def testBladeOuterScaleMultplicative():
# multiplicative scaling
B1 = Blade(np.array([2, 0]))
B2 = Blade(np.array([0, 1]))
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
print 'B1 outer B2 blade: ', result1.blade
assert la.det(result1.blade) > 0, la.det(result1.blade)
assert result1.blade.shape == (2, 2), result1.blade.shape
assert np.allclose(np.eye(2), np.dot(result1.blade.T, result1.blade))
assert np.allclose(bd.inner(result1, result1).s,
-4), bd.inner(result1, result1).s
result2 = bd.outer(B2, B1)
print 'B2 outer B1 s: ', result2.s
print 'B2 outer B1 blade: ', result2.blade
assert la.det(result2.blade) < 0, la.det(result2.blade)
assert result2.blade.shape == (2, 2), result2.blade.shape
assert np.allclose(np.eye(2), np.dot(result2.blade.T, result2.blade))
assert np.allclose(bd.inner(result2, result2).s,
-4), bd.inner(result2, result2).s
# result1 == inverse(result2)
assert np.allclose(bd.inner(result1, result2).s,
4), bd.inner(result1, result2).s
def testBladeOuterScalar():
# scalar to blade/ scalar to scalar
B1 = Blade(np.eye(2))
B2 = Blade(1, s=2)
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
assert np.allclose(2, result1.s), 'value'
assert result1.blade.shape == (2, 2), 'shape'
result2 = bd.outer(B2, B1)
print 'B2 outer B1 s: ', result2.s
assert np.allclose(2, result2.s), 'value'
assert result2.blade.shape == (2, 2), 'shape'
result3 = bd.outer(B2, B2)
print 'B2 outer B2 s: ', result3.s
assert np.allclose(result3.s, 4)
B1 = Blade(1, s=2)
B2 = Blade(1, s=-.5)
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
assert np.allclose(-1, result1.s), 'value'
assert result1.blade.shape == (1, 0), 'shape'
def testBladeOuterScalar():
# scalar to blade/ scalar to scalar
B1 = Blade(np.eye(2))
B2 = Blade(1, s=2)
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
assert np.allclose(2, result1.s), 'value'
assert result1.blade.shape == (2, 2), 'shape'
result2 = bd.outer(B2, B1)
print 'B2 outer B1 s: ', result2.s
assert np.allclose(2, result2.s), 'value'
assert result2.blade.shape == (2, 2), 'shape'
result3 = bd.outer(B2, B2)
print 'B2 outer B2 s: ', result3.s
assert np.allclose(result3.s, 4)
B1 = Blade(1, s=2)
B2 = Blade(1, s=-.5)
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
assert np.allclose(-1, result1.s), 'value'
assert result1.blade.shape == (1, 0), 'shape'
def testBladeDualEuclidean():
# s=0
B = Blade(0)
try:
D = bd.dual(B)
except AttributeError as err:
assert err[0] == 'Not dualizable, s=0'
except:
assert False, err
# scalar, expect a 2-blade
A = Blade(2)
D = bd.dual(A, n=2)
revPseudo = bd.outer(bd.inverse(A), D)
shouldBeOne = bd.inner(revPseudo, bd.pseudoScalar(2))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
# 1-blade, expect 1-blade back
A = Blade(np.array([1, 0]))
D = bd.dual(A)
revPseudo = bd.outer(bd.inverse(A), D)
shouldBeOne = bd.inner(revPseudo, bd.pseudoScalar(2))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
# 2-blade, expect 0-blade back
A = Blade(np.eye(2))
D = bd.dual(A)
revPseudo = bd.outer(bd.inverse(A), D)
shouldBeOne = bd.inner(revPseudo, bd.pseudoScalar(2))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
def testBladeDualEuclidean():
# s=0
B = Blade(0)
try:
D = bd.dual(B)
except AttributeError as err:
assert err[0] == 'Not dualizable, s=0'
except:
assert False, err
# scalar, expect a 2-blade
A = Blade(2)
D = bd.dual(A, n=2)
revPseudo = bd.outer(bd.inverse(A), D)
shouldBeOne = bd.inner(revPseudo, bd.pseudoScalar(2))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
# 1-blade, expect 1-blade back
A = Blade(np.array([1, 0]))
