X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTetrahedron.hs;h=9f68364042e7563a3b10d23f01e72f2845ba74af;hb=4d695b8d0b05a02d562bdee3d2a1b98ce6a5e747;hp=ad164d2426dce01e85cbf3d22cab0d0a19380410;hpb=7d6bba8440f4327de6272e5c513959425f841c5d;p=spline3.git

diff --git a/src/Tetrahedron.hs b/src/Tetrahedron.hs
index ad164d2..9f68364 100644
--- a/src/Tetrahedron.hs
+++ b/src/Tetrahedron.hs
@@ -1,56 +1,109 @@
 module Tetrahedron
 where
 
+import qualified Data.Vector as V (
+  singleton,
+  snoc,
+  sum
+  )
 import Numeric.LinearAlgebra hiding (i, scale)
+import Prelude hiding (LT)
+import Test.QuickCheck (Arbitrary(..), Gen)
 
-import Cube (back,
-             Cube(datum),
-             down,
-             front,
-             Gridded,
-             left,
-             right,
-             top)
+import Cardinal
+import Comparisons (nearly_ge)
+import FunctionValues
 import Misc (factorial)
 import Point
 import RealFunction
 import ThreeDimensional
 
-data Tetrahedron = Tetrahedron { cube :: Cube,
-                                 v0   :: Point,
-                                 v1   :: Point,
-                                 v2   :: Point,
-                                 v3   :: Point }
-                   deriving (Eq)
+data Tetrahedron =
+  Tetrahedron { fv :: FunctionValues,
+                v0 :: Point,
+                v1 :: Point,
+                v2 :: Point,
+                v3 :: Point,
+                precomputed_volume :: Double
+              }
+    deriving (Eq)
 
-instance Show Tetrahedron where
-    show t = "Tetrahedron (Cube: " ++ (show (cube t)) ++ ") " ++
-             "(v0: " ++ (show (v0 t)) ++ ") (v1: " ++ (show (v1 t)) ++
-             ") (v2: " ++ (show (v2 t)) ++ ") (v3: " ++ (show (v3 t)) ++
-             ")\n\n"
 
-instance Gridded Tetrahedron where
-    back t = back (cube t)
-    down t = down (cube t)
-    front t = front (cube t)
-    left t = left (cube t)
-    right t = right (cube t)
-    top t = top (cube t)
+instance Arbitrary Tetrahedron where
+    arbitrary = do
+      rnd_v0 <- arbitrary :: Gen Point
+      rnd_v1 <- arbitrary :: Gen Point
+      rnd_v2 <- arbitrary :: Gen Point
+      rnd_v3 <- arbitrary :: Gen Point
+      rnd_fv <- arbitrary :: Gen FunctionValues
+
+      -- We can't assign an incorrect precomputed volume,
+      -- so we have to calculate the correct one here.
+      let t' = Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 0
+      let vol = volume t'
+      return (Tetrahedron rnd_fv rnd_v0 rnd_v1 rnd_v2 rnd_v3 vol)
+
+
+instance Show Tetrahedron where
+    show t = "Tetrahedron:\n" ++
+             "  fv: " ++ (show (fv t)) ++ "\n" ++
+             "  v0: " ++ (show (v0 t)) ++ "\n" ++
+             "  v1: " ++ (show (v1 t)) ++ "\n" ++
+             "  v2: " ++ (show (v2 t)) ++ "\n" ++
+             "  v3: " ++ (show (v3 t)) ++ "\n"
 
 
 instance ThreeDimensional Tetrahedron where
-    center t = ((v0 t) + (v1 t) + (v2 t) + (v3 t)) `scale` (1/4)
+    center (Tetrahedron _ v0' v1' v2' v3' _) =
+        (v0' + v1' + v2' + v3') `scale` (1/4)
+
     contains_point t p =
-        (b0 t p) >= 0 && (b1 t p) >= 0 && (b2 t p) >= 0 && (b3 t p) >= 0
+      b0_unscaled `nearly_ge` 0 &&
+      b1_unscaled `nearly_ge` 0 &&
+      b2_unscaled `nearly_ge` 0 &&
+      b3_unscaled `nearly_ge` 0
+      where
+        -- Drop the useless division and volume calculation that we
+        -- would do if we used the regular b0,..b3 functions.
+        b0_unscaled :: Double
+        b0_unscaled = volume inner_tetrahedron
+          where inner_tetrahedron = t { v0 = p }
+
+        b1_unscaled :: Double
+        b1_unscaled = volume inner_tetrahedron
+          where inner_tetrahedron = t { v1 = p }
+
+        b2_unscaled :: Double
+        b2_unscaled = volume inner_tetrahedron
+          where inner_tetrahedron = t { v2 = p }
+
+        b3_unscaled :: Double
+        b3_unscaled = volume inner_tetrahedron
+          where inner_tetrahedron = t { v3 = p }
 
