1 -- | The FunctionValues module contains the 'FunctionValues' type and
2 -- the functions used to manipulate it.
3 module FunctionValues (
10 function_values_properties,
15 import Prelude hiding (LT)
17 import Test.Framework (Test, testGroup)
18 import Test.Framework.Providers.HUnit (testCase)
19 import Test.Framework.Providers.QuickCheck2 (testProperty)
20 import Test.QuickCheck (Arbitrary(..), choose)
22 import Assertions (assertTrue)
23 import Cardinal ( Cardinal(..), cwx, cwy, cwz )
24 import Examples (trilinear)
25 import Values (Values3D, dims, idx)
27 -- | The FunctionValues type represents the value of our function f at
28 -- the 27 points surrounding (and including) the center of a
29 -- cube. Each value of f can be accessed by the name of its
32 FunctionValues { front :: Double,
39 front_right :: Double,
50 front_left_down :: Double,
51 front_left_top :: Double,
52 front_right_down :: Double,
53 front_right_top :: Double,
54 back_left_down :: Double,
55 back_left_top :: Double,
56 back_right_down :: Double,
57 back_right_top :: Double,
62 instance Arbitrary FunctionValues where
64 front' <- choose (min_double, max_double)
65 back' <- choose (min_double, max_double)
66 left' <- choose (min_double, max_double)
67 right' <- choose (min_double, max_double)
68 top' <- choose (min_double, max_double)
69 down' <- choose (min_double, max_double)
70 front_left' <- choose (min_double, max_double)
71 front_right' <- choose (min_double, max_double)
72 front_top' <- choose (min_double, max_double)
73 front_down' <- choose (min_double, max_double)
74 back_left' <- choose (min_double, max_double)
75 back_right' <- choose (min_double, max_double)
76 back_top' <- choose (min_double, max_double)
77 back_down' <- choose (min_double, max_double)
78 left_top' <- choose (min_double, max_double)
79 left_down' <- choose (min_double, max_double)
80 right_top' <- choose (min_double, max_double)
81 right_down' <- choose (min_double, max_double)
82 front_left_top' <- choose (min_double, max_double)
83 front_left_down' <- choose (min_double, max_double)
84 front_right_top' <- choose (min_double, max_double)
85 front_right_down' <- choose (min_double, max_double)
86 back_left_top' <- choose (min_double, max_double)
87 back_left_down' <- choose (min_double, max_double)
88 back_right_top' <- choose (min_double, max_double)
89 back_right_down' <- choose (min_double, max_double)
90 interior' <- choose (min_double, max_double)
92 return empty_values { front = front',
98 front_left = front_left',
99 front_right = front_right',
100 front_top = front_top',
101 front_down = front_down',
102 back_left = back_left',
103 back_right = back_right',
104 back_top = back_top',
105 back_down = back_down',
106 left_top = left_top',
107 left_down = left_down',
108 right_top = right_top',
109 right_down = right_down',
110 front_left_top = front_left_top',
111 front_left_down = front_left_down',
112 front_right_top = front_right_top',
113 front_right_down = front_right_down',
114 back_left_top = back_left_top',
115 back_left_down = back_left_down',
116 back_right_top = back_right_top',
117 back_right_down = back_right_down',
118 interior = interior' }
120 -- | We perform addition with the function values contained in a
121 -- FunctionValues object. If we choose random doubles near the machine
122 -- min/max, we risk overflowing or underflowing the 'Double'. This
123 -- places a reasonably safe limit on the maximum size of our generated
128 -- | See 'max_double'.
130 min_double = (-1) * max_double
133 -- | Return a 'FunctionValues' with a bunch of zeros for data points.
134 empty_values :: FunctionValues
136 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
138 -- | The eval function is where the magic happens for the
139 -- FunctionValues type. Given a 'Cardinal' direction and a
140 -- 'FunctionValues' object, eval will return the value of the
141 -- function f in that 'Cardinal' direction. Note that 'Cardinal' can
142 -- be a composite type; eval is what performs the \"arithmetic\" on
143 -- 'Cardinal' directions.