D = bd.dual(A)
revPseudo = bd.outer(bd.inverse(A), D)
shouldBeOne = bd.inner(revPseudo, bd.pseudoScalar(2))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
# 2-blade, expect 0-blade back
A = Blade(np.eye(2))
D = bd.dual(A)
revPseudo = bd.outer(bd.inverse(A), D)
shouldBeOne = bd.inner(revPseudo, bd.pseudoScalar(2))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
def testBladeOuterSameSubspace():
B1 = Blade(np.eye(2))
B2 = Blade(np.eye(2)[:, 0])
result = bd.outer(B1, B2)
print 'B1 outer B2: ', result.s
assert np.allclose(0, result.s)
B1 = Blade(np.eye(2))
B2 = Blade(np.eye(2))
result = bd.outer(B1, B2)
print 'B1 outer B2: ', result.s
assert np.allclose(0, result.s)
def testBladeOuterSameSubspace():
B1 = Blade(np.eye(2))
B2 = Blade(np.eye(2)[:, 0])
result = bd.outer(B1, B2)
print 'B1 outer B2: ', result.s
assert np.allclose(0, result.s)
B1 = Blade(np.eye(2))
B2 = Blade(np.eye(2))
result = bd.outer(B1, B2)
print 'B1 outer B2: ', result.s
assert np.allclose(0, result.s)
def testBladeUnDualEuclidean():
# s=0
B = Blade(0)
try:
D = bd.undual(B)
except AttributeError as err:
assert err[0] == 'Not dualizable, s=0'
except:
assert False, err
# scalar, expect a 2-blade
A = Blade(2)
Aprime = bd.dual(bd.undual(A, n=2), n=2)
print 'A: ', A.blade, A.s
print 'APrimeInv: ', bd.inverse(Aprime).blade, bd.inverse(Aprime).s
shouldBeOne = bd.outer(A, bd.inverse(Aprime))
print 'shouldBeOne: ', shouldBeOne.blade, shouldBeOne.s
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
# 1-blade, expect 1-blade back
A = Blade(np.array([1, 0]))
Aprime = bd.undual(bd.dual(A, n=2), n=2)
shouldBeOne = bd.inner(A, bd.inverse(Aprime))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
# 2-blade, expect 0-blade back
A = Blade(np.eye(2))
Aprime = bd.undual(bd.dual(A, n=2), n=2)
print 'A: ', A.blade, A.s
print 'APrimeInv: ', bd.inverse(Aprime).blade, bd.inverse(Aprime).s
shouldBeOne = bd.inner(A, bd.inverse(Aprime))
print 'shouldBeOne: ', shouldBeOne.blade, shouldBeOne.s
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
def testBladeUnDualEuclidean():
# s=0
B = Blade(0)
try:
D = bd.undual(B)
except AttributeError as err:
assert err[0] == 'Not dualizable, s=0'
except:
assert False, err
# scalar, expect a 2-blade
A = Blade(2)
Aprime = bd.dual(bd.undual(A, n=2), n=2)
print 'A: ', A.blade, A.s
print 'APrimeInv: ', bd.inverse(Aprime).blade, bd.inverse(Aprime).s
shouldBeOne = bd.outer(A, bd.inverse(Aprime))
print 'shouldBeOne: ', shouldBeOne.blade, shouldBeOne.s
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
# 1-blade, expect 1-blade back
A = Blade(np.array([1, 0]))
Aprime = bd.undual(bd.dual(A, n=2), n=2)
shouldBeOne = bd.inner(A, bd.inverse(Aprime))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
# 2-blade, expect 0-blade back
A = Blade(np.eye(2))
Aprime = bd.undual(bd.dual(A, n=2), n=2)
print 'A: ', A.blade, A.s
print 'APrimeInv: ', bd.inverse(Aprime).blade, bd.inverse(Aprime).s
shouldBeOne = bd.inner(A, bd.inverse(Aprime))
print 'shouldBeOne: ', shouldBeOne.blade, shouldBeOne.s
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
def testBladeOuterScaleParallelopiped():
# test outer(B1, B2).s == 'area spanned by parallelopiped between B1, B2'
B1 = Blade(np.array([1, 0]))
B2 = Blade(np.array([1, 1])) # should have same area as [0, 1] when outered with B1
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
print 'B1 outer B2 blade: ', result1.blade
assert la.det(result1.blade) > 0, la.det(result1.blade)
assert result1.blade.shape == (2, 2), result1.blade.shape
