def csea(a=1, e=0):
r"""
Return a function object that wraps the ISEA projection and its inverse
of an ellipsoid with major radius `a` and eccentricity `e`.
OUTPUT:
- A function object of the form f(u, v, radians=False, inverse=False).
"""
R_A = auth_rad(a, e)
def f(u, v, radians=False, inverse=False):
if not inverse:
lam, phi = u, v
if not radians:
# Convert to radians.
lam, phi = deg2rad([lam, phi])
return tuple(R_A * array(csea_ellipsoid(lam, phi, e=e)))
else:
# Scale down to R_A = 1.
x, y = array((u, v)) / R_A
lam, phi = array(csea_ellipsoid_inverse(x, y, e=e))
if not radians:
# Convert to degrees.
lam, phi = rad2deg([lam, phi])
return lam, phi
return f
def test_rhealpix(self):
inputs = [(-pi, pi / 3), (0, pi / 4), (pi / 2, -pi / 6)]
# Should agree with rhealpix_ellipsoid and rhealpix_ellipsoid_inverse.
e = 0.5
a = 7
R_A = auth_rad(a, e)
f = pjr.rhealpix(a=a, e=e)
for p in inputs:
get = f(*p, radians=True)
expect = tuple(R_A * array(pjr.rhealpix_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 = pjr.rhealpix_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_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(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=True)
q = g(lam, beta, radians=True)
expect.append(tuple(q))
f = Proj(proj='rhealpix', a=a, e=e,
north_square=ns, south_square=ss)
get = [f(*p, radians=True) 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=True), e, radians=True,
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(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=True)
get = f(*q, radians=True, inverse=True)
print q, expect, get
self.assertTrue(rel_err(get, expect) < error)
def rhealpix(a=1, e=0, north_square=0, south_square=0):
r"""
Return a function object that wraps the rHEALPix projection and its inverse
of an ellipsoid with major radius `a` and eccentricity `e`.
EXAMPLES::
>>> f = rhealpix(a=2, e=0, north_square=1, south_square=2)
>>> print(f(0, pi/3, radians=True))
(-0.57495135977821477, 2.1457476865731113)
>>> p = (0, 60)
>>> q = f(*p, radians=False); print(q)
(-0.57495135977821477, 2.1457476865731113)
>>> print(f(*q, radians=False, inverse=True), p)
(6.3611093629270335e-15, 59.999999999999986) (0, 60)
OUTPUT:
- A function object of the form f(u, v, radians=False, inverse=False).
"""
R_A = auth_rad(a, e)
def f(u, v, radians=False, inverse=False):
if not inverse:
lam, phi = u, v
if not radians:
# Convert to radians.
lam, phi = deg2rad([lam, phi])
return tuple(R_A * array(
rhealpix_ellipsoid(lam,
phi,
e=e,
north_square=north_square,
south_square=south_square)))
else:
# Scale down to R_A = 1.
x, y = array((u, v)) / R_A
lam, phi = array(
rhealpix_ellipsoid_inverse(x,
y,
e=e,
north_square=north_square,
south_square=south_square))
if not radians:
# Convert to degrees.
lam, phi = rad2deg([lam, phi])
return lam, phi
return f
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_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=True)
q = g(lam, beta, radians=True)
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=True) 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=True, 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=True), e, radians=True,
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 isea(a=1, e=0):
r"""
Return a function object that wraps the ISEA projection and its inverse
of an ellipsoid with major radius `a` and eccentricity `e`.
EXAMPLES::
>>> f = isea(a=2, e=0)
>>> print f(0, pi/3, radians=True)
(-4.7097959155097699, 2.9195761776404252)
>>> g = isea(a=2, e=0.1)
>>> print g(0, 60, radians=False)
(-4.6978889550868494, 2.9179977608222689)
OUTPUT:
- A function object of the form f(u, v, radians=False, inverse=False).
"""
R_A = auth_rad(a, e)
def f(u, v, radians=False, inverse=False):
if not inverse:
lam, phi = u, v
if not radians:
# Convert to radians.
lam, phi = deg2rad([lam, phi])
return tuple(R_A * array(isea_ellipsoid(lam, phi, e=e)))
else:
# Scale down to R_A = 1.
x, y = array((u, v)) / R_A
lam, phi = array(isea_ellipsoid_inverse(x, y, e=e))
if not radians:
# Convert to degrees.
lam, phi = rad2deg([lam, phi])
return lam, phi
return f
def rhealpix_diagram(a=1,
e=0,
north_square=0,
south_square=0,
shade_polar_region=True):
r"""
Return a Sage Graphics object diagramming the rHEALPix projection
boundary and polar triangles for the ellipsoid with major radius `a`
and eccentricity `e`.
Inessential graphics method.
Requires Sage graphics methods.
"""
from sage.all import Graphics, line2d, point, polygon, text, RealNumber, Integer
# Make Sage types compatible with Numpy.
RealNumber = float
Integer = int
R = auth_rad(a, e)
g = Graphics()
color = 'black' # Boundary color.
shade_color = 'blue' # Polar triangles color.
