def random_point(self, lam_min=None, lam_max=None, phi_min=None,
phi_max=None):
r"""
Return a point (given in geodetic coordinates) sampled uniformly at
random from the section of this ellipsoid with longitude in the range
`lam_min <= lam < lam_max` and latitude in the range
`phi_min <= phi < phi_max`.
But avoid the poles.
EXAMPLES::
>>> E = UNIT_SPHERE
>>> print(E.random_point()) # doctest: +SKIP
(-1.0999574573422948, 0.21029104897701129)
"""
PI = self.pi()
if lam_min is None:
lam_min = -PI
if lam_max is None:
lam_max = PI
if phi_min is None:
phi_min = -PI/2
if phi_max is None:
phi_max = PI/2
if not self.radians:
# Convert to radians.
lam_min, lam_max, phi_min, phi_max = deg2rad(
[lam_min, lam_max, phi_min, phi_max])
# Pick a longitude.
while True:
u = uniform(0, 1)
lam = (lam_max - lam_min)*u + lam_min
# Don't include lam_max.
if lam < lam_max:
# Success.
break
# Pick a latitude.
delta = pi/360
while True:
v = uniform(0, 1)
if self.sphere:
phi = arcsin( (sin(phi_max) - sin(phi_min))*v + sin(phi_min) )
else:
# Sample from the authalic sphere.
# The map from the ellipsoid to the authalic sphere is
# an equiareal diffeomorphism.
# So a uniform distribution on the authalic sphere gives
# rise to a uniform distribution on the ellipsoid.
beta0 = auth_lat(phi_min, e=self.e, radians=True)
beta1 = auth_lat(phi_max, e=self.e, radians=True)
beta = arcsin( (sin(beta1) - sin(beta0))*v + sin(beta0) )
phi = auth_lat(beta, e=self.e, radians=True, inverse=True)
# Avoid the poles.
if abs(phi) <= pi/2 - delta:
# Success.
break
if not self.radians:
# Convert back to degrees.
lam, phi = rad2deg([lam, phi])
return lam, phi
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_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_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):
# Ellipsoid parameters.
a = 5
e = 0.8
R_A = auth_rad(a, e=e)
# Expected output of healpix_ellipsoid() applied to inputs.
healpix_ellipsoid_outputs = []
g = Proj(proj='healpix', R=R_A)
for p in inputs:
lam, phi = p
beta = auth_lat(phi, e=e, radians=RADIANS)
q = g(lam, beta, radians=RADIANS)
healpix_ellipsoid_outputs.append(q)
# Forward projection should be correct on test points.
f = Proj(proj='healpix', a=a, e=e)
given = inputs
get = [f(*p, radians=RADIANS) for p in given]
expect = healpix_ellipsoid_outputs
# Fuzz to allow for rounding errors:
error = 1e-12
print('=' * 80)
print(
'HEALPix forward projection, ellipsoid with major radius a = %s and eccentricity e = %s'
% (a, e))
print('input (radians) / expected output (meters) / received output')
print('=' * 80)
for i in range(len(get)):
print(given[i], expect[i], get[i])
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse of projection of a point p should yield p.
given = get
get = [f(*q, radians=RADIANS, inverse=True) 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=RADIANS),
e,
radians=RADIANS,
inverse=True)
error = 10 * rel_err(alpha_, alpha)
print('=' * 80)
print(
'HEALPix inverse projection, ellipsoid with major radius a = %s and eccentricity e = %s'
% (a, e))
print('input (meters) / expected output (radians) / received output')
print('=' * 80)
for i in range(len(get)):
print(given[i], expect[i], get[i])
self.assertTrue(rel_err(get[i], expect[i]) < error)
def test_rhealpix_ellipsoid(self):
# Ellipsoid parameters.
a = 5
e = 0.8
R_A = auth_rad(a, e=e)
