+{-# LANGUAGE BangPatterns #-}
+
-- | The FunctionValues module contains the 'FunctionValues' type and
-- the functions used to manipulate it.
-module FunctionValues
+--
+module FunctionValues (
+ FunctionValues(..),
+ empty_values,
+ eval,
+ make_values,
+ rotate,
+ function_values_tests,
+ function_values_properties,
+ value_at )
where
-import Prelude hiding (LT)
+import Prelude hiding ( LT )
+import Test.Tasty ( TestTree, testGroup )
+import Test.Tasty.HUnit ( Assertion, testCase )
+import Test.Tasty.QuickCheck ( Arbitrary( arbitrary ), choose, testProperty )
-import Cardinal
+import Assertions ( assertTrue )
+import Cardinal (
+ Cardinal(F, B, L, R, D, T, FL, FR, FD, FT, BL, BR, BD, BT, LD, LT, RD,
+ RT, FLD, FLT, FRD, FRT, BLD, BLT, BRD, BRT, I, Scalar, Sum,
+ Difference, Product, Quotient ),
+ cwx,
+ cwy,
+ cwz )
+import Examples ( trilinear )
+import Values ( Values3D, dims, idx )
-- | The FunctionValues type represents the value of our function f at
-- the 27 points surrounding (and including) the center of a
-- cube. Each value of f can be accessed by the name of its
-- direction.
+--
data FunctionValues =
- FunctionValues { front :: Double,
- back :: Double,
- left :: Double,
- right :: Double,
- top :: Double,
- down :: Double,
- front_left :: Double,
- front_right :: Double,
- front_top :: Double,
- front_down :: Double,
- back_left :: Double,
- back_right :: Double,
- back_top :: Double,
- back_down :: Double,
- left_top :: Double,
- left_down :: Double,
- right_top :: Double,
- right_down :: Double,
- front_left_top :: Double,
- front_left_down :: Double,
- front_right_top :: Double,
- front_right_down :: Double,
- back_left_top :: Double,
- back_left_down :: Double,
- back_right_top :: Double,
- back_right_down :: Double,
- interior :: Double }
+ FunctionValues { front :: !Double,
+ back :: !Double,
+ left :: !Double,
+ right :: !Double,
+ top :: !Double,
+ down :: !Double,
+ front_left :: !Double,
+ front_right :: !Double,
+ front_down :: !Double,
+ front_top :: !Double,
+ back_left :: !Double,
+ back_right :: !Double,
+ back_down :: !Double,
+ back_top :: !Double,
+ left_down :: !Double,
+ left_top :: !Double,
+ right_down :: !Double,
+ right_top :: !Double,
+ front_left_down :: !Double,
+ front_left_top :: !Double,
+ front_right_down :: !Double,
+ front_right_top :: !Double,
+ back_left_down :: !Double,
+ back_left_top :: !Double,
+ back_right_down :: !Double,
+ back_right_top :: !Double,
+ interior :: !Double }
deriving (Eq, Show)
+
+instance Arbitrary FunctionValues where
+ arbitrary = do
+ front' <- choose (min_double, max_double)
+ back' <- choose (min_double, max_double)
+ left' <- choose (min_double, max_double)
+ right' <- choose (min_double, max_double)
+ top' <- choose (min_double, max_double)
+ down' <- choose (min_double, max_double)
+ front_left' <- choose (min_double, max_double)
+ front_right' <- choose (min_double, max_double)
+ front_top' <- choose (min_double, max_double)
+ front_down' <- choose (min_double, max_double)
+ back_left' <- choose (min_double, max_double)
+ back_right' <- choose (min_double, max_double)
+ back_top' <- choose (min_double, max_double)
+ back_down' <- choose (min_double, max_double)
+ left_top' <- choose (min_double, max_double)
+ left_down' <- choose (min_double, max_double)
+ right_top' <- choose (min_double, max_double)
+ right_down' <- choose (min_double, max_double)
+ front_left_top' <- choose (min_double, max_double)
+ front_left_down' <- choose (min_double, max_double)