assert np.allclose(np.eye(2), np.dot(result1.blade.T, result1.blade))
assert np.allclose(bd.inner(result1, result1).s, -1), bd.inner(result1, result1).s
B1 = Blade(np.array([1, 0]), s=2.0)
B2 = Blade(np.array([1, 1]), s=3.0)
result2 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result2.s
print 'B1 outer B2 blade: ', result2.blade
assert la.det(result2.blade) > 0, la.det(result2.blade)
assert result2.blade.shape == (2, 2), result2.blade.shape
assert np.allclose(np.eye(2), np.dot(result2.blade.T, result2.blade))
assert np.allclose(bd.inner(result2, result2).s, -36), bd.inner(result2, result2).s
def testBladeOuterScaleMultplicative():
# multiplicative scaling
B1 = Blade(np.array([2, 0]))
B2 = Blade(np.array([0, 1]))
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
print 'B1 outer B2 blade: ', result1.blade
assert la.det(result1.blade) > 0, la.det(result1.blade)
assert result1.blade.shape == (2, 2), result1.blade.shape
assert np.allclose(np.eye(2), np.dot(result1.blade.T, result1.blade))
assert np.allclose(bd.inner(result1, result1).s, -4), bd.inner(result1, result1).s
result2 = bd.outer(B2, B1)
print 'B2 outer B1 s: ', result2.s
print 'B2 outer B1 blade: ', result2.blade
assert la.det(result2.blade) < 0, la.det(result2.blade)
assert result2.blade.shape == (2, 2), result2.blade.shape
assert np.allclose(np.eye(2), np.dot(result2.blade.T, result2.blade))
assert np.allclose(bd.inner(result2, result2).s, -4), bd.inner(result2, result2).s
# result1 == inverse(result2)
assert np.allclose(bd.inner(result1, result2).s, 4), bd.inner(result1, result2).s
def testBladeOuterSign():
# sign/antisymmetric
B1 = Blade(np.array([1, 0]))
B2 = Blade(np.array([0, 1]))
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
print 'B1 outer B2 blade: ', result1.blade
assert la.det(result1.blade) > 0, la.det(result1.blade)
assert result1.blade.shape == (2, 2), result1.blade.shape
assert np.allclose(np.eye(2), np.dot(result1.blade.T, result1.blade))
assert np.allclose(bd.inner(result1, result1).s, -1), bd.inner(result1, result1).s
result2 = bd.outer(B2, B1)
print 'B2 outer B1 s: ', result2.s
print 'B2 outer B1 blade: ', result2.blade
assert la.det(result2.blade) < 0, la.det(result2.blade)
assert result2.blade.shape == (2, 2), result2.blade.shape
assert np.allclose(np.eye(2), np.dot(result2.blade.T, result2.blade))
assert np.allclose(bd.inner(result2, result2).s, -1), bd.inner(result2, result2).s
# result1 == inverse(result2)
assert bd.inner(result1, result2).s > 0, bd.inner(result1, result2).s
def testBladeOuterScaleParallelopiped():
# test outer(B1, B2).s == 'area spanned by parallelopiped between B1, B2'
B1 = Blade(np.array([1, 0]))
B2 = Blade(np.array(
[1, 1])) # should have same area as [0, 1] when outered with B1
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
print 'B1 outer B2 blade: ', result1.blade
assert la.det(result1.blade) > 0, la.det(result1.blade)
assert result1.blade.shape == (2, 2), result1.blade.shape
assert np.allclose(np.eye(2), np.dot(result1.blade.T, result1.blade))
assert np.allclose(bd.inner(result1, result1).s,
-1), bd.inner(result1, result1).s
B1 = Blade(np.array([1, 0]), s=2.0)
B2 = Blade(np.array([1, 1]), s=3.0)
result2 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result2.s
print 'B1 outer B2 blade: ', result2.blade
assert la.det(result2.blade) > 0, la.det(result2.blade)
assert result2.blade.shape == (2, 2), result2.blade.shape
assert np.allclose(np.eye(2), np.dot(result2.blade.T, result2.blade))
assert np.allclose(bd.inner(result2, result2).s,