# Forward projection should be correct on test points.
print('='*80)
print('rHEALPix forward projection, ellipsoid with major radius a = %s and eccentricity e = %s' % (a, e))
print('input (radians) / expected output (meters) / received output')
print('='*80)
given = inputs
# Fuzz to allow for rounding errors:
error = 1e-12
for (ns, ss) in product(list(range(4)), repeat=2):
print('_____ north_square = %s, south_square = %s' % (ns, ss))
expect = []
g = Proj(proj='rhealpix', R=R_A,
north_square=ns, south_square=ss)
for p in given:
lam, phi = p
beta = auth_lat(phi, e=e, radians=RADIANS)
q = g(lam, beta, radians=RADIANS)
expect.append(tuple(q))
f = Proj(proj='rhealpix', a=a, e=e,
north_square=ns, south_square=ss)
get = [f(*p, radians=RADIANS) for p in given]
for i in range(len(given)):
print(given[i], expect[i], get[i])
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse of projection of point a 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, radians=RADIANS), e, radians=RADIANS,
inverse=True)
error = 10*rel_err(alpha_, alpha)
print('='*80)
print('HEALPix inverse projection, ellipsoid with major radius a = %s and eccentricity e = %s' % (a, e))
print('input (meters) / expected output (radians) / received output')
print('='*80)
# The inverse of the projection of a point p should yield p.
for (ns, ss) in product(list(range(4)), repeat=2):
print('_____ north_square = %s, south_square = %s' % (ns, ss))
f = Proj(proj='rhealpix', a=a, e=e, north_square=ns,
south_square=ss)
for p in inputs:
expect = p
q = f(*p, radians=RADIANS)
get = f(*q, radians=RADIANS, inverse=True)
print(q, expect, get)
self.assertTrue(rel_err(get, expect) < error)
def test_healpix_ellipsoid(self):
# Ellipsoid parameters.
a = 5
e = 0.8
R_A = auth_rad(a, e=e)
# Expected output of healpix_ellipsoid() applied to inputs.
healpix_ellipsoid_outputs = []
g = Proj(proj='healpix', R=R_A)
for p in inputs:
lam, phi = p
beta = auth_lat(phi, e=e, radians=RADIANS)
q = g(lam, beta, radians=RADIANS)
healpix_ellipsoid_outputs.append(q)
# Forward projection should be correct on test points.
f = Proj(proj='healpix', a=a, e=e)
given = inputs
get = [f(*p, radians=RADIANS) for p in given]
expect = healpix_ellipsoid_outputs
# Fuzz to allow for rounding errors:
error = 1e-12
print('='*80)
print('HEALPix forward projection, ellipsoid with major radius a = %s and eccentricity e = %s' % (a, e))
print('input (radians) / expected output (meters) / received output')
print('='*80)
for i in range(len(get)):
print(given[i], expect[i], get[i])
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse of projection of a point p should yield p.
given = get
get = [f(*q, radians=RADIANS, inverse=True) 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=RADIANS), e, radians=RADIANS,
inverse=True)
error = 10*rel_err(alpha_, alpha)
print('='*80)
print('HEALPix inverse projection, ellipsoid with major radius a = %s and eccentricity e = %s' % (a, e))
print('input (meters) / expected output (radians) / received output')
print('='*80)
for i in range(len(get)):
print(given[i], expect[i], get[i])
self.assertTrue(rel_err(get[i], expect[i]) < error)
def csea_ellipsoid(lam, phi, e=0):
r"""
Compute the signature functions of the ISEA may projection of an oblate
ellipsoid with eccentricity `e` whose authalic sphere is the unit sphere.
INPUT:
- `lam, phi` - Geodetic longitude-latitude coordinates in radians.
Assume -pi <= `lam` < pi and -pi/2 <= `phi` <= pi/2.
- `e` - Eccentricity of the oblate ellipsoid.
"""
beta = auth_lat(phi, e, radians=True)
return csea_sphere(lam, beta)
def isea_ellipsoid(lam, phi, e=0):
r"""
Compute the signature functions of the ISEA may projection of an oblate
ellipsoid with eccentricity `e` whose authalic sphere is the unit sphere.
INPUT:
- `lam, phi` - Geodetic longitude-latitude coordinates in radians.
Assume -pi <= `lam` < pi and -pi/2 <= `phi` <= pi/2.
- `e` - Eccentricity of the oblate ellipsoid.