 
 polynomial :: Tetrahedron -> (RealFunction Point)
 polynomial t =
-    sum [ (c t i j k l) `cmult` (beta t i j k l) | i <- [0..3],
-                                                   j <- [0..3],
-                                                   k <- [0..3],
-                                                   l <- [0..3],
-                                                   i + j + k + l == 3]
+    V.sum $ V.singleton ((c t 0 0 0 3) `cmult` (beta t 0 0 0 3)) `V.snoc`
+            ((c t 0 0 1 2) `cmult` (beta t 0 0 1 2)) `V.snoc`
+            ((c t 0 0 2 1) `cmult` (beta t 0 0 2 1)) `V.snoc`
+            ((c t 0 0 3 0) `cmult` (beta t 0 0 3 0)) `V.snoc`
+            ((c t 0 1 0 2) `cmult` (beta t 0 1 0 2)) `V.snoc`
+            ((c t 0 1 1 1) `cmult` (beta t 0 1 1 1)) `V.snoc`
+            ((c t 0 1 2 0) `cmult` (beta t 0 1 2 0)) `V.snoc`
+            ((c t 0 2 0 1) `cmult` (beta t 0 2 0 1)) `V.snoc`
+            ((c t 0 2 1 0) `cmult` (beta t 0 2 1 0)) `V.snoc`
+            ((c t 0 3 0 0) `cmult` (beta t 0 3 0 0)) `V.snoc`
+            ((c t 1 0 0 2) `cmult` (beta t 1 0 0 2)) `V.snoc`
+            ((c t 1 0 1 1) `cmult` (beta t 1 0 1 1)) `V.snoc`
+            ((c t 1 0 2 0) `cmult` (beta t 1 0 2 0)) `V.snoc`
+            ((c t 1 1 0 1) `cmult` (beta t 1 1 0 1)) `V.snoc`
+            ((c t 1 1 1 0) `cmult` (beta t 1 1 1 0)) `V.snoc`
+            ((c t 1 2 0 0) `cmult` (beta t 1 2 0 0)) `V.snoc`
+            ((c t 2 0 0 1) `cmult` (beta t 2 0 0 1)) `V.snoc`
+            ((c t 2 0 1 0) `cmult` (beta t 2 0 1 0)) `V.snoc`
+            ((c t 2 1 0 0) `cmult` (beta t 2 1 0 0)) `V.snoc`
+            ((c t 3 0 0 0) `cmult` (beta t 3 0 0 0))
 
 
 -- | Returns the domain point of t with indices i,j,k,l.
@@ -87,422 +140,148 @@ beta t i j k l
     b3_term = (b3 t) `fexp` l
 
