from __future__ import division
from past.utils import old_div
import numpy as np
from astropy.coordinates import SkyCoord, ICRS, BaseCoordinateFrame
from astropy.io import fits
from astropy import wcs
import astropy.units as u
from astromodels.functions.function import Function2D, FunctionMeta
from astromodels.utils.angular_distance import angular_distance
from astromodels.utils.vincenty import vincenty
import hashlib
from future.utils import with_metaclass
class Latitude_galactic_diffuse(with_metaclass(FunctionMeta, Function2D)):
r"""
description :
A Gaussian distribution in Galactic latitude around the Galactic plane
latex : $ K \exp{\left( \frac{-b^2}{2 \sigma_b^2} \right)} $
parameters :
K :
desc : normalization
initial value : 1
sigma_b :
desc : Sigma for
initial value : 1
l_min :
desc : min Longtitude
initial value : 10
l_max :
desc : max Longtitude
initial value : 30
"""
# This is optional, and it is only needed if we need more setup after the
# constructor provided by the meta class
def _setup(self):
self._frame = ICRS()
def set_frame(self, new_frame):
"""
Set a new frame for the coordinates (the default is ICRS J2000)
:param new_frame: a coordinate frame from astropy
:return: (none)
"""
assert isinstance(new_frame, BaseCoordinateFrame)
self._frame = new_frame
def _set_units(self, x_unit, y_unit, z_unit):
self.K.unit = z_unit
self.sigma_b.unit = x_unit
self.l_min.unit = y_unit
self.l_max.unit = y_unit
def evaluate(self, x, y, K, sigma_b, l_min, l_max):
# We assume x and y are R.A. and Dec
_coord = SkyCoord(ra=x, dec=y, frame=self._frame, unit="deg")
b = _coord.transform_to('galactic').b.value
l = _coord.transform_to('galactic').l.value
return K * np.exp(old_div(-b ** 2, (2 * sigma_b ** 2))) * np.logical_or(np.logical_and(l > l_min, l < l_max),np.logical_and(l_min > l_max, np.logical_or(l > l_min, l < l_max)))
def get_boundaries(self):
max_b = self.sigma_b.max_value
l_min = self.l_min.value
l_max = self.l_max.value
_coord = SkyCoord(l=[l_min, l_min, l_max, l_max], b=[max_b * -2., max_b * 2., max_b * 2., max_b * -2.], frame="galactic", unit="deg")
# no dealing with 0 360 overflow
min_lat = min(_coord.transform_to("icrs").dec.value)
max_lat = max(_coord.transform_to("icrs").dec.value)
min_lon = min(_coord.transform_to("icrs").ra.value)
max_lon = max(_coord.transform_to("icrs").ra.value)
return (min_lon, max_lon), (min_lat, max_lat)
def get_total_spatial_integral(self, z=None):
"""
Returns the total integral (for 2D functions) or the integral over the spatial components (for 3D functions).
needs to be implemented in subclasses.
:return: an array of values of the integral (same dimension as z).
"""
dL= self.l_max.value-self.l_min.value if self.l_max.value > self.l_min.value else 360 + self.l_max.value - self.l_max.value
#integral -inf to inf exp(-b**2 / 2*sigma_b**2 ) db = sqrt(2pi)*sigma_b
#Note that K refers to the peak diffuse flux (at b = 0) per square degree.
integral = np.sqrt( 2*np.pi ) * self.sigma_b.value * self.K.value * dL
if isinstance( z, u.Quantity):
z = z.value
return integral * np.power( 180. / np.pi, -2 ) * np.ones_like( z )
class Gaussian_on_sphere(with_metaclass(FunctionMeta, Function2D)):
r"""
description :
A bidimensional Gaussian function on a sphere (in spherical coordinates)
latex : $$ f(\vec{x}) = \left(\frac{180^\circ}{\pi}\right)^2 \frac{1}{2\pi \sqrt{\det{\Sigma}}} \, {\rm exp}\left( -\frac{1}{2} (\vec{x}-\vec{x}_0)^\intercal \cdot \Sigma^{-1}\cdot (\vec{x}-\vec{x}_0)\right) \\ \vec{x}_0 = ({\rm RA}_0,{\rm Dec}_0)\\ \Lambda = \left( \begin{array}{cc} \sigma^2 & 0 \\ 0 & \sigma^2 (1-e^2) \end{array}\right) \\ U = \left( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & cos \theta \end{array}\right) \\\Sigma = U\Lambda U^\intercal $$
parameters :
lon0 :
desc : Longitude of the center of the source
initial value : 0.0
min : 0.0
max : 360.0
lat0 :
desc : Latitude of the center of the source
initial value : 0.0
min : -90.0
max : 90.0
sigma :
desc : Standard deviation of the Gaussian distribution
initial value : 0.5
min : 0
max : 20
"""
def _set_units(self, x_unit, y_unit, z_unit):
# lon0 and lat0 and rdiff have most probably all units of degrees. However,
# let's set them up here just to save for the possibility of using the
# formula with other units (although it is probably never going to happen)
self.lon0.unit = x_unit
self.lat0.unit = y_unit
self.sigma.unit = x_unit
def evaluate(self, x, y, lon0, lat0, sigma):
lon, lat = x,y
angsep = angular_distance(lon0, lat0, lon, lat)
s2 = sigma**2
return (old_div(180, np.pi))**2 * 1 / (2.0 * np.pi * s2) * np.exp(-0.5 * angsep**2/s2)
def get_boundaries(self):
# Truncate the gaussian at 2 times the max of sigma allowed
max_sigma = self.sigma.max_value
min_lat = max(-90., self.lat0.value - 2 * max_sigma)
max_lat = min(90., self.lat0.value + 2 * max_sigma)
max_abs_lat = max(np.absolute(min_lat), np.absolute(max_lat))
if max_abs_lat > 89. or 2 * max_sigma / np.cos(max_abs_lat * np.pi / 180.) >= 180.:
min_lon = 0.