"""
beta = auth_lat(phi, e, radians=True)
return isea_sphere(lam, beta)
def __init__(self,
R=None,
a=WGS84_A,
b=None,
e=None,
f=WGS84_F,
lon_0=0,
lat_0=0,
radians=False):
self.lon_0 = lon_0
self.lat_0 = lat_0
self.radians = radians
if R is not None:
# The ellipsoid is a sphere.
# Override the other geometric parameters.
self.sphere = True
self.R = R
self.a = R
self.b = R
self.e = 0
self.f = 0
self.R_A = R
else:
self.sphere = False
self.a = a
if b is not None:
# Derive the other geometric parameters from a and b.
self.b = b
self.e = sqrt(1 - (b / a)**2)
self.f = (a - b) / a
elif e is not None:
# Derive the other geometric parameters from a and e.
self.e = e
self.b = a * sqrt(1 - e**2)
self.f = 1 - sqrt(1 - e**2)
else:
self.f = f
self.b = self.a * (1 - f)
self.e = sqrt(f * (1 - f))
self.R_A = auth_rad(self.a, self.e)
self.phi_0 = auth_lat(arcsin(2.0 / 3),
e=self.e,
radians=True,
inverse=True)
if not self.radians:
# Convert to degrees.
self.phi_0 = rad2deg(self.phi_0)
def healpix_ellipsoid(lam, phi, e=0):
r"""
Compute the signature functions of the HEALPix projection of an oblate
ellipsoid with eccentricity `e` whose authalic sphere is the unit sphere.
Works when `e` = 0 (spherical case) too.
INPUT:
- `lam, phi` - Geodetic longitude-latitude coordinates in radians.
Assume -pi <= `lam` < pi and -pi/2 <= `phi` <= pi/2.
- `e` - Eccentricity of the oblate ellipsoid.
EXAMPLES::
>>> print(my_round(healpix_ellipsoid(0, pi/7), 15))
(0, 0.51115723774642197)
>>> print(my_round(healpix_ellipsoid(0, pi/7, e=0.8), 15))
(0, 0.26848445085783701)
"""
beta = auth_lat(phi, e, radians=True)
return healpix_sphere(lam, beta)
def healpix_ellipsoid_inverse(x, y, e=0):
r"""
Compute the inverse of healpix_ellipsoid().
EXAMPLES::
>>> p = (0, pi/7)
>>> q = healpix_ellipsoid(*p)
>>> print(my_round(healpix_ellipsoid_inverse(*q), 15))
(0, 0.44879895051282798)
>>> print(my_round(p, 15))
(0, 0.448798950512828)
"""
# Throw error if input coordinates are out of bounds.
if not in_healpix_image(x, y):
print("Error: input coordinates (%f, %f) are out of bounds" % \
(x, y))
return
lam, beta = healpix_sphere_inverse(x, y)
phi = auth_lat(beta, e, radians=True, inverse=True)
return lam, phi
def __init__(self, R=None, a=WGS84_A, b=None, e=None, f=WGS84_F,
lon_0=0, lat_0=0, radians=False):
self.lon_0 = lon_0
self.lat_0 = lat_0
self.radians = radians
if R is not None:
# The ellipsoid is a sphere.
# Override the other geometric parameters.
self.sphere = True
self.R = R
self.a = R
self.b = R
self.e = 0
self.f = 0
self.R_A = R
else:
self.sphere = False
self.a = a
if b is not None:
# Derive the other geometric parameters from a and b.
self.b = b
self.e = sqrt(1 - (b/a)**2)
self.f = (a - b)/a
elif e is not None:
# Derive the other geometric parameters from a and e.
self.e = e
self.b = a*sqrt(1 - e**2)
self.f = 1 - sqrt(1 - e**2)
else:
self.f = f
self.b = self.a*(1 - f)
self.e = sqrt(f*(1 - f))
self.R_A = auth_rad(self.a, self.e)
self.phi_0 = auth_lat(arcsin(2.0/3), e=self.e, radians=True,
inverse=True)
if not self.radians:
