+tetrahedron cube 0 =
+ Tetrahedron (fv cube) v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (front_face cube)
+ v2' = Face.v0 (front_face cube)
+ v3' = Face.v1 (front_face cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 1 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (front_face cube)
+ v2' = Face.v1 (front_face cube)
+ v3' = Face.v2 (front_face cube)
+ fv' = rotate ccwx (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 2 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (front_face cube)
+ v2' = Face.v2 (front_face cube)
+ v3' = Face.v3 (front_face cube)
+ fv' = rotate ccwx $ rotate ccwx $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 3 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (front_face cube)
+ v2' = Face.v3 (front_face cube)
+ v3' = Face.v0 (front_face cube)
+ fv' = rotate cwx (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 4 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (top_face cube)
+ v2' = Face.v0 (top_face cube)
+ v3' = Face.v1 (top_face cube)
+ fv' = rotate cwy (fv cube)
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 5 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (top_face cube)
+ v2' = Face.v1 (top_face cube)
+ v3' = Face.v2 (top_face cube)
+ fv' = rotate cwy $ rotate cwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 6 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (top_face cube)
+ v2' = Face.v2 (top_face cube)
+ v3' = Face.v3 (top_face cube)
+ fv' = rotate cwy $ rotate cwz
+ $ rotate cwz
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 7 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (top_face cube)
+ v2' = Face.v3 (top_face cube)
+ v3' = Face.v0 (top_face cube)
+ fv' = rotate cwy $ rotate ccwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 8 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (back_face cube)
+ v2' = Face.v0 (back_face cube)
+ v3' = Face.v1 (back_face cube)
+ fv' = rotate cwy $ rotate cwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 9 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (back_face cube)
+ v2' = Face.v1 (back_face cube)
+ v3' = Face.v2 (back_face cube)
+ fv' = rotate cwy $ rotate cwy
+ $ rotate cwx
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 10 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (back_face cube)
+ v2' = Face.v2 (back_face cube)
+ v3' = Face.v3 (back_face cube)
+ fv' = rotate cwy $ rotate cwy
+ $ rotate cwx
+ $ rotate cwx
+ $ fv cube
+
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 11 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (back_face cube)
+ v2' = Face.v3 (back_face cube)
+ v3' = Face.v0 (back_face cube)
+ fv' = rotate cwy $ rotate cwy
+ $ rotate ccwx
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 12 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (down_face cube)
+ v2' = Face.v0 (down_face cube)
+ v3' = Face.v1 (down_face cube)
+ fv' = rotate ccwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 13 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (down_face cube)
+ v2' = Face.v1 (down_face cube)
+ v3' = Face.v2 (down_face cube)
+ fv' = rotate ccwy $ rotate ccwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 14 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (down_face cube)
+ v2' = Face.v2 (down_face cube)
+ v3' = Face.v3 (down_face cube)
+ fv' = rotate ccwy $ rotate ccwz
+ $ rotate ccwz
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 15 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (down_face cube)
+ v2' = Face.v3 (down_face cube)
+ v3' = Face.v0 (down_face cube)
+ fv' = rotate ccwy $ rotate cwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 16 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (right_face cube)
+ v2' = Face.v0 (right_face cube)
+ v3' = Face.v1 (right_face cube)
+ fv' = rotate ccwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 17 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (right_face cube)
+ v2' = Face.v1 (right_face cube)
+ v3' = Face.v2 (right_face cube)
+ fv' = rotate ccwz $ rotate cwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 18 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (right_face cube)
+ v2' = Face.v2 (right_face cube)
+ v3' = Face.v3 (right_face cube)
+ fv' = rotate ccwz $ rotate cwy
+ $ rotate cwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 19 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (right_face cube)
+ v2' = Face.v3 (right_face cube)
+ v3' = Face.v0 (right_face cube)
+ fv' = rotate ccwz $ rotate ccwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 20 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (left_face cube)
+ v2' = Face.v0 (left_face cube)
+ v3' = Face.v1 (left_face cube)
+ fv' = rotate cwz $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 21 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (left_face cube)
+ v2' = Face.v1 (left_face cube)
+ v3' = Face.v2 (left_face cube)
+ fv' = rotate cwz $ rotate ccwy $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 22 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (left_face cube)
+ v2' = Face.v2 (left_face cube)
+ v3' = Face.v3 (left_face cube)
+ fv' = rotate cwz $ rotate ccwy
+ $ rotate ccwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+tetrahedron cube 23 =
+ Tetrahedron fv' v0' v1' v2' v3' vol
+ where
+ v0' = center cube
+ v1' = center (left_face cube)
+ v2' = Face.v3 (left_face cube)
+ v3' = Face.v0 (left_face cube)
+ fv' = rotate cwz $ rotate cwy
+ $ fv cube
+ vol = tetrahedra_volume cube
+
+-- Feels dirty, but whatever.
