4 find_containing_tetrahedron,
10 import Data.Maybe (fromJust)
11 import qualified Data.Vector as V (
20 import Prelude hiding (LT)
21 import Test.Framework (Test, testGroup)
22 import Test.Framework.Providers.QuickCheck2 (testProperty)
23 import Test.QuickCheck (Arbitrary(..), Gen, Positive(..), choose)
26 import Comparisons ((~=), (~~=))
27 import qualified Face (Face(Face, v0, v1, v2, v3))
28 import FunctionValues (FunctionValues, eval, rotate)
29 import Misc (all_equal, disjoint)
30 import Point (Point(..), dot)
31 import Tetrahedron (Tetrahedron(..), c, volume)
32 import ThreeDimensional
34 data Cube = Cube { h :: !Double,
38 fv :: !FunctionValues,
39 tetrahedra_volume :: !Double }
43 instance Arbitrary Cube where
45 (Positive h') <- arbitrary :: Gen (Positive Double)
46 i' <- choose (coordmin, coordmax)
47 j' <- choose (coordmin, coordmax)
48 k' <- choose (coordmin, coordmax)
49 fv' <- arbitrary :: Gen FunctionValues
50 (Positive tet_vol) <- arbitrary :: Gen (Positive Double)
51 return (Cube h' i' j' k' fv' tet_vol)
53 -- The idea here is that, when cubed in the volume formula,
54 -- these numbers don't overflow 64 bits. This number is not
55 -- magic in any other sense than that it does not cause test
56 -- failures, while 2^23 does.
57 coordmax = 4194304 -- 2^22
61 instance Show Cube where
63 "Cube_" ++ subscript ++ "\n" ++
64 " h: " ++ (show (h cube)) ++ "\n" ++
65 " Center: " ++ (show (center cube)) ++ "\n" ++
66 " xmin: " ++ (show (xmin cube)) ++ "\n" ++
67 " xmax: " ++ (show (xmax cube)) ++ "\n" ++
68 " ymin: " ++ (show (ymin cube)) ++ "\n" ++
69 " ymax: " ++ (show (ymax cube)) ++ "\n" ++
70 " zmin: " ++ (show (zmin cube)) ++ "\n" ++
71 " zmax: " ++ (show (zmax cube)) ++ "\n"
74 (show (i cube)) ++ "," ++ (show (j cube)) ++ "," ++ (show (k cube))
77 -- | The left-side boundary of the cube. See Sorokina and Zeilfelder,
79 xmin :: Cube -> Double
80 xmin cube = (i' - 1/2)*delta
82 i' = fromIntegral (i cube) :: Double
85 -- | The right-side boundary of the cube. See Sorokina and Zeilfelder,
87 xmax :: Cube -> Double
88 xmax cube = (i' + 1/2)*delta
90 i' = fromIntegral (i cube) :: Double
93 -- | The front boundary of the cube. See Sorokina and Zeilfelder,
95 ymin :: Cube -> Double
96 ymin cube = (j' - 1/2)*delta
98 j' = fromIntegral (j cube) :: Double
101 -- | The back boundary of the cube. See Sorokina and Zeilfelder,
103 ymax :: Cube -> Double
104 ymax cube = (j' + 1/2)*delta
106 j' = fromIntegral (j cube) :: Double
109 -- | The bottom boundary of the cube. See Sorokina and Zeilfelder,
111 zmin :: Cube -> Double
112 zmin cube = (k' - 1/2)*delta
114 k' = fromIntegral (k cube) :: Double
117 -- | The top boundary of the cube. See Sorokina and Zeilfelder,
119 zmax :: Cube -> Double
120 zmax cube = (k' + 1/2)*delta
122 k' = fromIntegral (k cube) :: Double
125 instance ThreeDimensional Cube where
126 -- | The center of Cube_ijk coincides with v_ijk at
127 -- (ih, jh, kh). See Sorokina and Zeilfelder, p. 76.
128 center cube = Point x y z
131 i' = fromIntegral (i cube) :: Double
132 j' = fromIntegral (j cube) :: Double
133 k' = fromIntegral (k cube) :: Double
138 -- | It's easy to tell if a point is within a cube; just make sure
139 -- that it falls on the proper side of each of the cube's faces.
140 contains_point cube (Point x y z)
141 | x < (xmin cube) = False
142 | x > (xmax cube) = False
143 | y < (ymin cube) = False
144 | y > (ymax cube) = False
145 | z < (zmin cube) = False
146 | z > (zmax cube) = False
153 -- | The top (in the direction of z) face of the cube.
