X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=src%2FTetrahedron.hs;h=f3b53198362768b8fdbe83085cc55b8e6306b7cc;hb=1cd0b90dae4b2a0ea35447427e7962b6fe053308;hp=613b863a49953b57120884b15c83d7ebd733a187;hpb=31ec78e6db481a6410486a3279d62ba73ef14528;p=spline3.git

diff --git a/src/Tetrahedron.hs b/src/Tetrahedron.hs
index 613b863..f3b5319 100644
--- a/src/Tetrahedron.hs
+++ b/src/Tetrahedron.hs
@@ -17,7 +17,7 @@ import qualified Data.Vector as V (
   snoc,
   sum
   )
-import Numeric.LinearAlgebra hiding (i, scale)
+
 import Prelude hiding (LT)
 import Test.Framework (Test, testGroup)
 import Test.Framework.Providers.HUnit (testCase)
@@ -25,9 +25,8 @@ import Test.Framework.Providers.QuickCheck2 (testProperty)
 import Test.HUnit
 import Test.QuickCheck (Arbitrary(..), Gen, Property, (==>))
 
-import Cardinal
 import Comparisons ((~=), nearly_ge)
-import FunctionValues
+import FunctionValues (FunctionValues(..), empty_values)
 import Misc (factorial)
 import Point
 import RealFunction
@@ -161,151 +160,184 @@ beta t i j k l
 --   Zeilfelder, pp. 84-86. If incorrect indices are supplied, the
 --   function will simply error.
 c :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
-c t 0 0 3 0 = eval (function_values t) $
-              (1/8) * (I + F + L + T + LT + FL + FT + FLT)
-
-c t 0 0 0 3 = eval (function_values t) $
-              (1/8) * (I + F + R + T + RT + FR + FT + FRT)
-
-c t 0 0 2 1 = eval (function_values t) $
-              (5/24)*(I + F + T + FT) +
-              (1/24)*(L + FL + LT + FLT)
-
-c t 0 0 1 2 = eval (function_values t) $
-              (5/24)*(I + F + T + FT) +
-              (1/24)*(R + FR + RT + FRT)
-
-c t 0 1 2 0 = eval (function_values t) $
-              (5/24)*(I + F) +
-              (1/8)*(L + T + FL + FT) +
-              (1/24)*(LT + FLT)
-
-c t 0 1 0 2 = eval (function_values t) $
-              (5/24)*(I + F) +
-              (1/8)*(R + T + FR + FT) +
-              (1/24)*(RT + FRT)
-
-c t 0 1 1 1 = eval (function_values 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 (function_values 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 (function_values 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 (function_values 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 (function_values t) $
-              (1/4)*I +
-              (1/6)*(F + L + T) +
-              (1/12)*(LT + FL + FT)
-
-c t 1 0 0 2 = eval (function_values t) $
-              (1/4)*I +
-              (1/6)*(F + R + T) +
-              (1/12)*(RT + FR + FT)
-
-c t 1 0 1 1 = eval (function_values 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 (function_values 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 (function_values 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 (function_values 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 (function_values 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 (function_values 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 (function_values 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 (function_values 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)
-               [1,  1,  1,  1,
-                x1, x2, x3, x4,
-                y1, y2, y3, y4,
-                z1, z2, z3, z4 ]
-    where
-      (x1, y1, z1) = v0 t
-      (x2, y2, z2) = v1 t
-      (x3, y3, z3) = v2 t
-      (x4, y4, z4) = v3 t
+c t i j k l =
+  coefficient i j k l
+  where
+    fvs = function_values t
+    f = front fvs
+    b = back fvs
+    r = right fvs
+    l' = left fvs
+    t' = top fvs
+    d  = down fvs
+    fl = front_left fvs
+    fr = front_right fvs
+    fd = front_down fvs
+    ft = front_top fvs
+    bl = back_left fvs
+    br = back_right fvs
+    bd = back_down fvs
+    bt = back_top fvs
+    ld = left_down fvs
+    lt = left_top fvs
+    rd = right_down fvs
+    rt = right_top fvs
+    fld = front_left_down fvs
+    flt = front_left_top fvs
+    frd = front_right_down fvs
+    frt = front_right_top fvs
+    i' = interior fvs
+
+    coefficient :: Int -> Int -> Int -> Int -> Double
