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
510 -- Only used in tests, so we don't need the added speed
512 tetrahedra :: Cube -> [Tetrahedron]
513 tetrahedra cube = [ tetrahedron cube n | n <- [0..23] ]
515 front_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
516 front_left_top_tetrahedra cube =
517 V.singleton (tetrahedron cube 0) `V.snoc`
518 (tetrahedron cube 3) `V.snoc`
519 (tetrahedron cube 6) `V.snoc`
520 (tetrahedron cube 7) `V.snoc`
521 (tetrahedron cube 20) `V.snoc`
522 (tetrahedron cube 21)
524 front_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
525 front_left_down_tetrahedra cube =
526 V.singleton (tetrahedron cube 0) `V.snoc`
527 (tetrahedron cube 2) `V.snoc`
528 (tetrahedron cube 3) `V.snoc`
529 (tetrahedron cube 12) `V.snoc`
530 (tetrahedron cube 15) `V.snoc`
531 (tetrahedron cube 21)
533 front_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
534 front_right_top_tetrahedra cube =
535 V.singleton (tetrahedron cube 0) `V.snoc`
536 (tetrahedron cube 1) `V.snoc`
537 (tetrahedron cube 5) `V.snoc`
538 (tetrahedron cube 6) `V.snoc`
539 (tetrahedron cube 16) `V.snoc`
540 (tetrahedron cube 19)
542 front_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
543 front_right_down_tetrahedra cube =
544 V.singleton (tetrahedron cube 1) `V.snoc`
545 (tetrahedron cube 2) `V.snoc`
546 (tetrahedron cube 12) `V.snoc`
547 (tetrahedron cube 13) `V.snoc`
548 (tetrahedron cube 18) `V.snoc`
549 (tetrahedron cube 19)
551 back_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
552 back_left_top_tetrahedra cube =
553 V.singleton (tetrahedron cube 0) `V.snoc`
554 (tetrahedron cube 3) `V.snoc`
555 (tetrahedron cube 6) `V.snoc`
556 (tetrahedron cube 7) `V.snoc`
557 (tetrahedron cube 20) `V.snoc`
558 (tetrahedron cube 21)
560 back_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
561 back_left_down_tetrahedra cube =
562 V.singleton (tetrahedron cube 8) `V.snoc`
563 (tetrahedron cube 11) `V.snoc`
564 (tetrahedron cube 14) `V.snoc`
565 (tetrahedron cube 15) `V.snoc`
566 (tetrahedron cube 22) `V.snoc`
567 (tetrahedron cube 23)
569 back_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
570 back_right_top_tetrahedra cube =
571 V.singleton (tetrahedron cube 4) `V.snoc`
572 (tetrahedron cube 5) `V.snoc`
573 (tetrahedron cube 9) `V.snoc`
574 (tetrahedron cube 10) `V.snoc`
575 (tetrahedron cube 16) `V.snoc`
576 (tetrahedron cube 17)
578 back_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
579 back_right_down_tetrahedra cube =
580 V.singleton (tetrahedron cube 8) `V.snoc`
581 (tetrahedron cube 9) `V.snoc`
582 (tetrahedron cube 13) `V.snoc`
583 (tetrahedron cube 14) `V.snoc`
584 (tetrahedron cube 17) `V.snoc`
585 (tetrahedron cube 18)
587 in_top_half :: Cube -> Point -> Bool
588 in_top_half cube (Point _ _ z) =
589 distance_from_top <= distance_from_bottom
591 distance_from_top = abs $ (zmax cube) - z
592 distance_from_bottom = abs $ (zmin cube) - z
594 in_front_half :: Cube -> Point -> Bool
595 in_front_half cube (Point x _ _) =
596 distance_from_front <= distance_from_back
598 distance_from_front = abs $ (xmin cube) - x
599 distance_from_back = abs $ (xmax cube) - x
602 in_left_half :: Cube -> Point -> Bool
603 in_left_half cube (Point _ y _) =
604 distance_from_left <= distance_from_right
606 distance_from_left = abs $ (ymin cube) - y
607 distance_from_right = abs $ (ymax cube) - y
610 -- | Takes a 'Cube', and returns the Tetrahedra belonging to it that
611 -- contain the given 'Point'. This should be faster than checking
612 -- every tetrahedron individually, since we determine which half
613 -- (hemisphere?) of the cube the point lies in three times: once in
614 -- each dimension. This allows us to eliminate non-candidates
617 -- This can throw an exception, but the use of 'head' might
618 -- save us some unnecessary computations.