+ front_right_top' <- choose (min_double, max_double)
+ front_right_down' <- choose (min_double, max_double)
+ back_left_top' <- choose (min_double, max_double)
+ back_left_down' <- choose (min_double, max_double)
+ back_right_top' <- choose (min_double, max_double)
+ back_right_down' <- choose (min_double, max_double)
+ interior' <- choose (min_double, max_double)
+
+ return empty_values { front = front',
+ back = back',
+ left = left',
+ right = right',
+ top = top',
+ down = down',
+ front_left = front_left',
+ front_right = front_right',
+ front_top = front_top',
+ front_down = front_down',
+ back_left = back_left',
+ back_right = back_right',
+ back_top = back_top',
+ back_down = back_down',
+ left_top = left_top',
+ left_down = left_down',
+ right_top = right_top',
+ right_down = right_down',
+ front_left_top = front_left_top',
+ front_left_down = front_left_down',
+ front_right_top = front_right_top',
+ front_right_down = front_right_down',
+ back_left_top = back_left_top',
+ back_left_down = back_left_down',
+ back_right_top = back_right_top',
+ back_right_down = back_right_down',
+ interior = interior' }
+ where
+ -- | We perform addition with the function values contained in a
+ -- FunctionValues object. If we choose random doubles near the machine
+ -- min/max, we risk overflowing or underflowing the 'Double'. This
+ -- places a reasonably safe limit on the maximum size of our generated
+ -- 'Double' members.
+ max_double :: Double
+ max_double = 10000.0
+
+ -- | See 'max_double'.
+ min_double :: Double
+ min_double = (-1) * max_double
+
+
-- | Return a 'FunctionValues' with a bunch of zeros for data points.
empty_values :: FunctionValues
empty_values =
FunctionValues 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
+
-- | The eval function is where the magic happens for the
-- FunctionValues type. Given a 'Cardinal' direction and a
-- 'FunctionValues' object, eval will return the value of the
eval f (Product x y) = (eval f x) * (eval f y)
eval f (Quotient x y) = (eval f x) / (eval f y)
+
-- | Takes a three-dimensional list of 'Double' and a set of 3D
-- coordinates (i,j,k), and returns the value at (i,j,k) in the
--- supplied list. If there is no such value, zero is returned.
-value_at :: [[[Double]]] -> Int -> Int -> Int -> Double
-value_at values i j k
- | i < 0 = 0
- | j < 0 = 0
- | k < 0 = 0
- | length values <= k = 0
- | length (values !! k) <= j = 0
- | length ((values !! k) !! j) <= i = 0
- | otherwise = ((values !! k) !! j) !! i
+-- supplied list. If there is no such value, we calculate one
+-- according to Sorokina and Zeilfelder, remark 7.3, p. 99.
+--
+-- We specifically do not consider values more than one unit away
+-- from our grid.
+--
+-- Examples:
+--
+-- >>> value_at Examples.trilinear 0 0 0
+-- 1.0
+--
+-- >>> value_at Examples.trilinear (-1) 0 0
+-- 0.0
+--
+-- >>> value_at Examples.trilinear 0 0 4
+-- 1.0
+--
+-- >>> value_at Examples.trilinear 1 3 0
+-- 5.0
+--
+value_at :: Values3D -> Int -> Int -> Int -> Double
+value_at v3d !i !j !k
+ -- Put the most common case first!
+ | (valid_i i) && (valid_j j) && (valid_k k) =
+ idx v3d i j k
+
+ -- The next three are from the first line in (7.3). Analogous cases
+ -- have been added where the indices are one-too-big. These are the
+ -- "one index is bad" cases.
+ | not (valid_i i) =
+ if (dim_i == 1)
+ then
+ -- We're one-dimensional in our first coordinate, so just
+ -- return the data point that we do have. If we try to use
+ -- the formula from remark 7.3, we go into an infinite loop.
