-value_at :: [[[Double]]] -> Int -> Int -> Int -> Double
-value_at values i j k =
- ((values !! k) !! j) !! i
-
-make_values :: [[[Double]]] -> 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,
- right = value_at values i (j+1) k,
- down = value_at values i j (k-1),
- top = value_at values i j (k+1),
- front_left = value_at values (i-1) (j-1) k,
- front_right = value_at values (i-1) (j+1) k,
- front_down =value_at values (i-1) j (k-1),
- front_top = value_at values (i-1) j (k+1),
- back_left = value_at values (i+1) (j-1) k,
- back_right = value_at values (i+1) (j+1) k,
- back_down = value_at values (i+1) j (k-1),
- 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),
- 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_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),
+
+-- | 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, 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 :: 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,
+ right = value_at values i (j + 1) k,
+ down = value_at values i j (k - 1),
+ top = value_at values i j (k + 1),
+ front_left = value_at values (i - 1) (j - 1) k,
+ front_right = value_at values (i - 1) (j + 1) k,
+ front_down =value_at values (i - 1) j (k - 1),
+ front_top = value_at values (i - 1) j (k + 1),
+ back_left = value_at values (i + 1) (j - 1) k,
+ back_right = value_at values (i + 1) (j + 1) k,
+ back_down = value_at values (i + 1) j (k - 1),
+ 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_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_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),