620 {-# INLINE find_containing_tetrahedron #-}
621 find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
622 find_containing_tetrahedron cube p =
623 candidates `V.unsafeIndex` (fromJust lucky_idx)
625 front_half = in_front_half cube p
626 top_half = in_top_half cube p
627 left_half = in_left_half cube p
634 front_left_top_tetrahedra cube
636 front_left_down_tetrahedra cube
639 front_right_top_tetrahedra cube
641 front_right_down_tetrahedra cube
647 back_left_top_tetrahedra cube
649 back_left_down_tetrahedra cube
652 back_right_top_tetrahedra cube
654 back_right_down_tetrahedra cube
656 -- Use the dot product instead of Euclidean distance here to save
657 -- a sqrt(). So, "distances" below really means "distances
659 distances = V.map ((dot p) . center) candidates
660 shortest_distance = V.minimum distances
661 lucky_idx = V.findIndex
662 (\t -> (center t) `dot` p == shortest_distance)
674 prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
675 prop_opposite_octant_tetrahedra_disjoint1 cube =
676 disjoint (front_left_top_tetrahedra cube) (front_right_down_tetrahedra cube)
678 prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
679 prop_opposite_octant_tetrahedra_disjoint2 cube =
680 disjoint (back_left_top_tetrahedra cube) (back_right_down_tetrahedra cube)
682 prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
683 prop_opposite_octant_tetrahedra_disjoint3 cube =
684 disjoint (front_left_top_tetrahedra cube) (back_right_top_tetrahedra cube)
686 prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
687 prop_opposite_octant_tetrahedra_disjoint4 cube =
688 disjoint (front_left_down_tetrahedra cube) (back_right_down_tetrahedra cube)
690 prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
691 prop_opposite_octant_tetrahedra_disjoint5 cube =
692 disjoint (front_left_top_tetrahedra cube) (back_left_down_tetrahedra cube)
694 prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
695 prop_opposite_octant_tetrahedra_disjoint6 cube =
696 disjoint (front_right_top_tetrahedra cube) (back_right_down_tetrahedra cube)
699 -- | Since the grid size is necessarily positive, all tetrahedra
700 -- (which comprise cubes of positive volume) must have positive
702 prop_all_volumes_positive :: Cube -> Bool
703 prop_all_volumes_positive cube =
707 volumes = map volume ts
710 -- | In fact, since all of the tetrahedra are identical, we should
711 -- already know their volumes. There's 24 tetrahedra to a cube, so
712 -- we'd expect the volume of each one to be (1/24)*h^3.
713 prop_all_volumes_exact :: Cube -> Bool
714 prop_all_volumes_exact cube =
715 and [volume t ~~= (1/24)*(delta^(3::Int)) | t <- tetrahedra cube]
719 -- | All tetrahedron should have their v0 located at the center of the cube.
720 prop_v0_all_equal :: Cube -> Bool
721 prop_v0_all_equal cube = (v0 t0) == (v0 t1)
723 t0 = head (tetrahedra cube) -- Doesn't matter which two we choose.
724 t1 = head $ tail (tetrahedra cube)
727 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Note that the
728 -- third and fourth indices of c-t3 have been switched. This is
729 -- because we store the triangles oriented such that their volume is
730 -- positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde point
731 -- in opposite directions, one of them has to have negative volume!
732 prop_c0120_identity1 :: Cube -> Bool
733 prop_c0120_identity1 cube =
734 c t0 0 1 2 0 ~= (c t0 0 0 2 1 + c t3 0 0 1 2) / 2
736 t0 = tetrahedron cube 0
737 t3 = tetrahedron cube 3
740 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
741 -- 'prop_c0120_identity1' with tetrahedrons 1 and 2.
742 prop_c0120_identity2 :: Cube -> Bool
743 prop_c0120_identity2 cube =
744 c t1 0 1 2 0 ~= (c t1 0 0 2 1 + c t0 0 0 1 2) / 2
746 t0 = tetrahedron cube 0
747 t1 = tetrahedron cube 1
749 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
750 -- 'prop_c0120_identity1' with tetrahedrons 1 and 2.
