1 -- The "tetrahedron" function pattern matches on the integers zero
2 -- through twenty-three, but doesn't handle the "otherwise" case, for
3 -- performance reasons.
4 {-# OPTIONS_GHC -Wno-incomplete-patterns #-}
9 find_containing_tetrahedron,
14 import Data.Maybe ( fromJust )
15 import qualified Data.Vector as V (
23 import Prelude hiding ( LT )
24 import Test.Tasty ( TestTree, testGroup )
25 import Test.Tasty.QuickCheck (
26 Arbitrary( arbitrary ),
32 Cardinal(F, B, L, R, D, T, FL, FR, FD, FT,
33 BL, BR, BD, BT, LD, LT, RD, RT, I),
40 import Comparisons ( (~=), (~~=) )
41 import qualified Face ( Face(..), center )
42 import FunctionValues ( FunctionValues, eval, rotate )
43 import Misc ( all_equal, disjoint )
44 import Point ( Point( Point ), dot )
46 Tetrahedron(Tetrahedron, function_values, v0, v1, v2, v3),
51 data Cube = Cube { i :: !Int,
54 fv :: !FunctionValues,
55 tetrahedra_volume :: !Double }
59 instance Arbitrary Cube where
61 i' <- choose (coordmin, coordmax)
62 j' <- choose (coordmin, coordmax)
63 k' <- choose (coordmin, coordmax)
64 fv' <- arbitrary :: Gen FunctionValues
65 (Positive tet_vol) <- arbitrary :: Gen (Positive Double)
66 return (Cube i' j' k' fv' tet_vol)
68 -- The idea here is that, when cubed in the volume formula,
69 -- these numbers don't overflow 64 bits. This number is not
70 -- magic in any other sense than that it does not cause test
71 -- failures, while 2^23 does.
72 coordmax = 4194304 :: Int -- 2^22
76 instance Show Cube where
78 "Cube_" ++ subscript ++ "\n" ++
79 " Center: " ++ (show (center cube)) ++ "\n" ++
80 " xmin: " ++ (show (xmin cube)) ++ "\n" ++
81 " xmax: " ++ (show (xmax cube)) ++ "\n" ++
82 " ymin: " ++ (show (ymin cube)) ++ "\n" ++
83 " ymax: " ++ (show (ymax cube)) ++ "\n" ++
84 " zmin: " ++ (show (zmin cube)) ++ "\n" ++
85 " zmax: " ++ (show (zmax cube)) ++ "\n"
88 (show (i cube)) ++ "," ++ (show (j cube)) ++ "," ++ (show (k cube))
91 -- | The left-side boundary of the cube. See Sorokina and Zeilfelder,
93 xmin :: Cube -> Double
94 xmin cube = (i' - 1/2)
96 i' = fromIntegral (i cube) :: Double
98 -- | The right-side boundary of the cube. See Sorokina and Zeilfelder,
100 xmax :: Cube -> Double
101 xmax cube = (i' + 1/2)
103 i' = fromIntegral (i cube) :: Double
105 -- | The front boundary of the cube. See Sorokina and Zeilfelder,
107 ymin :: Cube -> Double
108 ymin cube = (j' - 1/2)
110 j' = fromIntegral (j cube) :: Double
112 -- | The back boundary of the cube. See Sorokina and Zeilfelder,
114 ymax :: Cube -> Double
115 ymax cube = (j' + 1/2)
117 j' = fromIntegral (j cube) :: Double
119 -- | The bottom boundary of the cube. See Sorokina and Zeilfelder,
121 zmin :: Cube -> Double
122 zmin cube = (k' - 1/2)
124 k' = fromIntegral (k cube) :: Double
126 -- | The top boundary of the cube. See Sorokina and Zeilfelder,
128 zmax :: Cube -> Double
129 zmax cube = (k' + 1/2)
131 k' = fromIntegral (k cube) :: Double
134 -- | The center of Cube_ijk coincides with v_ijk at
135 -- (i, j, k). See Sorokina and Zeilfelder, p. 76.