+ value_at v3d 0 j k
+ else
+ if (i == -1)
+ then
+ 2*(value_at v3d 0 j k) - (value_at v3d 1 j k)
+ else
+ 2*(value_at v3d (i-1) j k) - (value_at v3d (i-2) j k)
+
+ | not (valid_j j) =
+ if (dim_j == 1)
+ then
+ -- We're one-dimensional in our second coordinate, so just
+ -- return the data point that we do have. If we try to use
+ -- the formula from remark 7.3, we go into an infinite loop.
+ value_at v3d i 0 k
+ else
+ if (j == -1)
+ then
+ 2*(value_at v3d i 0 k) - (value_at v3d i 1 k)
+ else
+ 2*(value_at v3d i (j-1) k) - (value_at v3d i (j-2) k)
+
+ | not (valid_k k) =
+ if (dim_k == 1)
+ then
+ -- We're one-dimensional in our third coordinate, so just
+ -- return the data point that we do have. If we try to use
+ -- the formula from remark 7.3, we go into an infinite loop.
+ value_at v3d i j 0
+ else
+ if (k == -1)
+ then
+ 2*(value_at v3d i j 0) - (value_at v3d i j 1)
+ else
+ 2*(value_at v3d i j (k-1)) - (value_at v3d i j (k-2))
+ where
+ (dim_i, dim_j, dim_k) = dims v3d
+
+ valid_i :: Int -> Bool
+ valid_i i' = (i' >= 0) && (i' < dim_i)
+
+ valid_j :: Int -> Bool
+ valid_j j' = (j' >= 0) && (j' < dim_j)
+
+ valid_k :: Int -> Bool
+ valid_k k' = (k' >= 0) && (k' < dim_k)
+
-- | Given a three-dimensional list of 'Double' and a set of 3D
-- coordinates (i,j,k), constructs and returns the 'FunctionValues'
-- object centered at (i,j,k)
-make_values :: [[[Double]]] -> Int -> Int -> Int -> FunctionValues
-make_values values i j k =
+make_values :: Values3D -> Int -> Int -> Int -> FunctionValues
+make_values values !i !j !k =
empty_values { front = value_at values (i-1) j k,
back = value_at values (i+1) j k,
left = value_at values i (j-1) k,
back_top = value_at values (i+1) j (k+1),
left_down = value_at values i (j-1) (k-1),
left_top = value_at values i (j-1) (k+1),
- right_top = value_at values i (j+1) (k+1),
right_down = value_at values i (j+1) (k-1),
+ right_top = value_at values i (j+1) (k+1),
front_left_down = value_at values (i-1) (j-1) (k-1),
front_left_top = value_at values (i-1) (j-1) (k+1),
front_right_down = value_at values (i-1) (j+1) (k-1),
front_right_top = value_at values (i-1) (j+1) (k+1),
- back_left_down = value_at values (i-1) (j-1) (k-1),
+ back_left_down = value_at values (i+1) (j-1) (k-1),
back_left_top = value_at values (i+1) (j-1) (k+1),
back_right_down = value_at values (i+1) (j+1) (k-1),
back_right_top = value_at values (i+1) (j+1) (k+1),
-- 'Cardinal' to another (called a rotation). Then it applies the
-- rotation to each element of the 'FunctionValues' object, and
-- returns the result.
-rotate :: FunctionValues -> (Cardinal -> Cardinal) -> FunctionValues
-rotate fv rotation =
+rotate :: (Cardinal -> Cardinal) -> FunctionValues -> FunctionValues
+rotate rotation fv =
FunctionValues { front = eval fv (rotation F),
back = eval fv (rotation B),
left = eval fv (rotation L),
back_right_down = eval fv (rotation BRD),
back_right_top = eval fv (rotation BRT),
interior = interior fv }
+
+
+
+-- | Ensure that the trilinear values wind up where we think they
+-- should.