# Convert to degrees.
self.phi_0 = rad2deg(self.phi_0)
def test_rhealpix_ellipsoid(self):
# Ellipsoid parameters.
a = 5
e = 0.8
R_A = auth_rad(a, e=e)
# Forward projection should be correct on test points.
print('=' * 80)
print(
'rHEALPix forward projection, ellipsoid with major radius a = %s and eccentricity e = %s'
% (a, e))
print('input (radians) / expected output (meters) / received output')
print('=' * 80)
given = inputs
# Fuzz to allow for rounding errors:
error = 1e-12
for (ns, ss) in product(list(range(4)), repeat=2):
print('_____ north_square = %s, south_square = %s' % (ns, ss))
expect = []
g = Proj(proj='rhealpix', R=R_A, north_square=ns, south_square=ss)
for p in given:
lam, phi = p
beta = auth_lat(phi, e=e, radians=RADIANS)
q = g(lam, beta, radians=RADIANS)
expect.append(tuple(q))
f = Proj(proj='rhealpix',
a=a,
e=e,
north_square=ns,
south_square=ss)
get = [f(*p, radians=RADIANS) for p in given]
for i in range(len(given)):
print(given[i], expect[i], get[i])
self.assertTrue(rel_err(get[i], expect[i]) < error)
# Inverse of projection of point a 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, radians=RADIANS),
e,
radians=RADIANS,
inverse=True)
error = 10 * rel_err(alpha_, alpha)
print('=' * 80)
print(
'HEALPix inverse projection, ellipsoid with major radius a = %s and eccentricity e = %s'
% (a, e))
print('input (meters) / expected output (radians) / received output')
print('=' * 80)
# The inverse of the projection of a point p should yield p.
for (ns, ss) in product(list(range(4)), repeat=2):
print('_____ north_square = %s, south_square = %s' % (ns, ss))
f = Proj(proj='rhealpix',
a=a,
e=e,
north_square=ns,
south_square=ss)
for p in inputs:
expect = p
q = f(*p, radians=RADIANS)
get = f(*q, radians=RADIANS, inverse=True)
print(q, expect, get)
self.assertTrue(rel_err(get, expect) < error)
def random_point(self, lam_min=None, lam_max=None, phi_min=None,
phi_max=None):
r"""
Return a point (given in geodetic coordinates) sampled uniformly at
random from the section of this ellipsoid with longitude in the range
`lam_min <= lam < lam_max` and latitude in the range
`phi_min <= phi < phi_max`.
But avoid the poles.
EXAMPLES::
>>> E = UNIT_SPHERE
>>> print(E.random_point()) # doctest: +SKIP
(-1.0999574573422948, 0.21029104897701129)
"""
PI = self.pi()
if lam_min is None:
lam_min = -PI
if lam_max is None:
lam_max = PI
if phi_min is None:
phi_min = -PI/2
if phi_max is None:
phi_max = PI/2
if not self.radians:
# Convert to radians.
lam_min, lam_max, phi_min, phi_max = deg2rad(
[lam_min, lam_max, phi_min, phi_max])
# Pick a longitude.
while True:
u = uniform(0, 1)
lam = (lam_max - lam_min)*u + lam_min
# Don't include lam_max.
if lam < lam_max:
# Success.
break
# Pick a latitude.
delta = pi/360
while True:
v = uniform(0, 1)
if self.sphere:
phi = arcsin( (sin(phi_max) - sin(phi_min))*v + sin(phi_min) )
else:
# Sample from the authalic sphere.
# The map from the ellipsoid to the authalic sphere is
# an equiareal diffeomorphism.
# So a uniform distribution on the authalic sphere gives
# rise to a uniform distribution on the ellipsoid.
beta0 = auth_lat(phi_min, e=self.e, radians=True)
beta1 = auth_lat(phi_max, e=self.e, radians=True)
beta = arcsin( (sin(beta1) - sin(beta0))*v + sin(beta0) )
phi = auth_lat(beta, e=self.e, radians=True, inverse=True)
# Avoid the poles.
if abs(phi) <= pi/2 - delta:
# Success.
break
if not self.radians:
# Convert back to degrees.
lam, phi = rad2deg([lam, phi])
return lam, phi