snoc,
sum
)
-import Numeric.LinearAlgebra hiding (i, scale)
+
import Prelude hiding (LT)
import Test.Framework (Test, testGroup)
import Test.Framework.Providers.HUnit (testCase)
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
-- 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.
projects / spline3.git / blobdiff