def test_healpix_ellipsoid(self):
# Expected output of healpix_ellipsoid() applied to inputs.
e = 0.8
healpix_ellipsoid_outputs = []
for p in inputs:
lam, phi = p
beta = auth_lat(phi, e=e, radians=True)
q = pjh.healpix_sphere(lam, beta)
healpix_ellipsoid_outputs.append(q)
# Forward projection should be correct on test points.
given = inputs
get = [pjh.healpix_ellipsoid(*p, e=e) for p in given]
expect = healpix_ellipsoid_outputs
# Fuzz to allow for rounding errors:
error = 1e-12
for i in range(len(get)):
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse of projection of a point p should yield p.
given = get
get = [pjh.healpix_ellipsoid_inverse(*q, e=e) for q in given]
expect = inputs
# Fuzz for rounding errors based on the error of the approximation to
# the inverse authalic latitude function:
alpha = pi/4
alpha_ = auth_lat(auth_lat(alpha, e, radians=True), e, radians=True,
inverse=True)
error = 10*rel_err(alpha_, alpha)
for i in range(len(get)):
self.assertTrue(rel_err(get[i], expect[i]) < error)
def test_healpix_ellipsoid(self):
# Expected output of healpix_ellipsoid() applied to inputs.
e = 0.8
healpix_ellipsoid_outputs = []
for p in inputs:
lam, phi = p
beta = auth_lat(phi, e=e, radians=True)
q = pjh.healpix_sphere(lam, beta)
healpix_ellipsoid_outputs.append(q)
# Forward projection should be correct on test points.
given = inputs
get = [pjh.healpix_ellipsoid(*p, e=e) for p in given]
expect = healpix_ellipsoid_outputs
# Fuzz to allow for rounding errors:
error = 1e-12
for i in range(len(get)):
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse of projection of a point p should yield p.
given = get
get = [pjh.healpix_ellipsoid_inverse(*q, e=e) for q in given]
expect = inputs
# Fuzz for rounding errors based on the error of the approximation to
# the inverse authalic latitude function:
alpha = pi / 4
alpha_ = auth_lat(auth_lat(alpha, e, radians=True),
e,
radians=True,
inverse=True)
error = 10 * rel_err(alpha_, alpha)
for i in range(len(get)):
self.assertTrue(rel_err(get[i], expect[i]) < error)
def test_healpix_sphere(self):
# Expected outputs of healpix_sphere() applied to inputs.
sigma_a = sqrt(3 - 3*sin(a[1]))
ha = (pi/4*(1 - sigma_a), pi/4*(2 - sigma_a))
hb = (ha[0], -ha[1])
healpix_sphere_outputs = [
(0, 0),
(0, pi/4),
(0, -pi/4),
(pi/2, 0),
(-pi/2, 0),
(-pi, 0),
(-3*pi/4, pi/2),
(-3*pi/4, -pi/2),
ha,
hb
]
# Forward projection should be correct on test points.
given = inputs
get = [pjh.healpix_sphere(*p) for p in given]
expect = healpix_sphere_outputs
# Fuzz to allow for rounding errors:
error = 1e-12
for i in range(len(get)):
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse of projection of a point p should yield p.
given = get
get = [pjh.healpix_sphere_inverse(*q) for q in given]
expect = inputs
for i in range(len(get)):
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse projection of p below should return longitude of -pi.
# Previously, it was returning a number slightly less than pi
# because of a rounding error, which get magnified by
# wrap_lon())
p = (-7*pi/8, 3*pi/8)
get = pjh.healpix_sphere_inverse(*p)
expect = (-pi, arcsin(1 - 1.0/12))
self.assertEqual(get, expect)
q = (-5*pi/6, 5*pi/12)
get = pjh.healpix_sphere_inverse(*q)
expect = (-pi, arcsin(1 - 1.0/27))
self.assertEqual(get, expect)
def test_rhealpix_sphere(self):
# Test forward projection.
# Should return the same output as healpix_sphere() of the unit
# sphere, followed by combine_triangles(), followed by a scaling up.
for (ns, ss) in product(list(range(4)), repeat=2):
for p in inputs:
q = pjr.rhealpix_sphere(*p, north_square=ns, south_square=ss)
qq = pjr.combine_triangles(*pjh.healpix_sphere(*p),
north_square=ns, south_square=ss)
qq = tuple(qq)
self.assertEqual(q, qq)
# Test inverse projection.
# The inverse of the projection of a point p should yield p.
error = 1e-15 # Fuzz to handle small rounding errors.
for (ns, ss) in product(list(range(4)), repeat=2):
for p in inputs:
q = pjr.rhealpix_sphere(*p, north_square=ns, south_square=ss)
pp = pjr.rhealpix_sphere_inverse(*q, north_square=ns,
south_square=ss)
self.assertTrue(euclidean(p, pp) < error)
def rhealpix_sphere(lam, phi, north_square=0, south_square=0):
r"""
Compute the signature functions of the rHEALPix map projection of
the unit sphere.