+test_directions :: Assertion
+test_directions =
+ assertTrue "all direction functions work" (and equalities)
+ where
+ fvs = make_values trilinear 1 1 1
+ equalities = [ interior fvs == 4,
+ front fvs == 1,
+ back fvs == 7,
+ left fvs == 2,
+ right fvs == 6,
+ down fvs == 3,
+ top fvs == 5,
+ front_left fvs == 1,
+ front_right fvs == 1,
+ front_down fvs == 1,
+ front_top fvs == 1,
+ back_left fvs == 3,
+ back_right fvs == 11,
+ back_down fvs == 5,
+ back_top fvs == 9,
+ left_down fvs == 2,
+ left_top fvs == 2,
+ right_down fvs == 4,
+ right_top fvs == 8,
+ front_left_down fvs == 1,
+ front_left_top fvs == 1,
+ front_right_down fvs == 1,
+ front_right_top fvs == 1,
+ back_left_down fvs == 3,
+ back_left_top fvs == 3,
+ back_right_down fvs == 7,
+ back_right_top fvs == 15]
+
+
+function_values_tests :: TestTree
+function_values_tests =
+ testGroup "FunctionValues tests"
+ [ testCase "test directions" test_directions ]
+
+
+prop_x_rotation_doesnt_affect_front :: FunctionValues -> Bool
+prop_x_rotation_doesnt_affect_front fv0 =
+ expr1 == expr2
+ where
+ fv1 = rotate cwx fv0
+ expr1 = front fv0
+ expr2 = front fv1
+
+prop_x_rotation_doesnt_affect_back :: FunctionValues -> Bool
+prop_x_rotation_doesnt_affect_back fv0 =
+ expr1 == expr2
+ where
+ fv1 = rotate cwx fv0
+ expr1 = back fv0
+ expr2 = back fv1
+
+
+prop_y_rotation_doesnt_affect_left :: FunctionValues -> Bool
+prop_y_rotation_doesnt_affect_left fv0 =
+ expr1 == expr2
+ where
+ fv1 = rotate cwy fv0
+ expr1 = left fv0
+ expr2 = left fv1
+
+prop_y_rotation_doesnt_affect_right :: FunctionValues -> Bool
+prop_y_rotation_doesnt_affect_right fv0 =
+ expr1 == expr2
+ where
+ fv1 = rotate cwy fv0
+ expr1 = right fv0
+ expr2 = right fv1
+
+
+prop_z_rotation_doesnt_affect_down :: FunctionValues -> Bool
+prop_z_rotation_doesnt_affect_down fv0 =
+ expr1 == expr2
+ where
+ fv1 = rotate cwz fv0
+ expr1 = down fv0
+ expr2 = down fv1
+
+
+prop_z_rotation_doesnt_affect_top :: FunctionValues -> Bool
+prop_z_rotation_doesnt_affect_top fv0 =
+ expr1 == expr2
+ where
+ fv1 = rotate cwz fv0
+ expr1 = top fv0
+ expr2 = top fv1
+
+
+function_values_properties :: TestTree
+function_values_properties =
+ testGroup "FunctionValues properties" [
+ testProperty
+ "x rotation doesn't affect front"
+ prop_x_rotation_doesnt_affect_front,
+ testProperty
+ "x rotation doesn't affect back"
+ prop_x_rotation_doesnt_affect_back,
+ testProperty
+ "y rotation doesn't affect left"
+ prop_y_rotation_doesnt_affect_left,
+ testProperty
+ "y rotation doesn't affect right"
+ prop_y_rotation_doesnt_affect_right,
+ testProperty
+ "z rotation doesn't affect top"
+ prop_z_rotation_doesnt_affect_top,
+ testProperty
+ "z rotation doesn't affect down"
+ prop_z_rotation_doesnt_affect_down ]
projects / spline3.git / blobdiff