154 top_face :: Cube -> Face.Face
155 top_face cube = Face.Face v0' v1' v2' v3'
157 delta = (1/2)*(h cube)
159 v0' = cc + ( Point delta (-delta) delta )
160 v1' = cc + ( Point delta delta delta )
161 v2' = cc + ( Point (-delta) delta delta )
162 v3' = cc + ( Point (-delta) (-delta) delta )
166 -- | The back (in the direction of x) face of the cube.
167 back_face :: Cube -> Face.Face
168 back_face cube = Face.Face v0' v1' v2' v3'
170 delta = (1/2)*(h cube)
172 v0' = cc + ( Point delta (-delta) (-delta) )
173 v1' = cc + ( Point delta delta (-delta) )
174 v2' = cc + ( Point delta delta delta )
175 v3' = cc + ( Point delta (-delta) delta )
178 -- The bottom face (in the direction of -z) of the cube.
179 down_face :: Cube -> Face.Face
180 down_face cube = Face.Face v0' v1' v2' v3'
182 delta = (1/2)*(h cube)
184 v0' = cc + ( Point (-delta) (-delta) (-delta) )
185 v1' = cc + ( Point (-delta) delta (-delta) )
186 v2' = cc + ( Point delta delta (-delta) )
187 v3' = cc + ( Point delta (-delta) (-delta) )
191 -- | The front (in the direction of -x) face of the cube.
192 front_face :: Cube -> Face.Face
193 front_face cube = Face.Face v0' v1' v2' v3'
195 delta = (1/2)*(h cube)
197 v0' = cc + ( Point (-delta) (-delta) delta )
198 v1' = cc + ( Point (-delta) delta delta )
199 v2' = cc + ( Point (-delta) delta (-delta) )
200 v3' = cc + ( Point (-delta) (-delta) (-delta) )
202 -- | The left (in the direction of -y) face of the cube.
203 left_face :: Cube -> Face.Face
204 left_face cube = Face.Face v0' v1' v2' v3'
206 delta = (1/2)*(h cube)
208 v0' = cc + ( Point delta (-delta) delta )
209 v1' = cc + ( Point (-delta) (-delta) delta )
210 v2' = cc + ( Point (-delta) (-delta) (-delta) )
211 v3' = cc + ( Point delta (-delta) (-delta) )
214 -- | The right (in the direction of y) face of the cube.
215 right_face :: Cube -> Face.Face
216 right_face cube = Face.Face v0' v1' v2' v3'
218 delta = (1/2)*(h cube)
220 v0' = cc + ( Point (-delta) delta delta)
221 v1' = cc + ( Point delta delta delta )
222 v2' = cc + ( Point delta delta (-delta) )
223 v3' = cc + ( Point (-delta) delta (-delta) )
226 tetrahedron :: Cube -> Int -> Tetrahedron
229 Tetrahedron (fv cube) v0' v1' v2' v3' vol
236 vol = tetrahedra_volume cube
239 Tetrahedron fv' v0' v1' v2' v3' vol
246 fv' = rotate ccwx (fv cube)
247 vol = tetrahedra_volume cube
250 Tetrahedron fv' v0' v1' v2' v3' vol
257 fv' = rotate ccwx $ rotate ccwx $ fv cube
258 vol = tetrahedra_volume cube
261 Tetrahedron fv' v0' v1' v2' v3' vol
268 fv' = rotate cwx (fv cube)
269 vol = tetrahedra_volume cube
272 Tetrahedron fv' v0' v1' v2' v3' vol
279 fv' = rotate cwy (fv cube)
280 vol = tetrahedra_volume cube
283 Tetrahedron fv' v0' v1' v2' v3' vol
290 fv' = rotate cwy $ rotate cwz $ fv cube
291 vol = tetrahedra_volume cube
294 Tetrahedron fv' v0' v1' v2' v3' vol
301 fv' = rotate cwy $ rotate cwz
304 vol = tetrahedra_volume cube
307 Tetrahedron fv' v0' v1' v2' v3' vol
314 fv' = rotate cwy $ rotate ccwz $ fv cube
315 vol = tetrahedra_volume cube
318 Tetrahedron fv' v0' v1' v2' v3' vol
325 fv' = rotate cwy $ rotate cwy $ fv cube
326 vol = tetrahedra_volume cube
329 Tetrahedron fv' v0' v1' v2' v3' vol
336 fv' = rotate cwy $ rotate cwy
339 vol = tetrahedra_volume cube
341 tetrahedron cube 10 =
342 Tetrahedron fv' v0' v1' v2' v3' vol
349 fv' = rotate cwy $ rotate cwy
354 vol = tetrahedra_volume cube
356 tetrahedron cube 11 =
357 Tetrahedron fv' v0' v1' v2' v3' vol
364 fv' = rotate cwy $ rotate cwy
367 vol = tetrahedra_volume cube
369 tetrahedron cube 12 =
370 Tetrahedron fv' v0' v1' v2' v3' vol
377 fv' = rotate ccwy $ fv cube
378 vol = tetrahedra_volume cube
380 tetrahedron cube 13 =
381 Tetrahedron fv' v0' v1' v2' v3' vol
388 fv' = rotate ccwy $ rotate ccwz $ fv cube
389 vol = tetrahedra_volume cube
391 tetrahedron cube 14 =
392 Tetrahedron fv' v0' v1' v2' v3' vol