The north polar square is put in position `north_square`, and the
south polar square is put in position `south_square`.
INPUT:
- `lam, phi` -Geographic longitude-latitude coordinates in radians.
Assume -pi <= `lam` < pi and -pi/2 <= `phi` <= pi/2.
- `north_square, south_square` - (Optional; defaults = 0, 0) Integers
between 0 and 3 indicating positions of north polar and
south polar squares, respectively.
EXAMPLES::
>>> print(rhealpix_sphere(0, pi/4))
(-1.6199786334139372, 2.3070121835733044)
NOTE:
The polar squares are labeled 0, 1, 2, 3 from east to west like this::
east west
*---*---*---*---*
| 0 | 1 | 2 | 3 |
*---*---*---*---*
| | | | |
*---*---*---*---*
| 0 | 1 | 2 | 3 |
*---*---*---*---*
"""
x, y = healpix_sphere(lam, phi)
return combine_triangles(x,
y,
north_square=north_square,
south_square=south_square)
def test_healpix_sphere(self):
# Expected outputs of healpix_sphere() applied to inputs.
sigma_a = sqrt(3 - 3 * sin(a[1]))
ha = (pi / 4 * (1 - sigma_a), pi / 4 * (2 - sigma_a))
hb = (ha[0], -ha[1])
healpix_sphere_outputs = [(0, 0), (0, pi / 4),
(0, -pi / 4), (pi / 2, 0), (-pi / 2, 0),
(-pi, 0), (-3 * pi / 4, pi / 2),
(-3 * pi / 4, -pi / 2), ha, hb]
# Forward projection should be correct on test points.
given = inputs
get = [pjh.healpix_sphere(*p) for p in given]
expect = healpix_sphere_outputs
# Fuzz to allow for rounding errors:
error = 1e-12
for i in range(len(get)):
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse of projection of a point p should yield p.
given = get
get = [pjh.healpix_sphere_inverse(*q) for q in given]
expect = inputs
for i in range(len(get)):
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse projection of p below should return longitude of -pi.
# Previously, it was returning a number slightly less than pi
# because of a rounding error, which get magnified by
# wrap_lon())
p = (-7 * pi / 8, 3 * pi / 8)
get = pjh.healpix_sphere_inverse(*p)
expect = (-pi, arcsin(1 - 1.0 / 12))
self.assertEqual(get, expect)
q = (-5 * pi / 6, 5 * pi / 12)
get = pjh.healpix_sphere_inverse(*q)
expect = (-pi, arcsin(1 - 1.0 / 27))
self.assertEqual(get, expect)
def rhealpix_sphere(lam, phi, north_square=0, south_square=0):
r"""
Compute the signature functions of the rHEALPix map projection of
the unit sphere.
The north polar square is put in position `north_square`, and the
south polar square is put in position `south_square`.
INPUT:
- `lam, phi` -Geographic longitude-latitude coordinates in radians.
Assume -pi <= `lam` < pi and -pi/2 <= `phi` <= pi/2.
- `north_square, south_square` - (Optional; defaults = 0, 0) Integers
between 0 and 3 indicating positions of north polar and
south polar squares, respectively.
EXAMPLES::
>>> print(my_round(rhealpix_sphere(0, pi/4), 15))
(-1.619978633413937, 2.307012183573304)
NOTE:
The polar squares are labeled 0, 1, 2, 3 from east to west like this::
east west
*---*---*---*---*
| 0 | 1 | 2 | 3 |
*---*---*---*---*
| | | | |
*---*---*---*---*
| 0 | 1 | 2 | 3 |
*---*---*---*---*
"""
x, y = healpix_sphere(lam, phi)
return combine_triangles(x, y, north_square=north_square,
south_square=south_square)