751 prop_c0120_identity3 :: Cube -> Bool
752 prop_c0120_identity3 cube =
753 c t2 0 1 2 0 ~= (c t2 0 0 2 1 + c t1 0 0 1 2) / 2
755 t1 = tetrahedron cube 1
756 t2 = tetrahedron cube 2
758 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
759 -- 'prop_c0120_identity1' with tetrahedrons 2 and 3.
760 prop_c0120_identity4 :: Cube -> Bool
761 prop_c0120_identity4 cube =
762 c t3 0 1 2 0 ~= (c t3 0 0 2 1 + c t2 0 0 1 2) / 2
764 t2 = tetrahedron cube 2
765 t3 = tetrahedron cube 3
768 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
769 -- 'prop_c0120_identity1' with tetrahedrons 4 and 5.
770 prop_c0120_identity5 :: Cube -> Bool
771 prop_c0120_identity5 cube =
772 c t5 0 1 2 0 ~= (c t5 0 0 2 1 + c t4 0 0 1 2) / 2
774 t4 = tetrahedron cube 4
775 t5 = tetrahedron cube 5
777 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
778 -- 'prop_c0120_identity1' with tetrahedrons 5 and 6.
779 prop_c0120_identity6 :: Cube -> Bool
780 prop_c0120_identity6 cube =
781 c t6 0 1 2 0 ~= (c t6 0 0 2 1 + c t5 0 0 1 2) / 2
783 t5 = tetrahedron cube 5
784 t6 = tetrahedron cube 6
787 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). Repeats
788 -- 'prop_c0120_identity1' with tetrahedrons 6 and 7.
789 prop_c0120_identity7 :: Cube -> Bool
790 prop_c0120_identity7 cube =
791 c t7 0 1 2 0 ~= (c t7 0 0 2 1 + c t6 0 0 1 2) / 2
793 t6 = tetrahedron cube 6
794 t7 = tetrahedron cube 7
797 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
798 -- 'prop_c0120_identity1'.
799 prop_c0210_identity1 :: Cube -> Bool
800 prop_c0210_identity1 cube =
801 c t0 0 2 1 0 ~= (c t0 0 1 1 1 + c t3 0 1 1 1) / 2
803 t0 = tetrahedron cube 0
804 t3 = tetrahedron cube 3
807 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
808 -- 'prop_c0120_identity1'.
809 prop_c0300_identity1 :: Cube -> Bool
810 prop_c0300_identity1 cube =
811 c t0 0 3 0 0 ~= (c t0 0 2 0 1 + c t3 0 2 1 0) / 2
813 t0 = tetrahedron cube 0
814 t3 = tetrahedron cube 3
817 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
818 -- 'prop_c0120_identity1'.
819 prop_c1110_identity :: Cube -> Bool
820 prop_c1110_identity cube =
821 c t0 1 1 1 0 ~= (c t0 1 0 1 1 + c t3 1 0 1 1) / 2
823 t0 = tetrahedron cube 0
824 t3 = tetrahedron cube 3
827 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
828 -- 'prop_c0120_identity1'.
829 prop_c1200_identity1 :: Cube -> Bool
830 prop_c1200_identity1 cube =
831 c t0 1 2 0 0 ~= (c t0 1 1 0 1 + c t3 1 1 1 0) / 2
833 t0 = tetrahedron cube 0
834 t3 = tetrahedron cube 3
837 -- | Given in Sorokina and Zeilfelder, p. 79, (2.6). See
838 -- 'prop_c0120_identity1'.
839 prop_c2100_identity1 :: Cube -> Bool
840 prop_c2100_identity1 cube =
841 c t0 2 1 0 0 ~= (c t0 2 0 0 1 + c t3 2 0 1 0) / 2
843 t0 = tetrahedron cube 0
844 t3 = tetrahedron cube 3
848 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). Note that the
849 -- third and fourth indices of c-t3 have been switched. This is
850 -- because we store the triangles oriented such that their volume is
851 -- positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
852 -- point in opposite directions, one of them has to have negative
854 prop_c0102_identity1 :: Cube -> Bool
855 prop_c0102_identity1 cube =
856 c t0 0 1 0 2 ~= (c t0 0 0 1 2 + c t1 0 0 2 1) / 2
858 t0 = tetrahedron cube 0
859 t1 = tetrahedron cube 1
862 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
863 -- 'prop_c0102_identity1'.