136 center :: Cube -> Point
140 x = fromIntegral (i cube) :: Double
141 y = fromIntegral (j cube) :: Double
142 z = fromIntegral (k cube) :: Double
147 -- | The top (in the direction of z) face of the cube.
148 top_face :: Cube -> Face.Face
149 top_face cube = Face.Face v0' v1' v2' v3'
151 delta = (1/2) :: Double
153 v0' = cc + ( Point delta (-delta) delta )
154 v1' = cc + ( Point delta delta delta )
155 v2' = cc + ( Point (-delta) delta delta )
156 v3' = cc + ( Point (-delta) (-delta) delta )
160 -- | The back (in the direction of x) face of the cube.
161 back_face :: Cube -> Face.Face
162 back_face cube = Face.Face v0' v1' v2' v3'
164 delta = (1/2) :: Double
166 v0' = cc + ( Point delta (-delta) (-delta) )
167 v1' = cc + ( Point delta delta (-delta) )
168 v2' = cc + ( Point delta delta delta )
169 v3' = cc + ( Point delta (-delta) delta )
172 -- The bottom face (in the direction of -z) of the cube.
173 down_face :: Cube -> Face.Face
174 down_face cube = Face.Face v0' v1' v2' v3'
176 delta = (1/2) :: Double
178 v0' = cc + ( Point (-delta) (-delta) (-delta) )
179 v1' = cc + ( Point (-delta) delta (-delta) )
180 v2' = cc + ( Point delta delta (-delta) )
181 v3' = cc + ( Point delta (-delta) (-delta) )
185 -- | The front (in the direction of -x) face of the cube.
186 front_face :: Cube -> Face.Face
187 front_face cube = Face.Face v0' v1' v2' v3'
189 delta = (1/2) :: Double
191 v0' = cc + ( Point (-delta) (-delta) delta )
192 v1' = cc + ( Point (-delta) delta delta )
193 v2' = cc + ( Point (-delta) delta (-delta) )
194 v3' = cc + ( Point (-delta) (-delta) (-delta) )
196 -- | The left (in the direction of -y) face of the cube.
197 left_face :: Cube -> Face.Face
198 left_face cube = Face.Face v0' v1' v2' v3'
200 delta = (1/2) :: Double
202 v0' = cc + ( Point delta (-delta) delta )
203 v1' = cc + ( Point (-delta) (-delta) delta )
204 v2' = cc + ( Point (-delta) (-delta) (-delta) )
205 v3' = cc + ( Point delta (-delta) (-delta) )
208 -- | The right (in the direction of y) face of the cube.
209 right_face :: Cube -> Face.Face
210 right_face cube = Face.Face v0' v1' v2' v3'
212 delta = (1/2) :: Double
214 v0' = cc + ( Point (-delta) delta delta)
215 v1' = cc + ( Point delta delta delta )
216 v2' = cc + ( Point delta delta (-delta) )
217 v3' = cc + ( Point (-delta) delta (-delta) )
220 tetrahedron :: Cube -> Int -> Tetrahedron
223 Tetrahedron (fv cube) v0' v1' v2' v3' vol
230 vol = tetrahedra_volume cube
233 Tetrahedron fv' v0' v1' v2' v3' vol
240 fv' = rotate ccwx (fv cube)
241 vol = tetrahedra_volume cube
244 Tetrahedron fv' v0' v1' v2' v3' vol
251 fv' = rotate ccwx $ rotate ccwx $ fv cube
252 vol = tetrahedra_volume cube
255 Tetrahedron fv' v0' v1' v2' v3' vol
262 fv' = rotate cwx (fv cube)
263 vol = tetrahedra_volume cube
266 Tetrahedron fv' v0' v1' v2' v3' vol
273 fv' = rotate cwy (fv cube)
274 vol = tetrahedra_volume cube
277 Tetrahedron fv' v0' v1' v2' v3' vol
284 fv' = rotate cwy $ rotate cwz $ fv cube
285 vol = tetrahedra_volume cube
288 Tetrahedron fv' v0' v1' v2' v3' vol
