+
+
+-- | Rotate a cardinal direction counter-clockwise about the z-axis.
+ccwz :: Cardinal -> Cardinal
+ccwz F = R
+ccwz B = L
+ccwz L = F
+ccwz R = B
+ccwz D = D
+ccwz T = T
+ccwz FL = FR
+ccwz FR = BR
+ccwz FD = RD
+ccwz FT = RT
+ccwz BL = FL
+ccwz BR = BL
+ccwz BD = LD
+ccwz BT = LT
+ccwz LD = FD
+ccwz LT = FT
+ccwz RD = BD
+ccwz RT = BT
+ccwz FLD = FRD
+ccwz FLT = FRT
+ccwz FRD = BRD
+ccwz FRT = BRT
+ccwz BLD = FLD
+ccwz BLT = FLT
+ccwz BRD = BLD
+ccwz BRT = BLT
+ccwz I = I
+ccwz (Scalar s) = (Scalar s)
+ccwz (Sum c0 c1) = Sum (ccwz c0) (ccwz c1)
+ccwz (Difference c0 c1) = Difference (ccwz c0) (ccwz c1)
+ccwz (Product c0 c1) = Product (ccwz c0) (ccwz c1)
+ccwz (Quotient c0 c1) = Quotient (ccwz c0) (ccwz c1)
+
+-- | Rotate a cardinal direction clockwise about the z-axis.
+cwz :: Cardinal -> Cardinal
+cwz = ccwz . ccwz . ccwz
+
+
+
+
+-- | We know what (c t6 2 1 0 0) should be from Sorokina and
+-- Zeilfelder, p. 87. This test checks that the directions are
+-- rotated properly. The order of the letters has to be just right
+-- since I haven't defined a proper Eq instance for Cardinals.
+test_c_tilde_2100_rotation_correct :: Assertion
+test_c_tilde_2100_rotation_correct =
+ assertEqual "auto-rotate equals manual rotate" ((ccwz . ccwz . cwy) expr1) expr2
+ where
+ expr1 =
+ (3/8)*I +
+ (1/12)*(T + R + L + D) +
+ (1/64)*(FT + FR + FL + FD) +
+ (7/48)*F +
+ (1/48)*B +
+ (1/96)*(RT + LD + LT + RD) +
+ (1/192)*(BT + BR + BL + BD)
+
+ expr2 =
+ (3/8)*I +
+ (1/12)*(F + L + R + B) +
+ (1/64)*(FT + LT + RT + BT) +
+ (7/48)*T +
+ (1/48)*D +
+ (1/96)*(FL + BR + FR + BL) +
+ (1/192)*(FD + LD + RD + BD)
+
+-- | A list of all directions, sans the interior and composite types.
+all_directions :: [Cardinal]
+all_directions = [L, R, F, B, D, T, FL, FR, FD, FT,
+ BL, BR, BD, BT, LD, LT, RD, RT, FLD,
+ FLT, FRD, FRT, BLD, BLT, BRD, BRT]
+
+
+-- | If we rotate a direction (other than front or back)
+-- counter-clockwise with respect to the x-axis, we should get a new
+-- direction.
+prop_ccwx_rotation_changes_direction :: Cardinal -> Property
+prop_ccwx_rotation_changes_direction c =
+ c `elem` [L, R, D, T, FL, FR, FD, FT, BL, BR, BD, BT, LD, LT,
+ RD, RT, FLD, FLT, FRD, FRT, BLD, BLT, BRD, BRT]
+ ==> ccwx c /= c
+
+-- | If we rotate a direction (other than front or back) clockwise
+-- with respect to the x-axis, we should get a new direction.
+prop_cwx_rotation_changes_direction :: Cardinal -> Property
+prop_cwx_rotation_changes_direction c =
+ -- The front and back faces are unchanged by x-rotation.
+ c `elem` [L, R, D, T, FL, FR, FD, FT, BL, BR, BD, BT, LD, LT,
+ RD, RT, FLD, FLT, FRD, FRT, BLD, BLT, BRD, BRT]
+ ==> cwx c /= c
+
+-- | If we rotate a direction (other than left or right)
+-- counter-clockwise with respect to the y-axis, we should get a new
+-- direction.