864 prop_c0201_identity1 :: Cube -> Bool
865 prop_c0201_identity1 cube =
866 c t0 0 2 0 1 ~= (c t0 0 1 1 1 + c t1 0 1 1 1) / 2
868 t0 = tetrahedron cube 0
869 t1 = tetrahedron cube 1
872 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
873 -- 'prop_c0102_identity1'.
874 prop_c0300_identity2 :: Cube -> Bool
875 prop_c0300_identity2 cube =
876 c t0 0 3 0 0 ~= (c t0 0 2 1 0 + c t1 0 2 0 1) / 2
878 t0 = tetrahedron cube 0
879 t1 = tetrahedron cube 1
882 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
883 -- 'prop_c0102_identity1'.
884 prop_c1101_identity :: Cube -> Bool
885 prop_c1101_identity cube =
886 c t0 1 1 0 1 ~= (c t0 1 0 1 1 + c t1 1 0 1 1) / 2
888 t0 = tetrahedron cube 0
889 t1 = tetrahedron cube 1
892 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
893 -- 'prop_c0102_identity1'.
894 prop_c1200_identity2 :: Cube -> Bool
895 prop_c1200_identity2 cube =
896 c t0 1 2 0 0 ~= (c t0 1 1 1 0 + c t1 1 1 0 1) / 2
898 t0 = tetrahedron cube 0
899 t1 = tetrahedron cube 1
902 -- | Given in Sorokina and Zeilfelder, p. 79, (2.7). See
903 -- 'prop_c0102_identity1'.
904 prop_c2100_identity2 :: Cube -> Bool
905 prop_c2100_identity2 cube =
906 c t0 2 1 0 0 ~= (c t0 2 0 1 0 + c t1 2 0 0 1) / 2
908 t0 = tetrahedron cube 0
909 t1 = tetrahedron cube 1
912 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). The third and
913 -- fourth indices of c-t6 have been switched. This is because we
914 -- store the triangles oriented such that their volume is
915 -- positive. If T and T-tilde share \<v0,v1,v2\> and v3,v3-tilde
916 -- point in opposite directions, one of them has to have negative
918 prop_c3000_identity :: Cube -> Bool
919 prop_c3000_identity cube =
920 c t0 3 0 0 0 ~= c t0 2 1 0 0 + c t6 2 1 0 0
921 - ((c t0 2 0 1 0 + c t0 2 0 0 1)/ 2)
923 t0 = tetrahedron cube 0
924 t6 = tetrahedron cube 6
927 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
928 -- 'prop_c3000_identity'.
929 prop_c2010_identity :: Cube -> Bool
930 prop_c2010_identity cube =
931 c t0 2 0 1 0 ~= c t0 1 1 1 0 + c t6 1 1 0 1
932 - ((c t0 1 0 2 0 + c t0 1 0 1 1)/ 2)
934 t0 = tetrahedron cube 0
935 t6 = tetrahedron cube 6
938 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
939 -- 'prop_c3000_identity'.
940 prop_c2001_identity :: Cube -> Bool
941 prop_c2001_identity cube =
942 c t0 2 0 0 1 ~= c t0 1 1 0 1 + c t6 1 1 1 0
943 - ((c t0 1 0 0 2 + c t0 1 0 1 1)/ 2)
945 t0 = tetrahedron cube 0
946 t6 = tetrahedron cube 6
949 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
950 -- 'prop_c3000_identity'.
951 prop_c1020_identity :: Cube -> Bool
952 prop_c1020_identity cube =
953 c t0 1 0 2 0 ~= c t0 0 1 2 0 + c t6 0 1 0 2
954 - ((c t0 0 0 3 0 + c t0 0 0 2 1)/ 2)
956 t0 = tetrahedron cube 0
957 t6 = tetrahedron cube 6
960 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
961 -- 'prop_c3000_identity'.