144 eval :: FunctionValues -> Cardinal -> Double
151 eval f FL = front_left f
152 eval f FR = front_right f
153 eval f FD = front_down f
154 eval f FT = front_top f
155 eval f BL = back_left f
156 eval f BR = back_right f
157 eval f BD = back_down f
158 eval f BT = back_top f
159 eval f LD = left_down f
160 eval f LT = left_top f
161 eval f RD = right_down f
162 eval f RT = right_top f
163 eval f FLD = front_left_down f
164 eval f FLT = front_left_top f
165 eval f FRD = front_right_down f
166 eval f FRT = front_right_top f
167 eval f BLD = back_left_down f
168 eval f BLT = back_left_top f
169 eval f BRD = back_right_down f
170 eval f BRT = back_right_top f
171 eval f I = interior f
172 eval _ (Scalar x) = x
173 eval f (Sum x y) = (eval f x) + (eval f y)
174 eval f (Difference x y) = (eval f x) - (eval f y)
175 eval f (Product x y) = (eval f x) * (eval f y)
176 eval f (Quotient x y) = (eval f x) / (eval f y)
178 -- | Takes a three-dimensional list of 'Double' and a set of 3D
179 -- coordinates (i,j,k), and returns the value at (i,j,k) in the
180 -- supplied list. If there is no such value, we calculate one
181 -- according to Sorokina and Zeilfelder, remark 7.3, p. 99.
183 -- We specifically do not consider values more than one unit away
188 -- >>> value_at Examples.trilinear 0 0 0
191 -- >>> value_at Examples.trilinear (-1) 0 0
194 -- >>> value_at Examples.trilinear 0 0 4
197 -- >>> value_at Examples.trilinear 1 3 0
200 value_at :: Values3D -> Int -> Int -> Int -> Double
202 -- Put the most common case first!
203 | (valid_i i) && (valid_j j) && (valid_k k) =
206 -- The next three are from the first line in (7.3). Analogous cases
207 -- have been added where the indices are one-too-big. These are the
208 -- "one index is bad" cases.
212 2*(value_at v3d 0 j k) - (value_at v3d 1 j k)
214 2*(value_at v3d (i-1) j k) - (value_at v3d (i-2) j k)
219 2*(value_at v3d i 0 k) - (value_at v3d i 1 k)
221 2*(value_at v3d i (j-1) k) - (value_at v3d i (j-2) k)
226 2*(value_at v3d i j 0) - (value_at v3d i j 1)
228 2*(value_at v3d i j (k-1)) - (value_at v3d i j (k-2))
234 coordstr = "(" ++ istr ++ "," ++ jstr ++ "," ++ kstr ++ ")"
236 error $ "value_at called outside of domain: " ++ coordstr
238 (xsize, ysize, zsize) = dims v3d
240 valid_i :: Int -> Bool
241 valid_i i' = (i' >= 0) && (i' < xsize)
243 valid_j :: Int -> Bool
244 valid_j j' = (j' >= 0) && (j' < ysize)
246 valid_k :: Int -> Bool
247 valid_k k' = (k' >= 0) && (k' < zsize)
251 -- | Given a three-dimensional list of 'Double' and a set of 3D
252 -- coordinates (i,j,k), constructs and returns the 'FunctionValues'
253 -- object centered at (i,j,k)
254 make_values :: Values3D -> Int -> Int -> Int -> FunctionValues
255 make_values values i j k =
256 empty_values { front = value_at values (i-1) j k,
257 back = value_at values (i+1) j k,
258 left = value_at values i (j-1) k,
259 right = value_at values i (j+1) k,
260 down = value_at values i j (k-1),
261 top = value_at values i j (k+1),
262 front_left = value_at values (i-1) (j-1) k,
263 front_right = value_at values (i-1) (j+1) k,
264 front_down =value_at values (i-1) j (k-1),
265 front_top = value_at values (i-1) j (k+1),
266 back_left = value_at values (i+1) (j-1) k,
267 back_right = value_at values (i+1) (j+1) k,
268 back_down = value_at values (i+1) j (k-1),
269 back_top = value_at values (i+1) j (k+1),
270 left_down = value_at values i (j-1) (k-1),
271 left_top = value_at values i (j-1) (k+1),
272 right_down = value_at values i (j+1) (k-1),
273 right_top = value_at values i (j+1) (k+1),
274 front_left_down = value_at values (i-1) (j-1) (k-1),
275 front_left_top = value_at values (i-1) (j-1) (k+1),
276 front_right_down = value_at values (i-1) (j+1) (k-1),
277 front_right_top = value_at values (i-1) (j+1) (k+1),
278 back_left_down = value_at values (i+1) (j-1) (k-1),
279 back_left_top = value_at values (i+1) (j-1) (k+1),
280 back_right_down = value_at values (i+1) (j+1) (k-1),
281 back_right_top = value_at values (i+1) (j+1) (k+1),
282 interior = value_at values i j k }
284 -- | Takes a 'FunctionValues' and a function that transforms one
285 -- 'Cardinal' to another (called a rotation). Then it applies the
286 -- rotation to each element of the 'FunctionValues' object, and
287 -- returns the result.
