sum
)
-import Prelude hiding (LT)
import Test.Framework (Test, testGroup)
import Test.Framework.Providers.HUnit (testCase)
import Test.Framework.Providers.QuickCheck2 (testProperty)
-import Test.HUnit
+import Test.HUnit (Assertion, assertEqual)
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
-import ThreeDimensional
+import Point (Point, scale)
+import RealFunction (RealFunction, cmult, fexp)
+import ThreeDimensional (ThreeDimensional(..))
data Tetrahedron =
Tetrahedron { function_values :: FunctionValues,
((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.
--- Simply an alias for the domain_point function.
-xi :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
-xi = domain_point
-
--- | Returns the domain point of t with indices i,j,k,l.
-domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
-domain_point t i j k l
- | i + j + k + l == 3 = weighted_sum `scale` (1/3)
- | otherwise = error "domain point index out of bounds"
- where
- v0' = (v0 t) `scale` (fromIntegral i)
- v1' = (v1 t) `scale` (fromIntegral j)
- v2' = (v2 t) `scale` (fromIntegral k)
- v3' = (v3 t) `scale` (fromIntegral l)
- weighted_sum = v0' + v1' + v2' + v3'
-
-- | The Bernstein polynomial on t with indices i,j,k,l. Denoted by a
-- capital 'B' in the Sorokina/Zeilfelder paper.
-- 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"
-
-
-
+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 =
--- Both of these results are just copy/pasted from Sage. One of them
--- might be more numerically stable, faster, or both.
---
--- x1*y2*z4 - x1*y2*z3 + x1*y3*z2 - x1*y3*z4 - x1*y4*z2 + x1*y4*z3 +
--- x2*y1*z3 - x2*y1*z4 - x2*y3*z1 + x2*y3*z4 +
--- x2*y4*z1 - x2*y4*z3 - x3*y1*z2 + x3*y1*z4 + x3*y2*z1 - x3*y2*z4 - x3*y4*z1 +
--- x3*y4*z2 + x4*y1*z2 - x4*y1*z3 - x4*y2*z1 + x4*y2*z3 + x4*y3*z1 - x4*y3*z2
- -((x2 - x3)*y1 - (x1 - x3)*y2 + (x1 - x2)*y3)*z4 + ((x2 - x4)*y1 - (x1 - x4)*y2 + (x1 - x2)*y4)*z3 + ((x3 - x4)*y2 - (x2 - x4)*y3 + (x2 - x3)*y4)*z1 - ((x3 - x4)*y1 - (x1 - x4)*y3 + (x1 - x3)*y4)*z2
+ 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,
(volume t) > 0 ==> (b3 t) (v2 t) ~= 0
--- | Used for convenience in the next few tests; not a test itself.
-p :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
-p t i j k l = (polynomial t) (xi t i j k l)
-
--- | Given in Sorokina and Zeilfelder, p. 78.
-prop_c3000_identity :: Tetrahedron -> Property
-prop_c3000_identity t =
- (volume t) > 0 ==>
- c t 3 0 0 0 ~= p t 3 0 0 0
-
--- | Given in Sorokina and Zeilfelder, p. 78.
-prop_c2100_identity :: Tetrahedron -> Property
-prop_c2100_identity t =
- (volume t) > 0 ==>
- c t 2 1 0 0 ~= (term1 - term2 + term3 - term4)
- where
- term1 = (1/3)*(p t 0 3 0 0)
- term2 = (5/6)*(p t 3 0 0 0)
- term3 = 3*(p t 2 1 0 0)
- term4 = (3/2)*(p t 1 2 0 0)
-
--- | Given in Sorokina and Zeilfelder, p. 78.
-prop_c1110_identity :: Tetrahedron -> Property
-prop_c1110_identity t =
- (volume t) > 0 ==>
- c t 1 1 1 0 ~= (term1 + term2 - term3 - term4)
- where
- term1 = (1/3)*((p t 3 0 0 0) + (p t 0 3 0 0) + (p t 0 0 3 0))
- term2 = (9/2)*(p t 1 1 1 0)
- term3 = (3/4)*((p t 2 1 0 0) + (p t 1 2 0 0) + (p t 2 0 1 0))
- term4 = (3/4)*((p t 1 0 2 0) + (p t 0 2 1 0) + (p t 0 1 2 0))
-
-
prop_swapping_vertices_doesnt_affect_coefficients1 :: Tetrahedron -> Bool
prop_swapping_vertices_doesnt_affect_coefficients1 t =
c t 0 0 1 2 == c t' 0 0 1 2
p78_24_properties :: Test.Framework.Test
p78_24_properties =
- testGroup "p. 78, Section (2.4) Properties" [
- testProperty "c3000 identity" prop_c3000_identity,
- testProperty "c2100 identity" prop_c2100_identity,
- testProperty "c1110 identity" prop_c1110_identity]
+ testGroup "p. 78, Section (2.4) Properties" [
+ testProperty "c3000 identity" prop_c3000_identity,
+ testProperty "c2100 identity" prop_c2100_identity,
+ testProperty "c1110 identity" prop_c1110_identity]
+ where
+ -- | Returns the domain point of t with indices i,j,k,l.
+ domain_point :: Tetrahedron -> Int -> Int -> Int -> Int -> Point
+ domain_point t i j k l
+ | i + j + k + l == 3 = weighted_sum `scale` (1/3)
+ | otherwise = error "domain point index out of bounds"
+ where
+ v0' = (v0 t) `scale` (fromIntegral i)
+ v1' = (v1 t) `scale` (fromIntegral j)
+ v2' = (v2 t) `scale` (fromIntegral k)
+ v3' = (v3 t) `scale` (fromIntegral l)
+ weighted_sum = v0' + v1' + v2' + v3'
+
+
+ -- | Used for convenience in the next few tests.
+ p :: Tetrahedron -> Int -> Int -> Int -> Int -> Double
+ p t i j k l = (polynomial t) (domain_point t i j k l)
+
+
+ -- | Given in Sorokina and Zeilfelder, p. 78.
+ prop_c3000_identity :: Tetrahedron -> Property
+ prop_c3000_identity t =
+ (volume t) > 0 ==>
+ c t 3 0 0 0 ~= p t 3 0 0 0
+
+ -- | Given in Sorokina and Zeilfelder, p. 78.
+ prop_c2100_identity :: Tetrahedron -> Property
+ prop_c2100_identity t =
+ (volume t) > 0 ==>
+ c t 2 1 0 0 ~= (term1 - term2 + term3 - term4)
+ where
+ term1 = (1/3)*(p t 0 3 0 0)
+ term2 = (5/6)*(p t 3 0 0 0)
+ term3 = 3*(p t 2 1 0 0)
+ term4 = (3/2)*(p t 1 2 0 0)
+
+ -- | Given in Sorokina and Zeilfelder, p. 78.
+ prop_c1110_identity :: Tetrahedron -> Property
+ prop_c1110_identity t =
+ (volume t) > 0 ==>
+ c t 1 1 1 0 ~= (term1 + term2 - term3 - term4)
+ where
+ term1 = (1/3)*((p t 3 0 0 0) + (p t 0 3 0 0) + (p t 0 0 3 0))
+ term2 = (9/2)*(p t 1 1 1 0)
+ term3 = (3/4)*((p t 2 1 0 0) + (p t 1 2 0 0) + (p t 2 0 1 0))
+ term4 = (3/4)*((p t 1 0 2 0) + (p t 0 2 1 0) + (p t 0 1 2 0))
+
tetrahedron_properties :: Test.Framework.Test
projects / spline3.git / blobdiff