399 fv' = rotate ccwy $ rotate ccwz
402 vol = tetrahedra_volume cube
404 tetrahedron cube 15 =
405 Tetrahedron fv' v0' v1' v2' v3' vol
412 fv' = rotate ccwy $ rotate cwz $ fv cube
413 vol = tetrahedra_volume cube
415 tetrahedron cube 16 =
416 Tetrahedron fv' v0' v1' v2' v3' vol
423 fv' = rotate ccwz $ fv cube
424 vol = tetrahedra_volume cube
426 tetrahedron cube 17 =
427 Tetrahedron fv' v0' v1' v2' v3' vol
434 fv' = rotate ccwz $ rotate cwy $ fv cube
435 vol = tetrahedra_volume cube
437 tetrahedron cube 18 =
438 Tetrahedron fv' v0' v1' v2' v3' vol
445 fv' = rotate ccwz $ rotate cwy
448 vol = tetrahedra_volume cube
450 tetrahedron cube 19 =
451 Tetrahedron fv' v0' v1' v2' v3' vol
458 fv' = rotate ccwz $ rotate ccwy
460 vol = tetrahedra_volume cube
462 tetrahedron cube 20 =
463 Tetrahedron fv' v0' v1' v2' v3' vol
470 fv' = rotate cwz $ fv cube
471 vol = tetrahedra_volume cube
473 tetrahedron cube 21 =
474 Tetrahedron fv' v0' v1' v2' v3' vol
481 fv' = rotate cwz $ rotate ccwy $ fv cube
482 vol = tetrahedra_volume cube
484 tetrahedron cube 22 =
485 Tetrahedron fv' v0' v1' v2' v3' vol
492 fv' = rotate cwz $ rotate ccwy
495 vol = tetrahedra_volume cube
497 tetrahedron cube 23 =
498 Tetrahedron fv' v0' v1' v2' v3' vol
505 fv' = rotate cwz $ rotate cwy
507 vol = tetrahedra_volume cube
509 -- Feels dirty, but whatever.
510 tetrahedron _ _ = error "asked for a nonexistent tetrahedron"
513 -- Only used in tests, so we don't need the added speed
515 tetrahedra :: Cube -> [Tetrahedron]
516 tetrahedra cube = [ tetrahedron cube n | n <- [0..23] ]
518 front_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
519 front_left_top_tetrahedra cube =
520 V.singleton (tetrahedron cube 0) `V.snoc`
521 (tetrahedron cube 3) `V.snoc`
522 (tetrahedron cube 6) `V.snoc`
523 (tetrahedron cube 7) `V.snoc`
524 (tetrahedron cube 20) `V.snoc`
525 (tetrahedron cube 21)
527 front_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
528 front_left_down_tetrahedra cube =
529 V.singleton (tetrahedron cube 0) `V.snoc`
530 (tetrahedron cube 2) `V.snoc`
531 (tetrahedron cube 3) `V.snoc`
532 (tetrahedron cube 12) `V.snoc`
533 (tetrahedron cube 15) `V.snoc`
534 (tetrahedron cube 21)
536 front_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
537 front_right_top_tetrahedra cube =
538 V.singleton (tetrahedron cube 0) `V.snoc`
539 (tetrahedron cube 1) `V.snoc`
540 (tetrahedron cube 5) `V.snoc`
541 (tetrahedron cube 6) `V.snoc`
542 (tetrahedron cube 16) `V.snoc`
543 (tetrahedron cube 19)
545 front_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
546 front_right_down_tetrahedra cube =
547 V.singleton (tetrahedron cube 1) `V.snoc`
548 (tetrahedron cube 2) `V.snoc`
549 (tetrahedron cube 12) `V.snoc`
550 (tetrahedron cube 13) `V.snoc`
551 (tetrahedron cube 18) `V.snoc`
552 (tetrahedron cube 19)
554 back_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
555 back_left_top_tetrahedra cube =
556 V.singleton (tetrahedron cube 0) `V.snoc`
557 (tetrahedron cube 3) `V.snoc`
558 (tetrahedron cube 6) `V.snoc`
559 (tetrahedron cube 7) `V.snoc`
560 (tetrahedron cube 20) `V.snoc`
561 (tetrahedron cube 21)
563 back_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
564 back_left_down_tetrahedra cube =
565 V.singleton (tetrahedron cube 8) `V.snoc`
566 (tetrahedron cube 11) `V.snoc`
567 (tetrahedron cube 14) `V.snoc`
568 (tetrahedron cube 15) `V.snoc`
569 (tetrahedron cube 22) `V.snoc`
570 (tetrahedron cube 23)
572 back_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
573 back_right_top_tetrahedra cube =
574 V.singleton (tetrahedron cube 4) `V.snoc`
575 (tetrahedron cube 5) `V.snoc`
576 (tetrahedron cube 9) `V.snoc`
577 (tetrahedron cube 10) `V.snoc`
578 (tetrahedron cube 16) `V.snoc`
579 (tetrahedron cube 17)
581 back_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
582 back_right_down_tetrahedra cube =