+prop_ccwy_rotation_changes_direction :: Cardinal -> Property
+prop_ccwy_rotation_changes_direction c =
+ c `elem` [F, B, D, T, FL, FR, FD, FT, BL, BR, BD, BT, LD, LT,
+ RD, RT, FLD, FLT, FRD, FRT, BLD, BLT, BRD, BRT]
+ ==> ccwy c /= c
+
+
+-- | If we rotate a direction (other than left or right) clockwise
+-- with respect to the y-axis, we should get a new direction.
+prop_cwy_rotation_changes_direction :: Cardinal -> Property
+prop_cwy_rotation_changes_direction c =
+ c `elem` [F, B, D, T, FL, FR, FD, FT, BL, BR, BD, BT, LD, LT,
+ RD, RT, FLD, FLT, FRD, FRT, BLD, BLT, BRD, BRT]
+ ==> cwy c /= c
+
+
+-- | If we rotate a direction (other than top or down)
+-- counter-clockwise with respect to the z-axis, we should get a new
+-- direction.
+prop_ccwz_rotation_changes_direction :: Cardinal -> Property
+prop_ccwz_rotation_changes_direction c =
+ c `elem` [L, R, F, B, FL, FR, FD, FT, BL, BR, BD, BT, LD, LT,
+ RD, RT, FLD, FLT, FRD, FRT, BLD, BLT, BRD, BRT]
+ ==> ccwz c /= c
+
+
+-- | If we rotate a direction (other than top or down) clockwise with
+-- respect to the z-axis, we should get a new direction.
+prop_cwz_rotation_changes_direction :: Cardinal -> Property
+prop_cwz_rotation_changes_direction c =
+ c `elem` [L, R, F, B, FL, FR, FD, FT, BL, BR, BD, BT, LD, LT,
+ RD, RT, FLD, FLT, FRD, FRT, BLD, BLT, BRD, BRT]
+ ==> cwz c /= c
+
+
+-- | If we are given a direction c, there should only be one direction
+-- d which, when rotated counter-clockwise with respect to the
+-- x-axis, produces c.
+prop_ccwx_rotation_result_unique :: Cardinal -> Property
+prop_ccwx_rotation_result_unique c =
+ c `elem` all_directions ==>
+ (length [ d | d <- all_directions, ccwx d == c ]) == 1
+
+-- | If we are given a direction c, there should only be one direction
+-- d which, when rotated clockwise with respect to the x-axis,
+-- produces c.
+prop_cwx_rotation_result_unique :: Cardinal -> Property
+prop_cwx_rotation_result_unique c =
+ c `elem` all_directions ==>
+ (length [ d | d <- all_directions, cwx d == c ]) == 1
+
+
+-- | If we are given a direction c, there should only be one direction
+-- d which, when rotated counter-clockwise with respect to the
+-- y-axis, produces c.
+prop_ccwy_rotation_result_unique :: Cardinal -> Property
+prop_ccwy_rotation_result_unique c =
+ c `elem` all_directions ==>
+ (length [ d | d <- all_directions, ccwy d == c ]) == 1
+
+
+-- | If we are given a direction c, there should only be one direction
+-- d which, when rotated clockwise with respect to the y-axis,
+-- produces c.
+prop_cwy_rotation_result_unique :: Cardinal -> Property
+prop_cwy_rotation_result_unique c =
+ c `elem` all_directions ==>
+ (length [ d | d <- all_directions, cwy d == c ]) == 1
+
+
+-- | If we are given a direction c, there should only be one direction
+-- d which, when rotated counter-clockwise with respect to the
+-- z-axis, produces c.
+prop_ccwz_rotation_result_unique :: Cardinal -> Property
+prop_ccwz_rotation_result_unique c =
+ c `elem` all_directions ==>
+ (length [ d | d <- all_directions, ccwz d == c ]) == 1
+
+
+-- | If we are given a direction c, there should only be one direction
+-- d which, when rotated clockwise with respect to the z-axis,
+-- produces c.