max_lon = 360.
else:
min_lon = self.lon0.value - 2 * max_sigma / np.cos(max_abs_lat * np.pi / 180.)
max_lon = self.lon0.value + 2 * max_sigma / np.cos(max_abs_lat * np.pi / 180.)
if min_lon < 0.:
min_lon = min_lon + 360.
elif max_lon > 360.:
max_lon = max_lon - 360.
return (min_lon, max_lon), (min_lat, max_lat)
def get_total_spatial_integral(self, z=None):
"""
Returns the total integral (for 2D functions) or the integral over the spatial components (for 3D functions).
needs to be implemented in subclasses.
:return: an array of values of the integral (same dimension as z).
"""
if isinstance( z, u.Quantity):
z = z.value
return np.ones_like( z )
class Asymm_Gaussian_on_sphere(with_metaclass(FunctionMeta, Function2D)):
r"""
description :
A bidimensional Gaussian function on a sphere (in spherical coordinates)
see https://en.wikipedia.org/wiki/Gaussian_function#Two-dimensional_Gaussian_function
parameters :
lon0 :
desc : Longitude of the center of the source
initial value : 0.0
min : 0.0
max : 360.0
lat0 :
desc : Latitude of the center of the source
initial value : 0.0
min : -90.0
max : 90.0
a :
desc : Standard deviation of the Gaussian distribution (major axis)
initial value : 0.5
min : 0
max : 20
e :
desc : Excentricity of Gaussian ellipse
initial value : 0.5
min : 0
max : 1
theta :
desc : inclination of major axis to a line of constant latitude
initial value : 0.0
min : -90.0
max : 90.0
"""
def _set_units(self, x_unit, y_unit, z_unit):
# lon0 and lat0 and a have most probably all units of degrees. However,
# let's set them up here just to save for the possibility of using the
# formula with other units (although it is probably never going to happen)
self.lon0.unit = x_unit
self.lat0.unit = y_unit
self.a.unit = x_unit
self.e.unit = u.dimensionless_unscaled
self.theta.unit = u.degree
def evaluate(self, x, y, lon0, lat0, a, e, theta):
lon, lat = x,y
b = a * np.sqrt(1. - e**2)
dX = np.atleast_1d( angular_distance( lon0, lat0, lon, lat0) )
dY = np.atleast_1d( angular_distance( lon0, lat0, lon0, lat) )
dlon = lon - lon0
if isinstance( dlon, u.Quantity):
dlon = (dlon.to(u.degree)).value
idx=np.logical_and( np.logical_or( dlon < 0, dlon > 180), np.logical_or( dlon>-180, dlon < -360) )
dX[idx] = -dX[idx]
idx = lat < lat0
dY[idx]=-dY[idx]
if isinstance( theta, u.Quantity ):
phi = (theta.to(u.degree)).value + 90.0
else:
phi = theta + 90.
cos2_phi = np.power( np.cos( phi * np.pi/180.), 2)
sin2_phi = np.power( np.sin( phi * np.pi/180.), 2)
sin_2phi = np.sin( 2. * phi * np.pi/180.)
