
  10 import Point
  11 import RealFunction
  12 import ThreeDimensional
  13
  14 data Tetrahedron = Tetrahedron { fv :: FunctionValues,
  15                                  v0 :: Point,
  16                                  v1 :: Point,
  17                                  v2 :: Point,
  18                                  v3 :: Point }
  19                    deriving (Eq)
  20
  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"
  28
  29
  30 instance ThreeDimensional Tetrahedron where
  31     center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
  32     contains_point t p =
  33         (b0 t p) >= 0 && (b1 t p) >= 0 && (b2 t p) >= 0 && (b3 t p) >= 0
  34
  35
  36 polynomial :: Tetrahedron -> (RealFunction Point)
  37 polynomial t =
  38     sum [ (c t i j k l) `cmult` (beta t i j k l) | i <- [0..3],
  39                                                    j <- [0..3],
  40                                                    k <- [0..3],
  41                                                    l <- [0..3],
  42                                                    i + j + k + l == 3]
  43
  44
  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
  48 xi = domain_point
  49
  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"
  55    where
  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'
  61
  62
  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)
  66 beta t i j k l
  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"
  70   where
  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
  77
  78
  79 -- | The coefficient function. c t i j k l returns the coefficient
  80 --   c_ijkl with respect to the tetrahedron t. The definition uses
  81 --   pattern matching to mimic the definitions given in Sorokina and
  82 --   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
  83 --   function will simply error.
  84 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
  85 c t 0 0 3 0 = eval (fv t) $
  86               (1/8) * (I + F + L + T + LT + FL + FT + FLT)
  87
  88 c t 0 0 0 3 = eval (fv t) $
  89               (1/8) * (I + F + R + T + RT + FR + FT + FRT)
  90
  91 c t 0 0 2 1 = eval (fv t) $
  92               (5/24)*(I + F + T + FT) +
  93               (1/24)*(L + FL + LT + FLT)
  94
  95 c t 0 0 1 2 = eval (fv t) $
  96               (5/24)*(I + F + T + FT) +
  97               (1/24)*(R + FR + RT + FRT)
  98
  99 c t 0 1 2 0 = eval (fv t) $
 100               (5/24)*(I + F) +
 101               (1/8)*(L + T + FL + FT) +
 102               (1/24)*(LT + FLT)
 103
 104 c t 0 1 0 2 = eval (fv t) $
 105               (5/24)*(I + F) +
 106               (1/8)*(R + T + FR + FT) +
 107               (1/24)*(RT + FRT)
 108
 109 c t 0 1 1 1 = eval (fv t) $
 110           (13/48)*(I + F) +
 111           (7/48)*(T + FT) +
 112           (1/32)*(L + R + FL + FR) +
 113           (1/96)*(LT + RT + FLT + FRT)
 114
 115 c t 0 2 1 0 = eval (fv t) $
 116           (13/48)*(I + F) +
 117           (17/192)*(L + T + FL + FT) +
 118           (1/96)*(LT + FLT) +
 119           (1/64)*(R + D + FR + FD) +
 120           (1/192)*(RT + LD + FRT + FLD)
 121
 122 c t 0 2 0 1 = eval (fv t) $
 123           (13/48)*(I + F) +
 124           (17/192)*(R + T + FR + FT) +
 125           (1/96)*(RT + FRT) +
 126           (1/64)*(L + D + FL + FD) +
 127           (1/192)*(RD + LT + FLT + FRD)
 128
 129 c t 0 3 0 0 = eval (fv t) $
 130           (13/48)*(I + F) +
 131           (5/96)*(L + R + T + D + FL + FR + FT + FD) +
 132           (1/192)*(RT + RD + LT + LD + FRT + FRD + FLT + FLD)
 133
 134 c t 1 0 2 0 = eval (fv t) $
 135               (1/4)*I +
 136               (1/6)*(F + L + T) +
 137               (1/12)*(LT + FL + FT)
 138
 139 c t 1 0 0 2 = eval (fv t) $
 140               (1/4)*I +
 141               (1/6)*(F + R + T) +
 142               (1/12)*(RT + FR + FT)
 143
 144 c t 1 0 1 1 = eval (fv t) $
 145           (1/3)*I +
 146           (5/24)*(F + T) +