+prop_cwz_rotation_result_unique :: Cardinal -> Property
+prop_cwz_rotation_result_unique c =
+ c `elem` all_directions ==>
+ (length [ d | d <- all_directions, cwz d == c ]) == 1
+
+
+-- | If you rotate a cardinal direction four times in the clockwise
+-- (with respect to x) direction, you should wind up with the same
+-- direction.
+prop_four_cwx_is_identity :: Cardinal -> Bool
+prop_four_cwx_is_identity c =
+ (cwx . cwx . cwx . cwx) c == c
+
+-- | If you rotate a cardinal direction four times in the
+-- counter-clockwise (with respect to x) direction, you should wind up
+-- with the same direction.
+prop_four_ccwx_is_identity :: Cardinal -> Bool
+prop_four_ccwx_is_identity c =
+ (ccwx . ccwx . ccwx . ccwx) c == c
+
+-- | If you rotate a cardinal direction four times in the clockwise
+-- (with respect to y) direction, you should wind up with the same
+-- direction.
+prop_four_cwy_is_identity :: Cardinal -> Bool
+prop_four_cwy_is_identity c =
+ (cwy . cwy . cwy . cwy) c == c
+
+-- | If you rotate a cardinal direction four times in the counter-clockwise
+-- (with respect to y) direction, you should wind up with the same
+-- direction.
+prop_four_ccwy_is_identity :: Cardinal -> Bool
+prop_four_ccwy_is_identity c =
+ (ccwy . ccwy . ccwy . ccwy) c == c
+
+-- | If you rotate a cardinal direction four times in the clockwise
+-- (with respect to z) direction, you should wind up with the same
+-- direction.
+prop_four_cwz_is_identity :: Cardinal -> Bool
+prop_four_cwz_is_identity c =
+ (cwz . cwz . cwz . cwz) c == c
+
+-- | If you rotate a cardinal direction four times in the
+-- counter-clockwise (with respect to z) direction, you should wind up
+-- with the same direction.
+prop_four_ccwz_is_identity :: Cardinal -> Bool
+prop_four_ccwz_is_identity c =
+ (ccwz . ccwz . ccwz . ccwz) c == c
+
+
+cardinal_tests :: Test.Framework.Test
+cardinal_tests =
+ testGroup "Cardinal Tests" [
+ testCase "c-tilde_2100 rotation correct"test_c_tilde_2100_rotation_correct ]
+
+
+cardinal_properties :: Test.Framework.Test
+cardinal_properties =
+ let tp = testProperty
+ in
+ testGroup "Cardinal Properties" [
+ tp "ccwx rotation changes direction" prop_ccwx_rotation_changes_direction,
+ tp "cwx rotation changes direction" prop_cwx_rotation_changes_direction,
+ tp "ccwy rotation changes direction" prop_ccwy_rotation_changes_direction,
+ tp "cwy rotation changes direction" prop_cwy_rotation_changes_direction,
+ tp "ccwz rotation changes direction" prop_ccwz_rotation_changes_direction,
+ tp "cwz rotation changes direction" prop_cwz_rotation_changes_direction,
+ tp "ccwx rotation result unique" prop_ccwx_rotation_result_unique,
+ tp "cwx rotation result unique" prop_cwx_rotation_result_unique,
+ tp "ccwy rotation result unique" prop_ccwy_rotation_result_unique,
+ tp "cwy rotation result unique" prop_cwy_rotation_result_unique,
+ tp "ccwz rotation result unique" prop_ccwz_rotation_result_unique,
+ tp "cwz rotation result unique" prop_cwz_rotation_result_unique,
+ tp "four cwx is identity" prop_four_cwx_is_identity,
+ tp "four ccwx is identity" prop_four_ccwx_is_identity,
+ tp "four cwy is identity" prop_four_cwy_is_identity,
+ tp "four ccwy is identity" prop_four_ccwy_is_identity,
+ tp "four cwz is identity" prop_four_cwz_is_identity,
+ tp "four ccwz is identity" prop_four_ccwz_is_identity ]