+tetrahedron _ _ = error "asked for a nonexistent tetrahedron"
+
+
+-- Only used in tests, so we don't need the added speed
+-- of Data.Vector.
+tetrahedra :: Cube -> [Tetrahedron]
+tetrahedra cube = [ tetrahedron cube n | n <- [0..23] ]
+
+front_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_left_top_tetrahedra cube =
+ V.singleton (tetrahedron cube 0) `V.snoc`
+ (tetrahedron cube 3) `V.snoc`
+ (tetrahedron cube 6) `V.snoc`
+ (tetrahedron cube 7) `V.snoc`
+ (tetrahedron cube 20) `V.snoc`
+ (tetrahedron cube 21)
+
+front_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_left_down_tetrahedra cube =
+ V.singleton (tetrahedron cube 0) `V.snoc`
+ (tetrahedron cube 2) `V.snoc`
+ (tetrahedron cube 3) `V.snoc`
+ (tetrahedron cube 12) `V.snoc`
+ (tetrahedron cube 15) `V.snoc`
+ (tetrahedron cube 21)
+
+front_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_right_top_tetrahedra cube =
+ V.singleton (tetrahedron cube 0) `V.snoc`
+ (tetrahedron cube 1) `V.snoc`
+ (tetrahedron cube 5) `V.snoc`
+ (tetrahedron cube 6) `V.snoc`
+ (tetrahedron cube 16) `V.snoc`
+ (tetrahedron cube 19)
+
+front_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+front_right_down_tetrahedra cube =
+ V.singleton (tetrahedron cube 1) `V.snoc`
+ (tetrahedron cube 2) `V.snoc`
+ (tetrahedron cube 12) `V.snoc`
+ (tetrahedron cube 13) `V.snoc`
+ (tetrahedron cube 18) `V.snoc`
+ (tetrahedron cube 19)
+
+back_left_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_left_top_tetrahedra cube =
+ V.singleton (tetrahedron cube 0) `V.snoc`
+ (tetrahedron cube 3) `V.snoc`
+ (tetrahedron cube 6) `V.snoc`
+ (tetrahedron cube 7) `V.snoc`
+ (tetrahedron cube 20) `V.snoc`
+ (tetrahedron cube 21)
+
+back_left_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_left_down_tetrahedra cube =
+ V.singleton (tetrahedron cube 8) `V.snoc`
+ (tetrahedron cube 11) `V.snoc`
+ (tetrahedron cube 14) `V.snoc`
+ (tetrahedron cube 15) `V.snoc`
+ (tetrahedron cube 22) `V.snoc`
+ (tetrahedron cube 23)
+
+back_right_top_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_right_top_tetrahedra cube =
+ V.singleton (tetrahedron cube 4) `V.snoc`
+ (tetrahedron cube 5) `V.snoc`
+ (tetrahedron cube 9) `V.snoc`
+ (tetrahedron cube 10) `V.snoc`
+ (tetrahedron cube 16) `V.snoc`
+ (tetrahedron cube 17)
+
+back_right_down_tetrahedra :: Cube -> V.Vector Tetrahedron
+back_right_down_tetrahedra cube =
+ V.singleton (tetrahedron cube 8) `V.snoc`
+ (tetrahedron cube 9) `V.snoc`
+ (tetrahedron cube 13) `V.snoc`
+ (tetrahedron cube 14) `V.snoc`
+ (tetrahedron cube 17) `V.snoc`
+ (tetrahedron cube 18)
+
+in_top_half :: Cube -> Point -> Bool
+in_top_half cube (_,_,z) =
+ distance_from_top <= distance_from_bottom
+ where
+ distance_from_top = abs $ (zmax cube) - z
+ distance_from_bottom = abs $ (zmin cube) - z
+
+in_front_half :: Cube -> Point -> Bool
+in_front_half cube (x,_,_) =
+ distance_from_front <= distance_from_back
+ where
+ distance_from_front = abs $ (xmin cube) - x
+ distance_from_back = abs $ (xmax cube) - x
+
+
+in_left_half :: Cube -> Point -> Bool
+in_left_half cube (_,y,_) =
+ distance_from_left <= distance_from_right
+ where
+ distance_from_left = abs $ (ymin cube) - y
+ distance_from_right = abs $ (ymax cube) - y
+
+
+-- | Takes a 'Cube', and returns the Tetrahedra belonging to it that
+-- contain the given 'Point'. This should be faster than checking
+-- every tetrahedron individually, since we determine which half
+-- (hemisphere?) of the cube the point lies in three times: once in
+-- each dimension. This allows us to eliminate non-candidates
+-- quickly.