962 prop_c1002_identity :: Cube -> Bool
963 prop_c1002_identity cube =
964 c t0 1 0 0 2 ~= c t0 0 1 0 2 + c t6 0 1 2 0
965 - ((c t0 0 0 0 3 + c t0 0 0 1 2)/ 2)
967 t0 = tetrahedron cube 0
968 t6 = tetrahedron cube 6
971 -- | Given in Sorokina and Zeilfelder, p. 79, (2.8). See
972 -- 'prop_c3000_identity'.
973 prop_c1011_identity :: Cube -> Bool
974 prop_c1011_identity cube =
975 c t0 1 0 1 1 ~= c t0 0 1 1 1 + c t6 0 1 1 1 -
976 ((c t0 0 0 1 2 + c t0 0 0 2 1)/ 2)
978 t0 = tetrahedron cube 0
979 t6 = tetrahedron cube 6
982 -- | The function values at the interior should be the same for all
984 prop_interior_values_all_identical :: Cube -> Bool
985 prop_interior_values_all_identical cube =
986 all_equal [ eval (function_values tet) I | tet <- tetrahedra cube ]
989 -- | We know what (c t6 2 1 0 0) should be from Sorokina and Zeilfelder, p. 87.
990 -- This test checks the rotation works as expected.
991 prop_c_tilde_2100_rotation_correct :: Cube -> Bool
992 prop_c_tilde_2100_rotation_correct cube =
995 t0 = tetrahedron cube 0
996 t6 = tetrahedron cube 6
998 -- What gets computed for c2100 of t6.
999 expr1 = eval (function_values t6) $
1001 (1/12)*(T + R + L + D) +
1002 (1/64)*(FT + FR + FL + FD) +
1005 (1/96)*(RT + LD + LT + RD) +
1006 (1/192)*(BT + BR + BL + BD)
1008 -- What should be computed for c2100 of t6.
1009 expr2 = eval (function_values t0) $
1011 (1/12)*(F + R + L + B) +
1012 (1/64)*(FT + RT + LT + BT) +
1015 (1/96)*(FR + FL + BR + BL) +
1016 (1/192)*(FD + RD + LD + BD)
1019 -- | We know what (c t6 2 1 0 0) should be from Sorokina and
1020 -- Zeilfelder, p. 87. This test checks the actual value based on
1021 -- the FunctionValues of the cube.
1023 -- If 'prop_c_tilde_2100_rotation_correct' passes, then this test is
1025 prop_c_tilde_2100_correct :: Cube -> Bool
1026 prop_c_tilde_2100_correct cube =
1027 c t6 2 1 0 0 == expected
1029 t0 = tetrahedron cube 0
1030 t6 = tetrahedron cube 6
1031 fvs = function_values t0
1032 expected = eval fvs $
1034 (1/12)*(F + R + L + B) +
1035 (1/64)*(FT + RT + LT + BT) +
1038 (1/96)*(FR + FL + BR + BL) +
1039 (1/192)*(FD + RD + LD + BD)
1042 -- Tests to check that the correct edges are incidental.