-36), bd.inner(result2, result2).s
def testBladeOuterSign():
# sign/antisymmetric
B1 = Blade(np.array([1, 0]))
B2 = Blade(np.array([0, 1]))
result1 = bd.outer(B1, B2)
print 'B1 outer B2 s: ', result1.s
print 'B1 outer B2 blade: ', result1.blade
assert la.det(result1.blade) > 0, la.det(result1.blade)
assert result1.blade.shape == (2, 2), result1.blade.shape
assert np.allclose(np.eye(2), np.dot(result1.blade.T, result1.blade))
assert np.allclose(bd.inner(result1, result1).s,
-1), bd.inner(result1, result1).s
result2 = bd.outer(B2, B1)
print 'B2 outer B1 s: ', result2.s
print 'B2 outer B1 blade: ', result2.blade
assert la.det(result2.blade) < 0, la.det(result2.blade)
assert result2.blade.shape == (2, 2), result2.blade.shape
assert np.allclose(np.eye(2), np.dot(result2.blade.T, result2.blade))
assert np.allclose(bd.inner(result2, result2).s,
-1), bd.inner(result2, result2).s
# result1 == inverse(result2)
assert bd.inner(result1, result2).s > 0, bd.inner(result1, result2).s
def testBladeDualEuclideanRegression():
# regression test -- tests a bunch of functions and they all have to work
# for many input dimensions for this to succeed fully
for n in range(1, 10):
for k in range(0, n + 1):
if k == 0:
blade = Blade(1, s=float(rn.rand(1)[0]))
else:
blade = Blade(rn.randn(n, k))
print 'blade n, k: ', blade.n, blade.k
if np.allclose(blade.s, 0): # just skip this one for now...
continue # measure 0 event but could happen
D = bd.dual(blade, n=n)
print 'D n, k: ', D.n, D.k
revPseudo = bd.outer(bd.inverse(blade), D)
print 'n, k: ', n, k
print 'revPseudo blade, s: ', revPseudo.blade, revPseudo.s
shouldBeOne = bd.inner(revPseudo, bd.pseudoScalar(n))
assert np.allclose(shouldBeOne.s, 1.0), (shouldBeOne.blade, shouldBeOne.s)
def testBladeDualEuclideanRegression():
# regression test -- tests a bunch of functions and they all have to work
# for many input dimensions for this to succeed fully
for n in range(1, 10):
for k in range(0, n + 1):
if k == 0:
blade = Blade(1, s=float(rn.rand(1)[0]))
else:
blade = Blade(rn.randn(n, k))
print 'blade n, k: ', blade.n, blade.k
if np.allclose(blade.s, 0): # just skip this one for now...
continue # measure 0 event but could happen
D = bd.dual(blade, n=n)
print 'D n, k: ', D.n, D.k
revPseudo = bd.outer(bd.inverse(blade), D)
print 'n, k: ', n, k
print 'revPseudo blade, s: ', revPseudo.blade, revPseudo.s
shouldBeOne = bd.inner(revPseudo, bd.pseudoScalar(n))
assert np.allclose(shouldBeOne.s,
1.0), (shouldBeOne.blade, shouldBeOne.s)
def testBladeMeet():
# two bases
A = Blade(np.array([1, 0, 0]))
B = Blade(np.array([0, 1, 0]))
M = bd.meet(A, B)
assert M.blade.shape == (1, 0), M.blade.shape
assert M.s == 1
# linearly dependent
A = Blade(np.array([1, 0, 0]))
B = Blade(np.array([1, 0, 0]))
M = bd.meet(A, B)
assert M.blade.shape == (3, 1)
assert M.s == 1
# share one factor
A = bd.outer(Blade(np.array([1, 0, 0])), Blade(np.array([0, 1, 0])))
B = Blade(np.array([1, 0, 0]))
M = bd.meet(A, B)
assert M.blade.shape == (3, 1), M.blade.shape
assert M.s == 1
def testBladeMeet():
# two bases
A = Blade(np.array([1, 0, 0]))
B = Blade(np.array([0, 1, 0]))
M = bd.meet(A, B)
assert M.blade.shape == (1, 0), M.blade.shape
assert M.s == 1
# linearly dependent
A = Blade(np.array([1, 0, 0]))
B = Blade(np.array([1, 0, 0]))
M = bd.meet(A, B)
assert M.blade.shape == (3, 1)
assert M.s == 1
# share one factor
A = bd.outer(Blade(np.array([1, 0, 0])), Blade(np.array([0, 1, 0])))
B = Blade(np.array([1, 0, 0]))
M = bd.meet(A, B)
assert M.blade.shape == (3, 1), M.blade.shape
assert M.s == 1