 
+-- | The coefficient function. c t i j k l returns the coefficient
+--   c_ijkl with respect to the tetrahedron t. The definition uses
+--   pattern matching to mimic the definitions given in Sorokina and
+--   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
+--   function will simply error.
 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
-c x 0 0 3 0 = datum $ (1/8) * (i + f + l + t + lt + fl + ft + flt)
-    where
-      f = front x
-      flt = front (left (top x))
-      fl = front (left x)
-      ft =  front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      t = top x
-
-
-c x 0 0 0 3 = datum $ (1/8) * (i + f + r + t + rt + fr + ft + frt)
-    where
-      f = front x
-      fr = front (right x)
-      frt = front (right (top x))
-      ft = front (top x)
-      i = cube x
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 0 0 2 1 = datum $ (5/24)*(i + f + t + ft) + (1/24)*(l + fl + lt + flt)
-    where
-      f = front x
-      flt = front (left (top x))
-      fl = front (left x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      t = top x
-
-
-c x 0 0 1 2 = datum $ (5/24)*(i + f + t + ft) + (1/24)*(r + fr + rt + frt)
-    where
-      f = front x
-      frt = front (right (top x))
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 0 1 2 0 = datum $
-              (5/24)*(i + f) +
-              (1/8)*(l + t + fl + ft) +
-              (1/24)*(lt + flt)
-    where
-      f = front x
-      flt = front (left (top x))
-      fl = front (left x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      t = top x
-
-
-c x 0 1 0 2 = datum $
-              (5/24)*(i + f) +
-              (1/8)*(r + t + fr + ft) +
-              (1/24)*(rt + frt)
-    where
-      f = front x
-      fr = front (right x)
-      frt = front (right (top x))
-      ft = front (top x)
-      i = cube x
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 0 1 1 1 = datum $
-          (13/48)*(i + f) +
-          (7/48)*(t + ft) +
-          (1/32)*(l + r + fl + fr) +
-          (1/96)*(lt + rt + flt + frt)
-    where
-      f = front x
-      flt = front (left (top x))
-      fl = front (left x)
-      fr = front (right x)
-      frt = front (right (top x))
-      ft = front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 0 2 1 0 = datum $
-          (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)
-    where
-      d = down x
-      f = front x
-      fd = front (down x)
-      fld = front (left (down x))
-      flt = front (left (top x))
-      fl = front (left x)
-      fr = front (right x)
-      frt = front (right (top x))
-      ft = front (top x)
-      i = cube x
-      l = left x
-      ld = left (down x)
-      lt = left (top x)
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 0 2 0 1 = datum $
-          (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)
-    where
-      d = down x
-      f = front x
-      fd = front (down x)
-      flt = front (left (top x))
-      fl = front (left x)
-      frd = front (right (down x))
-      fr = front (right x)
-      frt = front (right (top x))
-      ft = front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      r = right x
-      rd = right (down x)
-      rt = right (top x)
-      t = top x
-
-
-c x 0 3 0 0 = datum $
-          (13/48)*(i + f) +
-          (5/96)*(l + r + t + d + fl + fr + ft + fd) +
-          (1/192)*(rt + rd + lt + ld + frt + frd + flt + fld)
-    where
-      d = down x
-      f = front x
-      fd = front (down x)
-      fld = front (left (down x))
-      flt = front (left (top x))
-      fl = front (left x)
-      frd = front (right (down x))
-      fr = front (right x)
-      frt = front (right (top x))
-      ft = front (top x)
-      i = cube x
-      l = left x
-      ld = left (down x)
-      lt = left (top x)
-      r = right x
-      rd = right (down x)
-      rt = right (top x)
-      t = top x
-
-c x 1 0 2 0 = datum $ (1/4)*i + (1/6)*(f + l + t) + (1/12)*(lt + fl + ft)
-    where
-      f = front x
-      fl = front (left x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      t = top x
-
-
-c x 1 0 0 2 = datum $ (1/4)*i + (1/6)*(f + r + t) + (1/12)*(rt + fr + ft)
-    where
-      f = front x
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 1 0 1 1 = datum $
-          (1/3)*i +
-          (5/24)*(f + t) +
-          (1/12)*ft +
-          (1/24)*(l + r) +
-          (1/48)*(lt + rt + fl + fr)
-    where
-      f = front x
-      fl = front (left x)
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 1 1 1 0 = datum $
-          (1/3)*i +
-          (5/24)*f +
-          (1/8)*(l + t) +
-          (5/96)*(fl + ft) +
-          (1/48)*(d + r + lt) +
-          (1/96)*(fd + ld + rt + fr)
-    where
-      d = down x
-      f = front x
-      fd = front (down x)
-      fl = front (left x)
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      ld = left (down x)
-      lt = left (top x)
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 1 1 0 1 = datum $
-          (1/3)*i +
-          (5/24)*f +
-          (1/8)*(r + t) +
-          (5/96)*(fr + ft) +
-          (1/48)*(d + l + rt) +
-          (1/96)*(fd + lt + rd + fl)
-    where
-      d = down x
-      f = front x
-      fd = front (down x)
-      fl = front (left x)