295 fv' = rotate cwy $ rotate cwz
298 vol = tetrahedra_volume cube
301 Tetrahedron fv' v0' v1' v2' v3' vol
308 fv' = rotate cwy $ rotate ccwz $ fv cube
309 vol = tetrahedra_volume cube
312 Tetrahedron fv' v0' v1' v2' v3' vol
319 fv' = rotate cwy $ rotate cwy $ fv cube
320 vol = tetrahedra_volume cube
323 Tetrahedron fv' v0' v1' v2' v3' vol
330 fv' = rotate cwy $ rotate cwy
333 vol = tetrahedra_volume cube
335 tetrahedron cube 10 =
336 Tetrahedron fv' v0' v1' v2' v3' vol
343 fv' = rotate cwy $ rotate cwy
348 vol = tetrahedra_volume cube
350 tetrahedron cube 11 =
351 Tetrahedron fv' v0' v1' v2' v3' vol
358 fv' = rotate cwy $ rotate cwy
361 vol = tetrahedra_volume cube
363 tetrahedron cube 12 =
364 Tetrahedron fv' v0' v1' v2' v3' vol
371 fv' = rotate ccwy $ fv cube
372 vol = tetrahedra_volume cube
374 tetrahedron cube 13 =
375 Tetrahedron fv' v0' v1' v2' v3' vol
382 fv' = rotate ccwy $ rotate ccwz $ fv cube
383 vol = tetrahedra_volume cube
385 tetrahedron cube 14 =
386 Tetrahedron fv' v0' v1' v2' v3' vol
393 fv' = rotate ccwy $ rotate ccwz
396 vol = tetrahedra_volume cube
398 tetrahedron cube 15 =
399 Tetrahedron fv' v0' v1' v2' v3' vol
406 fv' = rotate ccwy $ rotate cwz $ fv cube
407 vol = tetrahedra_volume cube
409 tetrahedron cube 16 =
410 Tetrahedron fv' v0' v1' v2' v3' vol
417 fv' = rotate ccwz $ fv cube
418 vol = tetrahedra_volume cube
420 tetrahedron cube 17 =
421 Tetrahedron fv' v0' v1' v2' v3' vol
428 fv' = rotate ccwz $ rotate cwy $ fv cube
429 vol = tetrahedra_volume cube
431 tetrahedron cube 18 =
432 Tetrahedron fv' v0' v1' v2' v3' vol
439 fv' = rotate ccwz $ rotate cwy
442 vol = tetrahedra_volume cube
444 tetrahedron cube 19 =
445 Tetrahedron fv' v0' v1' v2' v3' vol
452 fv' = rotate ccwz $ rotate ccwy
454 vol = tetrahedra_volume cube
456 tetrahedron cube 20 =
457 Tetrahedron fv' v0' v1' v2' v3' vol
464 fv' = rotate cwz $ fv cube
465 vol = tetrahedra_volume cube
467 tetrahedron cube 21 =
468 Tetrahedron fv' v0' v1' v2' v3' vol
475 fv' = rotate cwz $ rotate ccwy $ fv cube
476 vol = tetrahedra_volume cube
478 tetrahedron cube 22 =
479 Tetrahedron fv' v0' v1' v2' v3' vol
486 fv' = rotate cwz $ rotate ccwy
489 vol = tetrahedra_volume cube
491 tetrahedron cube 23 =
492 Tetrahedron fv' v0' v1' v2' v3' vol
499 fv' = rotate cwz $ rotate cwy
501 vol = tetrahedra_volume cube
504 -- Only used in tests, so we don't need the added speed
506 tetrahedra :: Cube -> [Tetrahedron]
507 tetrahedra cube = [ tetrahedron cube n | n <- [0..23] ]
509 front_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
510 front_left_top_tetrahedra cube =
511 V.singleton (tetrahedron cube 0) `V.snoc`
512 (tetrahedron cube 3) `V.snoc`
513 (tetrahedron cube 6) `V.snoc`
514 (tetrahedron cube 7) `V.snoc`
515 (tetrahedron cube 20) `V.snoc`
516 (tetrahedron cube 21)
518 front_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
519 front_left_down_tetrahedra cube =
520 V.singleton (tetrahedron cube 0) `V.snoc`
521 (tetrahedron cube 2) `V.snoc`
522 (tetrahedron cube 3) `V.snoc`
523 (tetrahedron cube 12) `V.snoc`
