def rhealpix_ellipsoid(lam, phi, e=0, north_square=0, south_square=0):
r"""
Compute the signature functions of the rHEALPix map
projection of an oblate ellipsoid with eccentricity `e` whose
authalic sphere is the unit sphere.
The north polar square is put in position `north_square`,
and the south polar square is put in position `south_square`.
Works when `e` = 0 (spherical case) too.
INPUT:
- `lam, phi` - Geographic longitude-latitude coordinates in radian.
Assume -pi <= `lam` < pi and -pi/2 <= `phi` <= pi/2.
- `e` - Eccentricity of the ellipsoid.
- `north_square, south_square` - (Optional; defaults = 0, 0) Integers
between 0 and 3 indicating positions of north polar and
south polar squares, respectively.
See rhealpix_sphere() docstring for a diagram.
EXAMPLES::
>>> from numpy import arcsin
>>> print(my_round(rhealpix_ellipsoid(0, arcsin(2.0/3)), 15))
(0, 0.78539816339744795)
"""
# Ensure north_square and south_square lie in {0, 1,2, 3}.
x, y = healpix_ellipsoid(lam, phi, e)
return combine_triangles(x, y, north_square=north_square,
south_square=south_square)
def rhealpix_ellipsoid(lam, phi, e=0, north_square=0, south_square=0):
r"""
Compute the signature functions of the rHEALPix map
projection of an oblate ellipsoid with eccentricity `e` whose
authalic sphere is the unit sphere.
The north polar square is put in position `north_square`,
and the south polar square is put in position `south_square`.
Works when `e` = 0 (spherical case) too.
INPUT:
- `lam, phi` - Geographic longitude-latitude coordinates in radian.
Assume -pi <= `lam` < pi and -pi/2 <= `phi` <= pi/2.
- `e` - Eccentricity of the ellipsoid.
- `north_square, south_square` - (Optional; defaults = 0, 0) Integers
between 0 and 3 indicating positions of north polar and
south polar squares, respectively.
See rhealpix_sphere() docstring for a diagram.
EXAMPLES::
>>> from numpy import arcsin
>>> print(rhealpix_ellipsoid(0, arcsin(2.0/3)))
(0, 0.78539816339744828)
"""
# Ensure north_square and south_square lie in {0, 1,2, 3}.
x, y = healpix_ellipsoid(lam, phi, e)
return combine_triangles(x,
y,
north_square=north_square,
south_square=south_square)
def test_rhealpix_ellipsoid(self):
# Test map projection.
# Should return the same output as healpix_ellipsoid(), followed
# by a scaling down, followed by combine_triangles(),
# followed by a scaling up.
e = 0.8
for (ns, ss) in product(list(range(4)), repeat=2):
for p in inputs:
q = pjr.rhealpix_ellipsoid(*p, north_square=ns,
south_square=ss, e=e)
qq = pjh.healpix_ellipsoid(*p, e=e)
qq = pjr.combine_triangles(*qq, north_square=ns,
south_square=ss)
self.assertEqual(q, qq)
# Test inverse projection.
# The inverse of the projection of a point p should yield p.
# 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), e, inverse=True)
error = 10*rel_err(alpha_, alpha)
for (ns, ss) in product(list(range(4)), repeat=2):
for p in inputs:
q = pjr.rhealpix_ellipsoid(*p, north_square=ns,
south_square=ss, e=e)
pp = pjr.rhealpix_ellipsoid_inverse(*q, north_square=ns,
south_square=ss, e=e)
self.assertTrue(rel_err(p, pp) < error)
def test_healpix(self):
inputs = [(-pi, pi/3), (0, pi/4), (pi/2, -pi/6)]
# Should agree with healpix_ellipsoid and healpix_ellipsoid_inverse.
e = 0.5
a = 7
R_A = auth_rad(a, e)
f = pjh.healpix(a=a, e=e)
for p in inputs:
get = f(*p, radians=True)
expect = tuple(R_A*array(pjh.healpix_ellipsoid(*p, e=e)))
for i in range(len(expect)):
self.assertAlmostEqual(get[i], expect[i])
get = f(*get, radians=True, inverse=True)
expect = tuple(array(expect)/R_A)
expect = pjh.healpix_ellipsoid_inverse(*expect, e=e)
for i in range(len(expect)):
self.assertAlmostEqual(get[i], expect[i])
# Should work in degrees mode.
for p in inputs:
get = f(*rad2deg(p), radians=False)
expect = f(*p, radians=True)
for i in range(len(expect)):
self.assertAlmostEqual(get[i], expect[i])
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(self):
inputs = [(-pi, pi / 3), (0, pi / 4), (pi / 2, -pi / 6)]
# Should agree with healpix_ellipsoid and healpix_ellipsoid_inverse.
e = 0.5
a = 7
R_A = auth_rad(a, e)
f = pjh.healpix(a=a, e=e)
for p in inputs:
get = f(*p, radians=True)
expect = tuple(R_A * array(pjh.healpix_ellipsoid(*p, e=e)))
for i in range(len(expect)):
self.assertAlmostEqual(get[i], expect[i])
get = f(*get, radians=True, inverse=True)
expect = tuple(array(expect) / R_A)
expect = pjh.healpix_ellipsoid_inverse(*expect, e=e)
for i in range(len(expect)):
self.assertAlmostEqual(get[i], expect[i])
# Should work in degrees mode.
for p in inputs:
get = f(*rad2deg(p), radians=False)
expect = f(*p, radians=True)
for i in range(len(expect)):
self.assertAlmostEqual(get[i], expect[i])