1043 prop_t0_shares_edge_with_t1 :: Cube -> Bool
1044 prop_t0_shares_edge_with_t1 cube =
1045 (v1 t0) == (v1 t1) && (v3 t0) == (v2 t1)
1047 t0 = tetrahedron cube 0
1048 t1 = tetrahedron cube 1
1050 prop_t0_shares_edge_with_t3 :: Cube -> Bool
1051 prop_t0_shares_edge_with_t3 cube =
1052 (v1 t0) == (v1 t3) && (v2 t0) == (v3 t3)
1054 t0 = tetrahedron cube 0
1055 t3 = tetrahedron cube 3
1057 prop_t0_shares_edge_with_t6 :: Cube -> Bool
1058 prop_t0_shares_edge_with_t6 cube =
1059 (v2 t0) == (v3 t6) && (v3 t0) == (v2 t6)
1061 t0 = tetrahedron cube 0
1062 t6 = tetrahedron cube 6
1064 prop_t1_shares_edge_with_t2 :: Cube -> Bool
1065 prop_t1_shares_edge_with_t2 cube =
1066 (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
1068 t1 = tetrahedron cube 1
1069 t2 = tetrahedron cube 2
1071 prop_t1_shares_edge_with_t19 :: Cube -> Bool
1072 prop_t1_shares_edge_with_t19 cube =
1073 (v2 t1) == (v3 t19) && (v3 t1) == (v2 t19)
1075 t1 = tetrahedron cube 1
1076 t19 = tetrahedron cube 19
1078 prop_t2_shares_edge_with_t3 :: Cube -> Bool
1079 prop_t2_shares_edge_with_t3 cube =
1080 (v1 t1) == (v1 t2) && (v3 t1) == (v2 t2)
1082 t1 = tetrahedron cube 1
1083 t2 = tetrahedron cube 2
1085 prop_t2_shares_edge_with_t12 :: Cube -> Bool
1086 prop_t2_shares_edge_with_t12 cube =
1087 (v2 t2) == (v3 t12) && (v3 t2) == (v2 t12)
1089 t2 = tetrahedron cube 2
1090 t12 = tetrahedron cube 12
1092 prop_t3_shares_edge_with_t21 :: Cube -> Bool
1093 prop_t3_shares_edge_with_t21 cube =
1094 (v2 t3) == (v3 t21) && (v3 t3) == (v2 t21)
1096 t3 = tetrahedron cube 3
1097 t21 = tetrahedron cube 21
1099 prop_t4_shares_edge_with_t5 :: Cube -> Bool
1100 prop_t4_shares_edge_with_t5 cube =
1101 (v1 t4) == (v1 t5) && (v3 t4) == (v2 t5)
1103 t4 = tetrahedron cube 4
1104 t5 = tetrahedron cube 5
1106 prop_t4_shares_edge_with_t7 :: Cube -> Bool
1107 prop_t4_shares_edge_with_t7 cube =
1108 (v1 t4) == (v1 t7) && (v2 t4) == (v3 t7)
1110 t4 = tetrahedron cube 4
1111 t7 = tetrahedron cube 7
1113 prop_t4_shares_edge_with_t10 :: Cube -> Bool
1114 prop_t4_shares_edge_with_t10 cube =
1115 (v2 t4) == (v3 t10) && (v3 t4) == (v2 t10)
1117 t4 = tetrahedron cube 4
1118 t10 = tetrahedron cube 10
1120 prop_t5_shares_edge_with_t6 :: Cube -> Bool
1121 prop_t5_shares_edge_with_t6 cube =
1122 (v1 t5) == (v1 t6) && (v3 t5) == (v2 t6)
1124 t5 = tetrahedron cube 5
1125 t6 = tetrahedron cube 6
1127 prop_t5_shares_edge_with_t16 :: Cube -> Bool
1128 prop_t5_shares_edge_with_t16 cube =
1129 (v2 t5) == (v3 t16) && (v3 t5) == (v2 t16)
1131 t5 = tetrahedron cube 5
1132 t16 = tetrahedron cube 16
1134 prop_t6_shares_edge_with_t7 :: Cube -> Bool
1135 prop_t6_shares_edge_with_t7 cube =
1136 (v1 t6) == (v1 t7) && (v3 t6) == (v2 t7)
1138 t6 = tetrahedron cube 6
1139 t7 = tetrahedron cube 7
1141 prop_t7_shares_edge_with_t20 :: Cube -> Bool
1142 prop_t7_shares_edge_with_t20 cube =
1143 (v2 t7) == (v3 t20) && (v2 t7) == (v3 t20)
1145 t7 = tetrahedron cube 7
1146 t20 = tetrahedron cube 20
1149 p79_26_properties :: Test.Framework.Test
1151 testGroup "p. 79, Section (2.6) Properties" [
1152 testProperty "c0120 identity1" prop_c0120_identity1,
1153 testProperty "c0120 identity2" prop_c0120_identity2,
1154 testProperty "c0120 identity3" prop_c0120_identity3,
1155 testProperty "c0120 identity4" prop_c0120_identity4,
1156 testProperty "c0120 identity5" prop_c0120_identity5,
1157 testProperty "c0120 identity6" prop_c0120_identity6,
1158 testProperty "c0120 identity7" prop_c0120_identity7,
1159 testProperty "c0210 identity1" prop_c0210_identity1,