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      r = right x
-      rd = right (down x)
-      rt = right (top x)
-      t = top x
-
-
-
-c x 1 2 0 0 = datum $
-          (1/3)*i +
-          (5/24)*f +
-          (7/96)*(l + r + t + d) +
-          (1/32)*(fl + fr + ft + fd) +
-          (1/96)*(rt + rd + lt + ld)
-    where
-      d = down x
-      f = front x
-      fd = front (down x)
-      fl = front (left x)
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      ld = left (down x)
-      lt = left (top x)
-      r = right x
-      rd = right (down x)
-      rt = right (top x)
-      t = top x
-
-
-c x 2 0 1 0 = datum $
-          (3/8)*i +
-          (7/48)*(f + t + l) +
-          (1/48)*(r + d + b + lt + fl + ft) +
-          (1/96)*(rt + bt + fr + fd + ld + bl)
-    where
-      b = back x
-      bl = back (left x)
-      bt = back (top x)
-      d = down x
-      f = front x
-      fd = front (down x)
-      fl = front (left x)
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      ld = left (down x)
-      lt = left (top x)
-      r = right x
-      rt = right (top x)
-      t = top x
-
-
-c x 2 0 0 1 = datum $
-          (3/8)*i +
-          (7/48)*(f + t + r) +
-          (1/48)*(l + d + b + rt + fr + ft) +
-          (1/96)*(lt + bt + fl + fd + rd + br)
-    where
-      b = back x
-      br = back (right x)
-      bt = back (top x)
-      d = down x
-      f = front x
-      fd = front (down x)
-      fl = front (left x)
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      lt = left (top x)
-      r = right x
-      rd = right (down x)
-      rt = right (top x)
-      t = top x
-
-
-
-c x 2 1 0 0 = datum $
-          (3/8)*i +
-          (1/12)*(t + r + l + d) +
-          (1/64)*(ft + fr + fl + fd) +
-          (7/48)*f +
-          (1/48)*b +
-          (1/96)*(rt + ld + lt + rd) +
-          (1/192)*(bt + br + bl + bd)
-    where
-      b = back x
-      bd = back (down x)
-      bl = back (left x)
-      br = back (right x)
-      bt = back (top x)
-      d = down x
-      f = front x
-      fd = front (down x)
-      fl = front (left x)
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      ld = left (down x)
-      lt = left (top x)
-      r = right x
-      rd = right (down x)
-      rt = right (top x)
-      t = top x
-
-
-c x 3 0 0 0 = datum $
-          (3/8)*i +
-          (1/12)*(t + f + l + r + d + b) +
-          (1/96)*(lt + fl + ft + rt + bt + fr) +
-          (1/96)*(fd + ld + bd + br + rd + bl)
-    where
-      b = back x
-      bd = back (down x)
-      bl = back (left x)
-      br = back (right x)
-      bt = back (top x)
-      d = down x
-      f = front x
-      fd = front (down x)
-      fl = front (left x)
-      fr = front (right x)
-      ft = front (top x)
-      i = cube x
-      l = left x
-      ld = left (down x)
-      lt = left (top x)
-      r = right x
-      rd = right (down x)
-      rt = right (top x)
-      t = top x
-
+c t 0 0 3 0 = eval (fv t) $
+              (1/8) * (I + F + L + T + LT + FL + FT + FLT)
+
+c t 0 0 0 3 = eval (fv t) $
+              (1/8) * (I + F + R + T + RT + FR + FT + FRT)
+
+c t 0 0 2 1 = eval (fv t) $
+              (5/24)*(I + F + T + FT) +
+              (1/24)*(L + FL + LT + FLT)
+
+c t 0 0 1 2 = eval (fv t) $
+              (5/24)*(I + F + T + FT) +
+              (1/24)*(R + FR + RT + FRT)
+
+c t 0 1 2 0 = eval (fv t) $
+              (5/24)*(I + F) +
+              (1/8)*(L + T + FL + FT) +
+              (1/24)*(LT + FLT)
+
+c t 0 1 0 2 = eval (fv t) $
+              (5/24)*(I + F) +
+              (1/8)*(R + T + FR + FT) +
+              (1/24)*(RT + FRT)
+
+c t 0 1 1 1 = eval (fv 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) $
+          (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) $
+          (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) $
+          (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) $
+              (1/4)*I +
+              (1/6)*(F + L + T) +
+              (1/12)*(LT + FL + FT)
+
+c t 1 0 0 2 = eval (fv t) $
+              (1/4)*I +
+              (1/6)*(F + R + T) +
+              (1/12)*(RT + FR + FT)
+
+c t 1 0 1 1 = eval (fv 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) $
+          (1/3)*I +
+          (5/24)*F +
+          (1/8)*(L + T) +
+          (5/96)*(FL + FT) +
+          (1/48)*(D + R + LT) +
+          (1/96)*(FD + LD + RT + FR)
+
+c t 1 1 0 1 = eval (fv t) $
+          (1/3)*I +
+          (5/24)*F +
+          (1/8)*(R + T) +
+          (5/96)*(FR + FT) +
+          (1/48)*(D + L + RT) +
+          (1/96)*(FD + LT + RD + FL)
+
+c t 1 2 0 0 = eval (fv 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) $
+          (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) $
+          (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) $
+          (3/8)*I +
+          (1/12)*(T + R + L + D) +
+          (1/64)*(FT + FR + FL + FD) +
+          (7/48)*F +
+          (1/48)*B +
+          (1/96)*(RT + LD + LT + RD) +
+          (1/192)*(BT + BR + BL + BD)
+
+c t 3 0 0 0 = eval (fv t) $
+          (3/8)*I +
+          (1/12)*(T + F + L + R + D + B) +
+          (1/96)*(LT + FL + FT + RT + BT + FR) +
+          (1/96)*(FD + LD + BD + BR + RD + BL)
 