583 V.singleton (tetrahedron cube 8) `V.snoc`
584 (tetrahedron cube 9) `V.snoc`
585 (tetrahedron cube 13) `V.snoc`
586 (tetrahedron cube 14) `V.snoc`
587 (tetrahedron cube 17) `V.snoc`
588 (tetrahedron cube 18)
590 in_top_half :: Cube -> Point -> Bool
591 in_top_half cube (Point _ _ z) =
592 distance_from_top <= distance_from_bottom
594 distance_from_top = abs $ (zmax cube) - z
595 distance_from_bottom = abs $ (zmin cube) - z
597 in_front_half :: Cube -> Point -> Bool
598 in_front_half cube (Point x _ _) =
599 distance_from_front <= distance_from_back
601 distance_from_front = abs $ (xmin cube) - x
602 distance_from_back = abs $ (xmax cube) - x
605 in_left_half :: Cube -> Point -> Bool
606 in_left_half cube (Point _ y _) =
607 distance_from_left <= distance_from_right
609 distance_from_left = abs $ (ymin cube) - y
610 distance_from_right = abs $ (ymax cube) - y
613 -- | Takes a 'Cube', and returns the Tetrahedra belonging to it that
614 -- contain the given 'Point'. This should be faster than checking
615 -- every tetrahedron individually, since we determine which half
616 -- (hemisphere?) of the cube the point lies in three times: once in
617 -- each dimension. This allows us to eliminate non-candidates
620 -- This can throw an exception, but the use of 'head' might
621 -- save us some unnecessary computations.
623 {-# INLINE find_containing_tetrahedron #-}
624 find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
625 find_containing_tetrahedron cube p =
626 candidates `V.unsafeIndex` (fromJust lucky_idx)
628 front_half = in_front_half cube p
629 top_half = in_top_half cube p
630 left_half = in_left_half cube p
637 front_left_top_tetrahedra cube
639 front_left_down_tetrahedra cube
642 front_right_top_tetrahedra cube
644 front_right_down_tetrahedra cube
650 back_left_top_tetrahedra cube
652 back_left_down_tetrahedra cube
655 back_right_top_tetrahedra cube
657 back_right_down_tetrahedra cube
659 -- Use the dot product instead of Euclidean distance here to save
660 -- a sqrt(). So, "distances" below really means "distances
662 distances = V.map ((dot p) . center) candidates
663 shortest_distance = V.minimum distances
664 lucky_idx = V.findIndex
665 (\t -> (center t) `dot` p == shortest_distance)
677 prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
678 prop_opposite_octant_tetrahedra_disjoint1 cube =
679 disjoint (front_left_top_tetrahedra cube) (front_right_down_tetrahedra cube)
681 prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
682 prop_opposite_octant_tetrahedra_disjoint2 cube =
683 disjoint (back_left_top_tetrahedra cube) (back_right_down_tetrahedra cube)
685 prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
686 prop_opposite_octant_tetrahedra_disjoint3 cube =
687 disjoint (front_left_top_tetrahedra cube) (back_right_top_tetrahedra cube)
689 prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
690 prop_opposite_octant_tetrahedra_disjoint4 cube =
691 disjoint (front_left_down_tetrahedra cube) (back_right_down_tetrahedra cube)
693 prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
694 prop_opposite_octant_tetrahedra_disjoint5 cube =
695 disjoint (front_left_top_tetrahedra cube) (back_left_down_tetrahedra cube)
697 prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
698 prop_opposite_octant_tetrahedra_disjoint6 cube =
699 disjoint (front_right_top_tetrahedra cube) (back_right_down_tetrahedra cube)
702 -- | Since the grid size is necessarily positive, all tetrahedra
703 -- (which comprise cubes of positive volume) must have positive
705 prop_all_volumes_positive :: Cube -> Bool
706 prop_all_volumes_positive cube =
710 volumes = map volume ts
713 -- | In fact, since all of the tetrahedra are identical, we should
714 -- already know their volumes. There's 24 tetrahedra to a cube, so
715 -- we'd expect the volume of each one to be (1/24)*h^3.