288 rotate :: (Cardinal -> Cardinal) -> FunctionValues -> FunctionValues
290 FunctionValues { front = eval fv (rotation F),
291 back = eval fv (rotation B),
292 left = eval fv (rotation L),
293 right = eval fv (rotation R),
294 down = eval fv (rotation D),
295 top = eval fv (rotation T),
296 front_left = eval fv (rotation FL),
297 front_right = eval fv (rotation FR),
298 front_down = eval fv (rotation FD),
299 front_top = eval fv (rotation FT),
300 back_left = eval fv (rotation BL),
301 back_right = eval fv (rotation BR),
302 back_down = eval fv (rotation BD),
303 back_top = eval fv (rotation BT),
304 left_down = eval fv (rotation LD),
305 left_top = eval fv (rotation LT),
306 right_down = eval fv (rotation RD),
307 right_top = eval fv (rotation RT),
308 front_left_down = eval fv (rotation FLD),
309 front_left_top = eval fv (rotation FLT),
310 front_right_down = eval fv (rotation FRD),
311 front_right_top = eval fv (rotation FRT),
312 back_left_down = eval fv (rotation BLD),
313 back_left_top = eval fv (rotation BLT),
314 back_right_down = eval fv (rotation BRD),
315 back_right_top = eval fv (rotation BRT),
316 interior = interior fv }
320 -- | Ensure that the trilinear values wind up where we think they
322 test_directions :: Assertion
324 assertTrue "all direction functions work" (and equalities)
326 fvs = make_values trilinear 1 1 1
327 equalities = [ interior fvs == 4,
335 front_right fvs == 1,
339 back_right fvs == 11,
346 front_left_down fvs == 1,
347 front_left_top fvs == 1,
348 front_right_down fvs == 1,
349 front_right_top fvs == 1,
350 back_left_down fvs == 3,
351 back_left_top fvs == 3,
352 back_right_down fvs == 7,
353 back_right_top fvs == 15]
356 function_values_tests :: Test.Framework.Test
357 function_values_tests =
358 testGroup "FunctionValues Tests"
359 [ testCase "test directions" test_directions ]
362 prop_x_rotation_doesnt_affect_front :: FunctionValues -> Bool
363 prop_x_rotation_doesnt_affect_front fv0 =
370 prop_x_rotation_doesnt_affect_back :: FunctionValues -> Bool
371 prop_x_rotation_doesnt_affect_back fv0 =
379 prop_y_rotation_doesnt_affect_left :: FunctionValues -> Bool
380 prop_y_rotation_doesnt_affect_left fv0 =
387 prop_y_rotation_doesnt_affect_right :: FunctionValues -> Bool
388 prop_y_rotation_doesnt_affect_right fv0 =
396 prop_z_rotation_doesnt_affect_down :: FunctionValues -> Bool
397 prop_z_rotation_doesnt_affect_down fv0 =
405 prop_z_rotation_doesnt_affect_top :: FunctionValues -> Bool
406 prop_z_rotation_doesnt_affect_top fv0 =
414 function_values_properties :: Test.Framework.Test
415 function_values_properties =
416 let tp = testProperty
418 testGroup "FunctionValues Properties" [
419 tp "x rotation doesn't affect front" prop_x_rotation_doesnt_affect_front,
420 tp "x rotation doesn't affect back" prop_x_rotation_doesnt_affect_back,
421 tp "y rotation doesn't affect left" prop_y_rotation_doesnt_affect_left,
422 tp "y rotation doesn't affect right" prop_y_rotation_doesnt_affect_right,
423 tp "z rotation doesn't affect top" prop_z_rotation_doesnt_affect_top,
424 tp "z rotation doesn't affect down" prop_z_rotation_doesnt_affect_down ]