 147           (1/12)*FT +
 148           (1/24)*(L + R) +
 149           (1/48)*(LT + RT + FL + FR)
 150
 151 c t 1 1 1 0 = eval (fv t) $
 152           (1/3)*I +
 153           (5/24)*F +
 154           (1/8)*(L + T) +
 155           (5/96)*(FL + FT) +
 156           (1/48)*(D + R + LT) +
 157           (1/96)*(FD + LD + RT + FR)
 158
 159 c t 1 1 0 1 = eval (fv t) $
 160           (1/3)*I +
 161           (5/24)*F +
 162           (1/8)*(R + T) +
 163           (5/96)*(FR + FT) +
 164           (1/48)*(D + L + RT) +
 165           (1/96)*(FD + LT + RD + FL)
 166
 167 c t 1 2 0 0 = eval (fv t) $
 168           (1/3)*I +
 169           (5/24)*F +
 170           (7/96)*(L + R + T + D) +
 171           (1/32)*(FL + FR + FT + FD) +
 172           (1/96)*(RT + RD + LT + LD)
 173
 174 c t 2 0 1 0 = eval (fv t) $
 175           (3/8)*I +
 176           (7/48)*(F + T + L) +
 177           (1/48)*(R + D + B + LT + FL + FT) +
 178           (1/96)*(RT + BT + FR + FD + LD + BL)
 179
 180 c t 2 0 0 1 = eval (fv t) $
 181           (3/8)*I +
 182           (7/48)*(F + T + R) +
 183           (1/48)*(L + D + B + RT + FR + FT) +
 184           (1/96)*(LT + BT + FL + FD + RD + BR)
 185
 186 c t 2 1 0 0 = eval (fv t) $
 187           (3/8)*I +
 188           (1/12)*(T + R + L + D) +
 189           (1/64)*(FT + FR + FL + FD) +
 190           (7/48)*F +
 191           (1/48)*B +
 192           (1/96)*(RT + LD + LT + RD) +
 193           (1/192)*(BT + BR + BL + BD)
 194
 195 c t 3 0 0 0 = eval (fv t) $
 196           (3/8)*I +
 197           (1/12)*(T + F + L + R + D + B) +
 198           (1/96)*(LT + FL + FT + RT + BT + FR) +
 199           (1/96)*(FD + LD + BD + BR + RD + BL)
 200
 201 c _ _ _ _ _ = error "coefficient index out of bounds"
 202
 203
 204
 205 -- | The matrix used in the tetrahedron volume calculation as given in
 206 --   Lai & Schumaker, Definition 15.4, page 436.
 207 vol_matrix :: Tetrahedron -> Matrix Double
 208 vol_matrix t = (4><4)
 209                [1,  1,  1,  1,
 210                 x1, x2, x3, x4,
 211                 y1, y2, y3, y4,
 212                 z1, z2, z3, z4 ]
 213     where
 214       x1 = x_coord (v0 t)
 215       x2 = x_coord (v1 t)
 216       x3 = x_coord (v2 t)
 217       x4 = x_coord (v3 t)
 218       y1 = y_coord (v0 t)
 219       y2 = y_coord (v1 t)
 220       y3 = y_coord (v2 t)
 221       y4 = y_coord (v3 t)
 222       z1 = z_coord (v0 t)
 223       z2 = z_coord (v1 t)
 224       z3 = z_coord (v2 t)
 225       z4 = z_coord (v3 t)
 226
 227 -- | Computed using the formula from Lai & Schumaker, Definition 15.4,
 228 --   page 436.
 229 volume :: Tetrahedron -> Double
 230 volume t
 231        | (v0 t) == (v1 t) = 0
 232        | (v0 t) == (v2 t) = 0
 233        | (v0 t) == (v3 t) = 0
 234        | (v1 t) == (v2 t) = 0
 235        | (v1 t) == (v3 t) = 0
 236        | (v2 t) == (v3 t) = 0
 237        | otherwise = (1/6)*(det (vol_matrix t))
 238
 239
 240 -- | The barycentric coordinates of a point with respect to v0.
 241 b0 :: Tetrahedron -> (RealFunction Point)
 242 b0 t point = (volume inner_tetrahedron) / (volume t)
 243              where
 244                inner_tetrahedron = t { v0 = point }
 245
 246
 247 -- | The barycentric coordinates of a point with respect to v1.
 248 b1 :: Tetrahedron -> (RealFunction Point)
 249 b1 t point = (volume inner_tetrahedron) / (volume t)
 250              where
 251                inner_tetrahedron = t { v1 = point }
 252
 253
 254 -- | The barycentric coordinates of a point with respect to v2.
 255 b2 :: Tetrahedron -> (RealFunction Point)
 256 b2 t point = (volume inner_tetrahedron) / (volume t)
 257              where
 258                inner_tetrahedron = t { v2 = point }
 259
 260
 261 -- | The barycentric coordinates of a point with respect to v3.
 262 b3 :: Tetrahedron -> (RealFunction Point)
 263 b3 t point = (volume inner_tetrahedron) / (volume t)
 264              where
 265                inner_tetrahedron = t { v3 = point }