716 prop_all_volumes_exact :: Cube -> Bool
717 prop_all_volumes_exact cube =
718 and [volume t ~~= (1/24)*(delta^(3::Int)) | t <- tetrahedra cube]
722 -- | All tetrahedron should have their v0 located at the center of the cube.
723 prop_v0_all_equal :: Cube -> Bool
724 prop_v0_all_equal cube = (v0 t0) == (v0 t1)
726 t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
727 t1 = head $ tail (tetrahedra cube)
730 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
731 -- third and fourth indices of c-t3 have been switched. This is
732 -- because we store the triangles oriented such that their volume is
733 -- positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
734 -- in opposite directions, one of them has to have negative volume!
735 prop_c0120_identity1 :: Cube -> Bool
736 prop_c0120_identity1 cube =
737 c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
739 t0 = tetrahedron cube 0
740 t3 = tetrahedron cube 3
743 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
744 -- 'prop_c0120_identity1' with tetrahedrons 1 and 2.
745 prop_c0120_identity2 :: Cube -> Bool
746 prop_c0120_identity2 cube =
747 c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
749 t0 = tetrahedron cube 0
750 t1 = tetrahedron cube 1
752 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
753 -- 'prop_c0120_identity1' with tetrahedrons 1 and 2.
754 prop_c0120_identity3 :: Cube -> Bool
755 prop_c0120_identity3 cube =
756 c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
758 t1 = tetrahedron cube 1
759 t2 = tetrahedron cube 2
761 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
762 -- 'prop_c0120_identity1' with tetrahedrons 2 and 3.
763 prop_c0120_identity4 :: Cube -> Bool
764 prop_c0120_identity4 cube =
765 c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
767 t2 = tetrahedron cube 2
768 t3 = tetrahedron cube 3
771 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
772 -- 'prop_c0120_identity1' with tetrahedrons 4 and 5.
773 prop_c0120_identity5 :: Cube -> Bool
774 prop_c0120_identity5 cube =
775 c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
777 t4 = tetrahedron cube 4
778 t5 = tetrahedron cube 5
780 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
781 -- 'prop_c0120_identity1' with tetrahedrons 5 and 6.
782 prop_c0120_identity6 :: Cube -> Bool
783 prop_c0120_identity6 cube =
784 c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
786 t5 = tetrahedron cube 5
787 t6 = tetrahedron cube 6
790 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
791 -- 'prop_c0120_identity1' with tetrahedrons 6 and 7.
792 prop_c0120_identity7 :: Cube -> Bool
793 prop_c0120_identity7 cube =
794 c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
796 t6 = tetrahedron cube 6
797 t7 = tetrahedron cube 7
800 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
801 -- 'prop_c0120_identity1'.
802 prop_c0210_identity1 :: Cube -> Bool
803 prop_c0210_identity1 cube =
804 c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
806 t0 = tetrahedron cube 0
807 t3 = tetrahedron cube 3
810 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
811 -- 'prop_c0120_identity1'.
812 prop_c0300_identity1 :: Cube -> Bool
813 prop_c0300_identity1 cube =
814 c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
816 t0 = tetrahedron cube 0
817 t3 = tetrahedron cube 3
820 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
821 -- 'prop_c0120_identity1'.
822 prop_c1110_identity :: Cube -> Bool
823 prop_c1110_identity cube =
824 c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
826 t0 = tetrahedron cube 0
827 t3 = tetrahedron cube 3
830 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
831 -- 'prop_c0120_identity1'.
832 prop_c1200_identity1 :: Cube -> Bool
833 prop_c1200_identity1 cube =
834 c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
836 t0 = tetrahedron cube 0
837 t3 = tetrahedron cube 3
840 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
841 -- 'prop_c0120_identity1'.
842 prop_c2100_identity1 :: Cube -> Bool
843 prop_c2100_identity1 cube =
844 c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
846 t0 = tetrahedron cube 0
847 t3 = tetrahedron cube 3
851 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). Note that the
852 -- third and fourth indices of c-t3 have been switched. This is
853 -- because we store the triangles oriented such that their volume is
854 -- positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
855 -- point in opposite directions, one of them has to have negative
857 prop_c0102_identity1 :: Cube -> Bool
858 prop_c0102_identity1 cube =
859 c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
861 t0 = tetrahedron cube 0
862 t1 = tetrahedron cube 1
865 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
866 -- 'prop_c0102_identity1'.