+--
+-- This can throw an exception, but the use of 'head' might
+-- save us some unnecessary computations.
+--
+find_containing_tetrahedron :: Cube -> Point -> Tetrahedron
+find_containing_tetrahedron cube p =
+ candidates `V.unsafeIndex` (fromJust lucky_idx)
+ where
+ front_half = in_front_half cube p
+ top_half = in_top_half cube p
+ left_half = in_left_half cube p
+
+ candidates =
+ if front_half then
+
+ if left_half then
+ if top_half then
+ front_left_top_tetrahedra cube
+ else
+ front_left_down_tetrahedra cube
+ else
+ if top_half then
+ front_right_top_tetrahedra cube
+ else
+ front_right_down_tetrahedra cube
+
+ else -- bottom half
+
+ if left_half then
+ if top_half then
+ back_left_top_tetrahedra cube
+ else
+ back_left_down_tetrahedra cube
+ else
+ if top_half then
+ back_right_top_tetrahedra cube
+ else
+ back_right_down_tetrahedra cube
+
+ -- Use the dot product instead of 'distance' here to save a
+ -- sqrt(). So, "distances" below really means "distances squared."
+ distances = V.map ((dot p) . center) candidates
+ shortest_distance = V.minimum distances
+ lucky_idx = V.findIndex
+ (\t -> (center t) `dot` p == shortest_distance)
+ candidates
+
+
+
+
+
+
+-- Tests
+
+-- Quickcheck tests.
+
+prop_opposite_octant_tetrahedra_disjoint1 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint1 cube =
+ disjoint (front_left_top_tetrahedra cube) (front_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint2 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint2 cube =
+ disjoint (back_left_top_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint3 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint3 cube =
+ disjoint (front_left_top_tetrahedra cube) (back_right_top_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint4 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint4 cube =
+ disjoint (front_left_down_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint5 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint5 cube =
+ disjoint (front_left_top_tetrahedra cube) (back_left_down_tetrahedra cube)
+
+prop_opposite_octant_tetrahedra_disjoint6 :: Cube -> Bool
+prop_opposite_octant_tetrahedra_disjoint6 cube =
+ disjoint (front_right_top_tetrahedra cube) (back_right_down_tetrahedra cube)
+
+
+-- | Since the grid size is necessarily positive, all tetrahedra
+-- (which comprise cubes of positive volume) must have positive volume
+-- as well.
+prop_all_volumes_positive :: Cube -> Bool
+prop_all_volumes_positive cube =
+ null nonpositive_volumes
+ where
+ ts = tetrahedra cube
+ volumes = map volume ts
+ nonpositive_volumes = filter (<= 0) volumes
+
+-- | In fact, since all of the tetrahedra are identical, we should
+-- already know their volumes. There's 24 tetrahedra to a cube, so
+-- we'd expect the volume of each one to be (1/24)*h^3.
+prop_all_volumes_exact :: Cube -> Bool
+prop_all_volumes_exact cube =
+ and [volume t ~~= (1/24)*(delta^(3::Int)) | t <- tetrahedra cube]
+ where
+ delta = h cube