+    coefficient 0 0 3 0 =
+      (1/8) * (i' + f + l' + t' + lt + fl + ft + flt)
+
+    coefficient 0 0 0 3 =
+      (1/8) * (i' + f + r + t' + rt + fr + ft + frt)
+
+    coefficient 0 0 2 1 =
+      (5/24)*(i' + f + t' + ft) + (1/24)*(l' + fl + lt + flt)
+
+    coefficient 0 0 1 2 =
+      (5/24)*(i' + f + t' + ft) + (1/24)*(r + fr + rt + frt)
+
+    coefficient 0 1 2 0 =
+      (5/24)*(i' + f) + (1/8)*(l' + t' + fl + ft)
+                      + (1/24)*(lt + flt)
+
+    coefficient 0 1 0 2 =
+      (5/24)*(i' + f) + (1/8)*(r + t' + fr + ft)
+                      + (1/24)*(rt + frt)
+
+    coefficient 0 1 1 1 =
+      (13/48)*(i' + f) + (7/48)*(t' + ft)
+                       + (1/32)*(l' + r + fl + fr)
+                       + (1/96)*(lt + rt + flt + frt)
+
+    coefficient 0 2 1 0 =
+      (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)
+
+    coefficient 0 2 0 1 =
+      (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)
+
+    coefficient 0 3 0 0 =
+     (13/48)*(i' + f) + (5/96)*(l' + r + t' + d + fl + fr + ft + fd)
+                      + (1/192)*(rt + rd + lt + ld + frt + frd + flt + fld)
+
+    coefficient 1 0 2 0 =
+      (1/4)*i' + (1/6)*(f + l' + t')
+               + (1/12)*(lt + fl + ft)
+
+    coefficient 1 0 0 2 =
+      (1/4)*i' + (1/6)*(f + r + t')
+               + (1/12)*(rt + fr + ft)
+
+    coefficient 1 0 1 1 =
+      (1/3)*i' + (5/24)*(f + t')
+               + (1/12)*ft
+               + (1/24)*(l' + r)
+               + (1/48)*(lt + rt + fl + fr)
+
+    coefficient 1 1 1 0 =
+      (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)
+
+    coefficient 1 1 0 1 =
+      (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)
+
+    coefficient 1 2 0 0 =
+      (1/3)*i' + (5/24)*f
+               + (7/96)*(l' + r + t' + d)
+               + (1/32)*(fl + fr + ft + fd)
+               + (1/96)*(rt + rd + lt + ld)
+
+    coefficient 2 0 1 0 =
+      (3/8)*i' + (7/48)*(f + t' + l')
+               + (1/48)*(r + d + b + lt + fl + ft)
+               + (1/96)*(rt + bt + fr + fd + ld + bl)
+
+    coefficient 2 0 0 1 =
+      (3/8)*i' + (7/48)*(f + t' + r)
+               + (1/48)*(l' + d + b + rt + fr + ft)
+               + (1/96)*(lt + bt + fl + fd + rd + br)
+
+    coefficient 2 1 0 0 =
+      (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)
+
+    coefficient 3 0 0 0 =
+      (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)
+
+    coefficient _ _ _ _ = error "coefficient index out of bounds"
+
+
+
+-- | Compute the determinant of the 4x4 matrix,
+--
+--   [1]
+--   [x]
+--   [y]
+--   [z]
+--
+--   where [1] = [1, 1, 1, 1],
+--         [x] = [x1,x2,x3,x4],
+--
+--   et cetera.
+--
+--   The termX nonsense is an attempt to prevent Double overflow.
+--   which has been observed to happen with large coordinates.
+--
+det :: Point -> Point -> Point -> Point -> Double
+det p0 p1 p2 p3 =
+  term5 + term6
+  where
+    (x1, y1, z1) = p0
+    (x2, y2, z2) = p1
+    (x3, y3, z3) = p2
+    (x4, y4, z4) = p3
+    term1 = ((x2 - x4)*y1 - (x1 - x4)*y2 + (x1 - x2)*y4)*z3
+    term2 = ((x2 - x3)*y1 - (x1 - x3)*y2 + (x1 - x2)*y3)*z4
+    term3 = ((x3 - x4)*y2 - (x2 - x4)*y3 + (x2 - x3)*y4)*z1
+    term4 = ((x3 - x4)*y1 - (x1 - x4)*y3 + (x1 - x3)*y4)*z2
+    term5 = term1 - term2
+    term6 = term3 - term4
+
 
 -- | Computed using the formula from Lai & Schumaker, Definition 15.4,
 --   page 436.
 volume :: Tetrahedron -> Double
 volume t
-       | (v0 t) == (v1 t) = 0
-       | (v0 t) == (v2 t) = 0
-       | (v0 t) == (v3 t) = 0
-       | (v1 t) == (v2 t) = 0
-       | (v1 t) == (v3 t) = 0
-       | (v2 t) == (v3 t) = 0
-       | otherwise = (1/6)*(det (vol_matrix t))
+       | v0' == v1' = 0
+       | v0' == v2' = 0
+       | v0' == v3' = 0
+       | v1' == v2' = 0
+       | v1' == v3' = 0
+       | v2' == v3' = 0
+       | otherwise = (1/6)*(det v0' v1' v2' v3')
+  where
+    v0' = v0 t
+    v1' = v1 t
+    v2' = v2 t
+    v3' = v3 t
 
 
 -- | The barycentric coordinates of a point with respect to v0.