867 prop_c0201_identity1 :: Cube -> Bool
868 prop_c0201_identity1 cube =
869 c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
871 t0 = tetrahedron cube 0
872 t1 = tetrahedron cube 1
875 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
876 -- 'prop_c0102_identity1'.
877 prop_c0300_identity2 :: Cube -> Bool
878 prop_c0300_identity2 cube =
879 c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
881 t0 = tetrahedron cube 0
882 t1 = tetrahedron cube 1
885 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
886 -- 'prop_c0102_identity1'.
887 prop_c1101_identity :: Cube -> Bool
888 prop_c1101_identity cube =
889 c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
891 t0 = tetrahedron cube 0
892 t1 = tetrahedron cube 1
895 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
896 -- 'prop_c0102_identity1'.
897 prop_c1200_identity2 :: Cube -> Bool
898 prop_c1200_identity2 cube =
899 c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
901 t0 = tetrahedron cube 0
902 t1 = tetrahedron cube 1
905 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
906 -- 'prop_c0102_identity1'.
907 prop_c2100_identity2 :: Cube -> Bool
908 prop_c2100_identity2 cube =
909 c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
911 t0 = tetrahedron cube 0
912 t1 = tetrahedron cube 1
915 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
916 -- fourth indices of c-t6 have been switched. This is because we
917 -- store the triangles oriented such that their volume is
918 -- positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
919 -- point in opposite directions, one of them has to have negative
921 prop_c3000_identity :: Cube -> Bool
922 prop_c3000_identity cube =
923 c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
924 - ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
926 t0 = tetrahedron cube 0
927 t6 = tetrahedron cube 6
930 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
931 -- 'prop_c3000_identity'.
932 prop_c2010_identity :: Cube -> Bool
933 prop_c2010_identity cube =
934 c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
935 - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
937 t0 = tetrahedron cube 0
938 t6 = tetrahedron cube 6
941 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
942 -- 'prop_c3000_identity'.
943 prop_c2001_identity :: Cube -> Bool
944 prop_c2001_identity cube =
945 c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
946 - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
948 t0 = tetrahedron cube 0
949 t6 = tetrahedron cube 6
952 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
953 -- 'prop_c3000_identity'.
954 prop_c1020_identity :: Cube -> Bool
955 prop_c1020_identity cube =
956 c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
957 - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
959 t0 = tetrahedron cube 0
960 t6 = tetrahedron cube 6
963 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
964 -- 'prop_c3000_identity'.
965 prop_c1002_identity :: Cube -> Bool
966 prop_c1002_identity cube =
967 c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
968 - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
970 t0 = tetrahedron cube 0
971 t6 = tetrahedron cube 6
974 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
975 -- 'prop_c3000_identity'.
976 prop_c1011_identity :: Cube -> Bool
977 prop_c1011_identity cube =
978 c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
979 ((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
981 t0 = tetrahedron cube 0
982 t6 = tetrahedron cube 6
985 -- | The function values at the interior should be the same for all
987 prop_interior_values_all_identical :: Cube -> Bool
988 prop_interior_values_all_identical cube =
989 all_equal [ eval (function_values tet) I | tet <- tetrahedra cube ]
992 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
993 -- This test checks the rotation works as expected.
994 prop_c_tilde_2100_rotation_correct :: Cube -> Bool
995 prop_c_tilde_2100_rotation_correct cube =
998 t0 = tetrahedron cube 0
999 t6 = tetrahedron cube 6
1001 -- What gets computed for c2100 of t6.
1002 expr1 = eval (function_values t6) $
1004 (1/12)*(T + R + L + D) +
1005 (1/64)*(FT + FR + FL + FD) +
1008 (1/96)*(RT + LD + LT + RD) +
1009 (1/192)*(BT + BR + BL + BD)
1011 -- What should be computed for c2100 of t6.
1012 expr2 = eval (function_values t0) $
1014 (1/12)*(F + R + L + B) +
1015 (1/64)*(FT + RT + LT + BT) +
1018 (1/96)*(FR + FL + BR + BL) +
1019 (1/192)*(FD + RD + LD + BD)
1022 -- | We know what (c t6 2 1 0 0) should be from Sorokina and
1023 -- Zeilfelder, p. 87. This test checks the actual value based on
1024 -- the FunctionValues of the cube.