1160 testProperty "c0300 identity1" prop_c0300_identity1,
1161 testProperty "c1110 identity" prop_c1110_identity,
1162 testProperty "c1200 identity1" prop_c1200_identity1,
1163 testProperty "c2100 identity1" prop_c2100_identity1]
1165 p79_27_properties :: Test.Framework.Test
1167 testGroup "p. 79, Section (2.7) Properties" [
1168 testProperty "c0102 identity1" prop_c0102_identity1,
1169 testProperty "c0201 identity1" prop_c0201_identity1,
1170 testProperty "c0300 identity2" prop_c0300_identity2,
1171 testProperty "c1101 identity" prop_c1101_identity,
1172 testProperty "c1200 identity2" prop_c1200_identity2,
1173 testProperty "c2100 identity2" prop_c2100_identity2 ]
1176 p79_28_properties :: Test.Framework.Test
1178 testGroup "p. 79, Section (2.8) Properties" [
1179 testProperty "c3000 identity" prop_c3000_identity,
1180 testProperty "c2010 identity" prop_c2010_identity,
1181 testProperty "c2001 identity" prop_c2001_identity,
1182 testProperty "c1020 identity" prop_c1020_identity,
1183 testProperty "c1002 identity" prop_c1002_identity,
1184 testProperty "c1011 identity" prop_c1011_identity ]
1187 edge_incidence_tests :: Test.Framework.Test
1188 edge_incidence_tests =
1189 testGroup "Edge Incidence Tests" [
1190 testProperty "t0 shares edge with t6" prop_t0_shares_edge_with_t6,
1191 testProperty "t0 shares edge with t1" prop_t0_shares_edge_with_t1,
1192 testProperty "t0 shares edge with t3" prop_t0_shares_edge_with_t3,
1193 testProperty "t1 shares edge with t2" prop_t1_shares_edge_with_t2,
1194 testProperty "t1 shares edge with t19" prop_t1_shares_edge_with_t19,
1195 testProperty "t2 shares edge with t3" prop_t2_shares_edge_with_t3,
1196 testProperty "t2 shares edge with t12" prop_t2_shares_edge_with_t12,
1197 testProperty "t3 shares edge with t21" prop_t3_shares_edge_with_t21,
1198 testProperty "t4 shares edge with t5" prop_t4_shares_edge_with_t5,
1199 testProperty "t4 shares edge with t7" prop_t4_shares_edge_with_t7,
1200 testProperty "t4 shares edge with t10" prop_t4_shares_edge_with_t10,
1201 testProperty "t5 shares edge with t6" prop_t5_shares_edge_with_t6,
1202 testProperty "t5 shares edge with t16" prop_t5_shares_edge_with_t16,
1203 testProperty "t6 shares edge with t7" prop_t6_shares_edge_with_t7,
1204 testProperty "t7 shares edge with t20" prop_t7_shares_edge_with_t20 ]
1206 cube_properties :: Test.Framework.Test
1208 testGroup "Cube Properties" [
1212 edge_incidence_tests,
1213 testProperty "opposite octant tetrahedra are disjoint (1)"
1214 prop_opposite_octant_tetrahedra_disjoint1,
1215 testProperty "opposite octant tetrahedra are disjoint (2)"
1216 prop_opposite_octant_tetrahedra_disjoint2,
1217 testProperty "opposite octant tetrahedra are disjoint (3)"
1218 prop_opposite_octant_tetrahedra_disjoint3,
1219 testProperty "opposite octant tetrahedra are disjoint (4)"
1220 prop_opposite_octant_tetrahedra_disjoint4,
1221 testProperty "opposite octant tetrahedra are disjoint (5)"
1222 prop_opposite_octant_tetrahedra_disjoint5,
1223 testProperty "opposite octant tetrahedra are disjoint (6)"
1224 prop_opposite_octant_tetrahedra_disjoint6,
1225 testProperty "all volumes positive" prop_all_volumes_positive,
1226 testProperty "all volumes exact" prop_all_volumes_exact,
1227 testProperty "v0 all equal" prop_v0_all_equal,
1228 testProperty "interior values all identical"
1229 prop_interior_values_all_identical,
1230 testProperty "c-tilde_2100 rotation correct"
1231 prop_c_tilde_2100_rotation_correct,
1232 testProperty "c-tilde_2100 correct"
1233 prop_c_tilde_2100_correct ]