A = old_div(cos2_phi, (2.*b**2)) + old_div(sin2_phi, (2.*a**2))
B = old_div(- sin_2phi, (4.*b**2)) + old_div(sin_2phi, (4.*a**2))
C = old_div(sin2_phi, (2.*b**2)) + old_div(cos2_phi, (2.*a**2))
E = -A*np.power(dX, 2) + 2.*B*dX*dY - C*np.power(dY, 2)
return np.power(old_div(180, np.pi), 2) * 1. / (2 * np.pi * a * b) * np.exp( E )
def get_boundaries(self):
# Truncate the gaussian at 2 times the max of sigma allowed
min_lat = max(-90., self.lat0.value - 2 * self.a.max_value)
max_lat = min(90., self.lat0.value + 2 * self.a.max_value)
max_abs_lat = max(np.absolute(min_lat), np.absolute(max_lat))
if max_abs_lat > 89. or 2 * self.a.max_value / np.cos(max_abs_lat * np.pi / 180.) >= 180.:
min_lon = 0.
max_lon = 360.
else:
min_lon = self.lon0.value - 2 * self.a.max_value / np.cos(max_abs_lat * np.pi / 180.)
max_lon = self.lon0.value + 2 * self.a.max_value / np.cos(max_abs_lat * np.pi / 180.)
if min_lon < 0.:
min_lon = min_lon + 360.
elif max_lon > 360.:
max_lon = max_lon - 360.
return (min_lon, max_lon), (min_lat, max_lat)
def get_total_spatial_integral(self, z=None):
"""
Returns the total integral (for 2D functions) or the integral over the spatial components (for 3D functions).
needs to be implemented in subclasses.
:return: an array of values of the integral (same dimension as z).
"""
if isinstance( z, u.Quantity):
z = z.value
return np.ones_like( z )
class Disk_on_sphere(with_metaclass(FunctionMeta, Function2D)):
r"""
description :
A bidimensional disk/tophat function on a sphere (in spherical coordinates)
latex : $$ f(\vec{x}) = \left(\frac{180}{\pi}\right)^2 \frac{1}{\pi~({\rm radius})^2} ~\left\{\begin{matrix} 1 & {\rm if}& {\rm | \vec{x} - \vec{x}_0| \le {\rm radius}} \\ 0 & {\rm if}& {\rm | \vec{x} - \vec{x}_0| > {\rm radius}} \end{matrix}\right. $$
parameters :
lon0 :
desc : Longitude of the center of the source
initial value : 0.0
min : 0.0
max : 360.0
lat0 :
desc : Latitude of the center of the source
initial value : 0.0
min : -90.0
max : 90.0
radius :
desc : Radius of the disk
initial value : 0.5
min : 0
max : 20
"""
def _set_units(self, x_unit, y_unit, z_unit):
# lon0 and lat0 and rdiff have most probably all units of degrees. However,
# let's set them up here just to save for the possibility of using the
# formula with other units (although it is probably never going to happen)
self.lon0.unit = x_unit
self.lat0.unit = y_unit
self.radius.unit = x_unit
def evaluate(self, x, y, lon0, lat0, radius):
lon, lat = x,y
angsep = angular_distance(lon0, lat0, lon, lat)
return np.power(old_div(180, np.pi), 2) * 1. / (np.pi * radius ** 2) * (angsep <= radius)
def get_boundaries(self):
# Truncate the disk at 2 times the max of radius allowed
max_radius = self.radius.max_value
min_lat = max(-90., self.lat0.value - 2 * max_radius)
max_lat = min(90., self.lat0.value + 2 * max_radius)
max_abs_lat = max(np.absolute(min_lat), np.absolute(max_lat))
if max_abs_lat > 89. or 2 * max_radius / np.cos(max_abs_lat * np.pi / 180.) >= 180.:
min_lon = 0.
max_lon = 360.
else:
min_lon = self.lon0.value - 2 * max_radius / np.cos(max_abs_lat * np.pi / 180.)
max_lon = self.lon0.value + 2 * max_radius / np.cos(max_abs_lat * np.pi / 180.)
if min_lon < 0.:
min_lon = min_lon + 360.
elif max_lon > 360.:
max_lon = max_lon - 360.
return (min_lon, max_lon), (min_lat, max_lat)
def get_total_spatial_integral(self, z=None):
"""
Returns the total integral (for 2D functions) or the integral over the spatial components (for 3D functions).
needs to be implemented in subclasses.
:return: an array of values of the integral (same dimension as z).