524 (tetrahedron cube 15) `V.snoc`
525 (tetrahedron cube 21)
527 front_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
528 front_right_top_tetrahedra cube =
529 V.singleton (tetrahedron cube 0) `V.snoc`
530 (tetrahedron cube 1) `V.snoc`
531 (tetrahedron cube 5) `V.snoc`
532 (tetrahedron cube 6) `V.snoc`
533 (tetrahedron cube 16) `V.snoc`
534 (tetrahedron cube 19)
536 front_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
537 front_right_down_tetrahedra cube =
538 V.singleton (tetrahedron cube 1) `V.snoc`
539 (tetrahedron cube 2) `V.snoc`
540 (tetrahedron cube 12) `V.snoc`
541 (tetrahedron cube 13) `V.snoc`
542 (tetrahedron cube 18) `V.snoc`
543 (tetrahedron cube 19)
545 back_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
546 back_left_top_tetrahedra cube =
547 V.singleton (tetrahedron cube 0) `V.snoc`
548 (tetrahedron cube 3) `V.snoc`
549 (tetrahedron cube 6) `V.snoc`
550 (tetrahedron cube 7) `V.snoc`
551 (tetrahedron cube 20) `V.snoc`
552 (tetrahedron cube 21)
554 back_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
555 back_left_down_tetrahedra cube =
556 V.singleton (tetrahedron cube 8) `V.snoc`
557 (tetrahedron cube 11) `V.snoc`
558 (tetrahedron cube 14) `V.snoc`
559 (tetrahedron cube 15) `V.snoc`
560 (tetrahedron cube 22) `V.snoc`
561 (tetrahedron cube 23)
563 back_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
564 back_right_top_tetrahedra cube =
565 V.singleton (tetrahedron cube 4) `V.snoc`
566 (tetrahedron cube 5) `V.snoc`
567 (tetrahedron cube 9) `V.snoc`
568 (tetrahedron cube 10) `V.snoc`
569 (tetrahedron cube 16) `V.snoc`
570 (tetrahedron cube 17)
572 back_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
573 back_right_down_tetrahedra cube =
574 V.singleton (tetrahedron cube 8) `V.snoc`
575 (tetrahedron cube 9) `V.snoc`
576 (tetrahedron cube 13) `V.snoc`
577 (tetrahedron cube 14) `V.snoc`
578 (tetrahedron cube 17) `V.snoc`
579 (tetrahedron cube 18)
581 in_top_half :: Cube -> Point -> Bool
582 in_top_half cube (Point _ _ z) =
583 distance_from_top <= distance_from_bottom
585 distance_from_top = abs $ (zmax cube) - z
586 distance_from_bottom = abs $ (zmin cube) - z
588 in_front_half :: Cube -> Point -> Bool
589 in_front_half cube (Point x _ _) =
590 distance_from_front <= distance_from_back
592 distance_from_front = abs $ (xmin cube) - x
593 distance_from_back = abs $ (xmax cube) - x
596 in_left_half :: Cube -> Point -> Bool
597 in_left_half cube (Point _ y _) =
598 distance_from_left <= distance_from_right
600 distance_from_left = abs $ (ymin cube) - y
601 distance_from_right = abs $ (ymax cube) - y
604 -- | Takes a 'Cube', and returns the Tetrahedra belonging to it that
605 -- contain the given 'Point'. This should be faster than checking
606 -- every tetrahedron individually, since we determine which half
607 -- (hemisphere?) of the cube the point lies in three times: once in
608 -- each dimension. This allows us to eliminate non-candidates
611 -- This can throw an exception, but the use of 'head' might
612 -- save us some unnecessary computations.