north = north_square
south = south_square
south_sq = [(-R * pi + R * south * pi / 2, -R * pi / 4),
(-R * pi + R * south * pi / 2, -R * 3 * pi / 4),
(-R * pi + R * (south + 1) * pi / 2, -R * 3 * pi / 4),
(-R * pi + R * (south + 1) * pi / 2, -R * pi / 4)]
north_sq = [(-R * pi + R * north * pi / 2, R * pi / 4),
(-R * pi + R * north * pi / 2, R * 3 * pi / 4),
(-R * pi + R * (north + 1) * pi / 2, R * 3 * pi / 4),
(-R * pi + R * (north + 1) * pi / 2, R * pi / 4)]
# Outline.
g += line2d(south_sq, linestyle='--', color=color)
g += line2d(north_sq, linestyle='--', color=color)
g += line2d([(R * pi, -R * pi / 4), (R * pi, R * pi / 4)],
linestyle='--',
color=color)
g += line2d([
north_sq[0], (-R * pi, R * pi / 4), (-R * pi, -R * pi / 4), south_sq[0]
],
color=color)
g += line2d([south_sq[3], (R * pi, -R * pi / 4)], color=color)
g += line2d([north_sq[3], (R * pi, R * pi / 4)], color=color)
g += point([south_sq[0], south_sq[3]], size=20, zorder=3, color=color)
g += point([north_sq[0], north_sq[3]], size=20, zorder=3, color=color)
g += point([(R * pi, -R * pi / 4), (R * pi, R * pi / 4)],
size=20,
zorder=3,
color=color)
g += point([(R * pi, -R * pi / 4), (R * pi, R * pi / 4)],
size=10,
color='white',
zorder=3)
if shade_polar_region:
# Shade.
g += polygon(south_sq, alpha=0.1, color=shade_color)
g += polygon(north_sq, alpha=0.1, color=shade_color)
# Slice square into polar triangles.
g += line2d([south_sq[0], south_sq[2]], color='lightgray')
g += line2d([south_sq[1], south_sq[3]], color='lightgray')
g += line2d([north_sq[0], north_sq[2]], color='lightgray')
g += line2d([north_sq[1], north_sq[3]], color='lightgray')
# Label polar triangles.
sp = south_sq[0] + R * array((pi / 4, -pi / 4))
np = north_sq[0] + R * array((pi / 4, pi / 4))
shift = R * 3 * pi / 16
g += text(str(south), sp + array((0, shift)), color='red', fontsize=20)
g += text(str((south + 1) % 4),
sp + array((shift, 0)),
color='red',
rotation=90,
fontsize=20)
g += text(str((south + 2) % 4),
sp + array((0, -shift)),
color='red',
rotation=180,
fontsize=20)
g += text(str((south + 3) % 4),
sp + array((-shift, 0)),
color='red',
rotation=270,
fontsize=20)
g += text(str(north), np + array((0, -shift)), color='red', fontsize=20)
g += text(str((north + 1) % 4),
np + array((shift, 0)),
color='red',
rotation=90,
fontsize=20)
g += text(str((north + 2) % 4),
np + array((0, shift)),
color='red',
rotation=180,
fontsize=20)
g += text(str((north + 3) % 4),
np + array((-shift, 0)),
color='red',
rotation=270,
fontsize=20)
return g
def healpix_diagram(a=1, e=0, shade_polar_region=True):
r"""
Return a Sage Graphics object diagramming the HEALPix projection
boundary and polar triangles for the ellipsoid with major radius `a`
and eccentricity `e`.
Inessential graphics method.
Requires Sage graphics methods.
"""
from sage.all import Graphics, line2d, point, polygon, text, RealNumber, Integer
# Make Sage types compatible with Numpy.
RealNumber = float
Integer = int
R = auth_rad(a, e)
g = Graphics()
color = 'black' # Boundary color.
shade_color = 'blue' # Polar triangles color.
dl = array((R * pi / 2, 0))
lu = [(-R * pi, R * pi / 4), (-R * 3 * pi / 4, R * pi / 2)]
ld = [(-R * 3 * pi / 4, R * pi / 2), (-R * pi / 2, R * pi / 4)]
g += line2d([(-R * pi, -R * pi / 4), (-R * pi, R * pi / 4)], color=color)
g += line2d([(R * pi, R * pi / 4), (R * pi, -R * pi / 4)],
linestyle='--',
color=color)
for k in range(4):
g += line2d([array(p) + k * dl for p in lu], color=color)
g += line2d([array(p) + k * dl for p in ld],
linestyle='--',
color=color)
g += line2d([
array(p) + array((k * R * pi / 2 - R * pi / 4, -R * 3 * pi / 4))
for p in ld
],
color=color)
g += line2d([
array(p) + array((k * R * pi / 2 + R * pi / 4, -R * 3 * pi / 4))
for p in lu
],
linestyle='--',
color=color)
pn = array((-R * 3 * pi / 4, R * pi / 2))
ps = array((-R * 3 * pi / 4, -R * pi / 2))
g += point([pn + k * dl
for k in range(4)] + [ps + k * dl for k in range(4)],
size=20,
color=color)
g += point([pn + k * dl
for k in range(1, 4)] + [ps + k * dl for k in range(1, 4)],
color='white',
size=10,
zorder=3)
npp = [(-R * pi, R * pi / 4), (-R * 3 * pi / 4, R * pi / 2),
(-R * pi / 2, R * pi / 4)]
spp = [(-R * pi, -R * pi / 4), (-R * 3 * pi / 4, -R * pi / 2),
(-R * pi / 2, -R * pi / 4)]
if shade_polar_region:
for k in range(4):
g += polygon([array(p) + k * dl for p in npp],
alpha=0.1,
color=shade_color)
g += text(str(k),
array((-R * 3 * pi / 4, R * 5 * pi / 16)) + k * dl,
color='red',
fontsize=20)
g += polygon([array(p) + k * dl for p in spp],
alpha=0.1,
color=shade_color)
g += text(str(k),
array((-R * 3 * pi / 4, -R * 5 * pi / 16)) + k * dl,
color='red',
fontsize=20)
return g