1026 -- If 'prop_c_tilde_2100_rotation_correct' passes, then this test is
1028 prop_c_tilde_2100_correct :: Cube -> Bool
1029 prop_c_tilde_2100_correct cube =
1030 c t6 2 1 0 0 == expected
1032 t0 = tetrahedron cube 0
1033 t6 = tetrahedron cube 6
1034 fvs = function_values t0
1035 expected = eval fvs $
1037 (1/12)*(F + R + L + B) +
1038 (1/64)*(FT + RT + LT + BT) +
1041 (1/96)*(FR + FL + BR + BL) +
1042 (1/192)*(FD + RD + LD + BD)
1045 -- Tests to check that the correct edges are incidental.
1046 prop_t0_shares_edge_with_t1 :: Cube -> Bool
1047 prop_t0_shares_edge_with_t1 cube =
1048 (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
1050 t0 = tetrahedron cube 0
1051 t1 = tetrahedron cube 1
1053 prop_t0_shares_edge_with_t3 :: Cube -> Bool
1054 prop_t0_shares_edge_with_t3 cube =
1055 (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
1057 t0 = tetrahedron cube 0
1058 t3 = tetrahedron cube 3
1060 prop_t0_shares_edge_with_t6 :: Cube -> Bool
1061 prop_t0_shares_edge_with_t6 cube =
1062 (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
1064 t0 = tetrahedron cube 0
1065 t6 = tetrahedron cube 6
1067 prop_t1_shares_edge_with_t2 :: Cube -> Bool
1068 prop_t1_shares_edge_with_t2 cube =
1069 (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
1071 t1 = tetrahedron cube 1
1072 t2 = tetrahedron cube 2
1074 prop_t1_shares_edge_with_t19 :: Cube -> Bool
1075 prop_t1_shares_edge_with_t19 cube =
1076 (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
1078 t1 = tetrahedron cube 1
1079 t19 = tetrahedron cube 19
1081 prop_t2_shares_edge_with_t3 :: Cube -> Bool
1082 prop_t2_shares_edge_with_t3 cube =
1083 (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
1085 t1 = tetrahedron cube 1
1086 t2 = tetrahedron cube 2
1088 prop_t2_shares_edge_with_t12 :: Cube -> Bool
1089 prop_t2_shares_edge_with_t12 cube =
1090 (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
1092 t2 = tetrahedron cube 2
1093 t12 = tetrahedron cube 12
1095 prop_t3_shares_edge_with_t21 :: Cube -> Bool
1096 prop_t3_shares_edge_with_t21 cube =
1097 (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
1099 t3 = tetrahedron cube 3
1100 t21 = tetrahedron cube 21
1102 prop_t4_shares_edge_with_t5 :: Cube -> Bool
1103 prop_t4_shares_edge_with_t5 cube =
1104 (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
1106 t4 = tetrahedron cube 4
1107 t5 = tetrahedron cube 5
1109 prop_t4_shares_edge_with_t7 :: Cube -> Bool
1110 prop_t4_shares_edge_with_t7 cube =
1111 (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
1113 t4 = tetrahedron cube 4
1114 t7 = tetrahedron cube 7
1116 prop_t4_shares_edge_with_t10 :: Cube -> Bool
1117 prop_t4_shares_edge_with_t10 cube =
1118 (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
1120 t4 = tetrahedron cube 4
1121 t10 = tetrahedron cube 10
1123 prop_t5_shares_edge_with_t6 :: Cube -> Bool
1124 prop_t5_shares_edge_with_t6 cube =
1125 (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
1127 t5 = tetrahedron cube 5
1128 t6 = tetrahedron cube 6
1130 prop_t5_shares_edge_with_t16 :: Cube -> Bool
1131 prop_t5_shares_edge_with_t16 cube =
1132 (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
1134 t5 = tetrahedron cube 5
1135 t16 = tetrahedron cube 16
1137 prop_t6_shares_edge_with_t7 :: Cube -> Bool
1138 prop_t6_shares_edge_with_t7 cube =
1139 (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
1141 t6 = tetrahedron cube 6
1142 t7 = tetrahedron cube 7
1144 prop_t7_shares_edge_with_t20 :: Cube -> Bool
1145 prop_t7_shares_edge_with_t20 cube =
1146 (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
1148 t7 = tetrahedron cube 7
1149 t20 = tetrahedron cube 20
1152 p79_26_properties :: Test.Framework.Test
1154 testGroup "p. 79, Section (2.6) Properties" [
1155 testProperty "c0120 identity1" prop_c0120_identity1,
1156 testProperty "c0120 identity2" prop_c0120_identity2,