614 {-# INLINE find_containing_tetrahedron #-}
615 find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
616 find_containing_tetrahedron cube p =
617 candidates `V.unsafeIndex` (fromJust lucky_idx)
619 front_half = in_front_half cube p
620 top_half = in_top_half cube p
621 left_half = in_left_half cube p
623 candidates :: V.Vector Tetrahedron
628 front_left_top_tetrahedra cube
630 front_left_down_tetrahedra cube
633 front_right_top_tetrahedra cube
635 front_right_down_tetrahedra cube
637 | otherwise = -- back half
640 back_left_top_tetrahedra cube
642 back_left_down_tetrahedra cube
645 back_right_top_tetrahedra cube
647 back_right_down_tetrahedra cube
649 -- Use the dot product instead of Euclidean distance here to save
650 -- a sqrt(). So, "distances" below really means "distances
652 distances :: V.Vector Double
653 distances = V.map ((dot p) . barycenter) candidates
655 shortest_distance :: Double
656 shortest_distance = V.minimum distances
658 -- Compute the index of the tetrahedron with the center closest to
659 -- p. This is a bad algorithm, but don't change it! If you make it
660 -- smarter by finding the index of shortest_distance in distances
661 -- (this should give the same answer and avoids recomputing the
662 -- dot product), the program gets slower. Seriously!
663 lucky_idx :: Maybe Int
664 lucky_idx = V.findIndex
665 (\t -> (barycenter t) `dot` p == shortest_distance)
675 prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
676 prop_opposite_octant_tetrahedra_disjoint1 cube =
677 disjoint (front_left_top_tetrahedra cube) (front_right_down_tetrahedra cube)
679 prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
680 prop_opposite_octant_tetrahedra_disjoint2 cube =
681 disjoint (back_left_top_tetrahedra cube) (back_right_down_tetrahedra cube)
683 prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
684 prop_opposite_octant_tetrahedra_disjoint3 cube =
685 disjoint (front_left_top_tetrahedra cube) (back_right_top_tetrahedra cube)
687 prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
688 prop_opposite_octant_tetrahedra_disjoint4 cube =
689 disjoint (front_left_down_tetrahedra cube) (back_right_down_tetrahedra cube)
691 prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
692 prop_opposite_octant_tetrahedra_disjoint5 cube =
693 disjoint (front_left_top_tetrahedra cube) (back_left_down_tetrahedra cube)
695 prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
696 prop_opposite_octant_tetrahedra_disjoint6 cube =
697 disjoint (front_right_top_tetrahedra cube) (back_right_down_tetrahedra cube)
700 -- | Since the grid size is necessarily positive, all tetrahedra
701 -- (which comprise cubes of positive volume) must have positive
703 prop_all_volumes_positive :: Cube -> Bool
704 prop_all_volumes_positive cube =
708 volumes = map volume ts
711 -- | In fact, since all of the tetrahedra are identical, we should
712 -- already know their volumes. There's 24 tetrahedra to a cube, so
713 -- we'd expect the volume of each one to be 1/24.
714 prop_all_volumes_exact :: Cube -> Bool
715 prop_all_volumes_exact cube =
716 and [volume t ~~= 1/24 | t <- tetrahedra cube]
718 -- | All tetrahedron should have their v0 located at the center of the
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 :: TestTree
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 :: TestTree
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 :: TestTree
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 :: TestTree
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 :: TestTree
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 ]