 c _ _ _ _ _ = error "coefficient index out of bounds"
 
 
 
+-- | 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) $
+vol_matrix t = (4><4)
                [1,  1,  1,  1,
                 x1, x2, x3, x4,
                 y1, y2, y3, y4,
                 z1, z2, z3, z4 ]
     where
-      x1 = x_coord (v0 t)
-      x2 = x_coord (v1 t)
-      x3 = x_coord (v2 t)
-      x4 = x_coord (v3 t)
-      y1 = y_coord (v0 t)
-      y2 = y_coord (v1 t)
-      y3 = y_coord (v2 t)
-      y4 = y_coord (v3 t)
-      z1 = z_coord (v0 t)
-      z2 = z_coord (v1 t)
-      z3 = z_coord (v2 t)
-      z4 = z_coord (v3 t)
-
--- Computed using the formula from Lai & Schumaker, Definition 15.4,
--- page 436.
+      (x1, y1, z1) = v0 t
+      (x2, y2, z2) = v1 t
+      (x3, y3, z3) = v2 t
+      (x4, y4, z4) = v3 t
+
+-- | Computed using the formula from Lai & Schumaker, Definition 15.4,
+--   page 436.
 volume :: Tetrahedron -> Double
 volume t
        | (v0 t) == (v1 t) = 0
@@ -514,22 +293,29 @@ volume t
        | otherwise = (1/6)*(det (vol_matrix t))
 
 
+-- | The barycentric coordinates of a point with respect to v0.
 b0 :: Tetrahedron -> (RealFunction Point)
-b0 t point = (volume inner_tetrahedron) / (volume t)
+b0 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v0 = point }
 
+
+-- | The barycentric coordinates of a point with respect to v1.
 b1 :: Tetrahedron -> (RealFunction Point)
-b1 t point = (volume inner_tetrahedron) / (volume t)
+b1 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v1 = point }
 
+
+-- | The barycentric coordinates of a point with respect to v2.
 b2 :: Tetrahedron -> (RealFunction Point)
-b2 t point = (volume inner_tetrahedron) / (volume t)
+b2 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v2 = point }
 
+
+-- | The barycentric coordinates of a point with respect to v3.
 b3 :: Tetrahedron -> (RealFunction Point)
-b3 t point = (volume inner_tetrahedron) / (volume t)
+b3 t point = (volume inner_tetrahedron) / (precomputed_volume t)
              where
                inner_tetrahedron = t { v3 = point }