1157 testProperty "c0120 identity3" prop_c0120_identity3,
1158 testProperty "c0120 identity4" prop_c0120_identity4,
1159 testProperty "c0120 identity5" prop_c0120_identity5,
1160 testProperty "c0120 identity6" prop_c0120_identity6,
1161 testProperty "c0120 identity7" prop_c0120_identity7,
1162 testProperty "c0210 identity1" prop_c0210_identity1,
1163 testProperty "c0300 identity1" prop_c0300_identity1,
1164 testProperty "c1110 identity" prop_c1110_identity,
1165 testProperty "c1200 identity1" prop_c1200_identity1,
1166 testProperty "c2100 identity1" prop_c2100_identity1]
1168 p79_27_properties :: Test.Framework.Test
1170 testGroup "p. 79, Section (2.7) Properties" [
1171 testProperty "c0102 identity1" prop_c0102_identity1,
1172 testProperty "c0201 identity1" prop_c0201_identity1,
1173 testProperty "c0300 identity2" prop_c0300_identity2,
1174 testProperty "c1101 identity" prop_c1101_identity,
1175 testProperty "c1200 identity2" prop_c1200_identity2,
1176 testProperty "c2100 identity2" prop_c2100_identity2 ]
1179 p79_28_properties :: Test.Framework.Test
1181 testGroup "p. 79, Section (2.8) Properties" [
1182 testProperty "c3000 identity" prop_c3000_identity,
1183 testProperty "c2010 identity" prop_c2010_identity,
1184 testProperty "c2001 identity" prop_c2001_identity,
1185 testProperty "c1020 identity" prop_c1020_identity,
1186 testProperty "c1002 identity" prop_c1002_identity,
1187 testProperty "c1011 identity" prop_c1011_identity ]
1190 edge_incidence_tests :: Test.Framework.Test
1191 edge_incidence_tests =
1192 testGroup "Edge Incidence Tests" [
1193 testProperty "t0 shares edge with t6" prop_t0_shares_edge_with_t6,
1194 testProperty "t0 shares edge with t1" prop_t0_shares_edge_with_t1,
1195 testProperty "t0 shares edge with t3" prop_t0_shares_edge_with_t3,
1196 testProperty "t1 shares edge with t2" prop_t1_shares_edge_with_t2,
1197 testProperty "t1 shares edge with t19" prop_t1_shares_edge_with_t19,
1198 testProperty "t2 shares edge with t3" prop_t2_shares_edge_with_t3,
1199 testProperty "t2 shares edge with t12" prop_t2_shares_edge_with_t12,
1200 testProperty "t3 shares edge with t21" prop_t3_shares_edge_with_t21,
1201 testProperty "t4 shares edge with t5" prop_t4_shares_edge_with_t5,
1202 testProperty "t4 shares edge with t7" prop_t4_shares_edge_with_t7,
1203 testProperty "t4 shares edge with t10" prop_t4_shares_edge_with_t10,
1204 testProperty "t5 shares edge with t6" prop_t5_shares_edge_with_t6,
1205 testProperty "t5 shares edge with t16" prop_t5_shares_edge_with_t16,
1206 testProperty "t6 shares edge with t7" prop_t6_shares_edge_with_t7,
1207 testProperty "t7 shares edge with t20" prop_t7_shares_edge_with_t20 ]
1209 cube_properties :: Test.Framework.Test
1211 testGroup "Cube Properties" [
1215 edge_incidence_tests,
1216 testProperty "opposite octant tetrahedra are disjoint (1)"
1217 prop_opposite_octant_tetrahedra_disjoint1,
1218 testProperty "opposite octant tetrahedra are disjoint (2)"
1219 prop_opposite_octant_tetrahedra_disjoint2,
1220 testProperty "opposite octant tetrahedra are disjoint (3)"
1221 prop_opposite_octant_tetrahedra_disjoint3,
1222 testProperty "opposite octant tetrahedra are disjoint (4)"
1223 prop_opposite_octant_tetrahedra_disjoint4,
1224 testProperty "opposite octant tetrahedra are disjoint (5)"
1225 prop_opposite_octant_tetrahedra_disjoint5,
1226 testProperty "opposite octant tetrahedra are disjoint (6)"
1227 prop_opposite_octant_tetrahedra_disjoint6,
1228 testProperty "all volumes positive" prop_all_volumes_positive,
1229 testProperty "all volumes exact" prop_all_volumes_exact,
1230 testProperty "v0 all equal" prop_v0_all_equal,
1231 testProperty "interior values all identical"
1232 prop_interior_values_all_identical,
1233 testProperty "c-tilde_2100 rotation correct"
1234 prop_c_tilde_2100_rotation_correct,
1235 testProperty "c-tilde_2100 correct"
1236 prop_c_tilde_2100_correct ]