"""
if isinstance( z, u.Quantity):
z = z.value
return np.ones_like( z )
class Ellipse_on_sphere(with_metaclass(FunctionMeta, Function2D)):
r"""
description :
An ellipse function on a sphere (in spherical coordinates)
latex : $$ f(\vec{x}) = \left(\frac{180}{\pi}\right)^2 \frac{1}{\pi~ a b} ~\left\{\begin{matrix} 1 & {\rm if}& {\rm | \vec{x} - \vec{x}_{f1}| + | \vec{x} - \vec{x}_{f2}| \le {\rm 2a}} \\ 0 & {\rm if}& {\rm | \vec{x} - \vec{x}_{f1}| + | \vec{x} - \vec{x}_{f2}| > {\rm 2a}} \end{matrix}\right. $$
parameters :
lon0 :
desc : Longitude of the center of the source
initial value : 0.0
min : 0.0
max : 360.0
lat0 :
desc : Latitude of the center of the source
initial value : 0.0
min : -90.0
max : 90.0
a :
desc : semimajor axis of the ellipse
initial value : 0.5
min : 0
max : 20
e :
desc : eccentricity of ellipse
initial value : 0.5
min : 0
max : 1
theta :
desc : inclination of semimajoraxis to a line of constant latitude
initial value : 0.0
min : -90.0
max : 90.0
"""
lon1 = None
lat1 = None
lon2 = None
lat2 = None
focal_pts = False
def _set_units(self, x_unit, y_unit, z_unit):
# lon0 and lat0 have most probably all units of degrees.
# However, let's set them up here just to save for the possibility of
# using the formula with other units (although it is probably never
# going to happen)
self.lon0.unit = x_unit
self.lat0.unit = y_unit
self.a.unit = x_unit
# eccentricity is dimensionless
self.e.unit = u.dimensionless_unscaled
self.theta.unit = u.degree
def calc_focal_pts(self, lon0, lat0, a, b, theta):
# focal distance
f = np.sqrt(a**2 - b**2)
if isinstance( theta, u.Quantity):
bearing = 90. - (theta.to(u.degree)).value
else:
bearing = 90. - theta
# coordinates of focal points
lon1, lat1 = vincenty(lon0, lat0, bearing, f)
lon2, lat2 = vincenty(lon0, lat0, bearing + 180., f)
return lon1, lat1, lon2, lat2
def evaluate(self, x, y, lon0, lat0, a, e, theta):
b = a * np.sqrt(1. - e**2)
# calculate focal points
self.lon1, self.lat1, self.lon2, self.lat2 = self.calc_focal_pts(lon0, lat0, a, b, theta)
self.focal_pts = True
# lon/lat of point in question
lon, lat = x, y
# sum of geodesic distances to focii (should be <= 2a to be in ellipse)
angsep1 = angular_distance(self.lon1, self.lat1, lon, lat)
angsep2 = angular_distance(self.lon2, self.lat2, lon, lat)
angsep = angsep1 + angsep2
return np.power(old_div(180, np.pi), 2) * 1. / (np.pi * a * b) * (angsep <= 2*a)
def get_boundaries(self):
# Truncate the ellipse at 2 times the max of semimajor axis allowed
max_radius = self.a.max_value
min_lat = max(-90., self.lat0.value - 2 * max_radius)
max_lat = min(90., self.lat0.value + 2 * max_radius)
max_abs_lat = max(np.absolute(min_lat), np.absolute(max_lat))
if max_abs_lat > 89. or 2 * max_radius / np.cos(max_abs_lat * np.pi / 180.) >= 180.:
min_lon = 0.
max_lon = 360.
else:
min_lon = self.lon0.value - 2 * max_radius / np.cos(max_abs_lat * np.pi / 180.)
max_lon = self.lon0.value + 2 * max_radius / np.cos(max_abs_lat * np.pi / 180.)
if min_lon < 0.:
min_lon = min_lon + 360.
elif max_lon > 360.:
max_lon = max_lon - 360.
return (min_lon, max_lon), (min_lat, max_lat)
def get_total_spatial_integral(self, z=None):
"""
Returns the total integral (for 2D functions) or the integral over the spatial components (for 3D functions).
needs to be implemented in subclasses.
:return: an array of values of the integral (same dimension as z).
"""
if isinstance( z, u.Quantity):
z = z.value
return np.ones_like( z )
class SpatialTemplate_2D(with_metaclass(FunctionMeta, Function2D)):
r"""
description :
User input Spatial Template. Expected to be normalized to 1/sr
latex : $ hi $
parameters :
K :
desc : normalization
initial value : 1
fix : yes
hash :
desc: hash of model map [needed for memoization]
initial value: 1
fix: yes
"""
def _set_units(self, x_unit, y_unit, z_unit):
self.K.unit = z_unit
# This is optional, and it is only needed if we need more setup after the
# constructor provided by the meta class
def _setup(self):
self._frame = "icrs"
self._fitsfile = None
self._map = None
def load_file(self,fitsfile, ihdu=0):
if fitsfile is None:
raise RuntimeError( "Need to specify a fits file with a template map." )
self._fitsfile=fitsfile
with fits.open(self._fitsfile) as f:
self._wcs = wcs.WCS( header = f[ihdu].header )
self._map = f[ihdu].data
self._nX = f[ihdu].header['NAXIS1']
self._nY = f[ihdu].header['NAXIS2']
#note: map coordinates are switched compared to header. NAXIS1 is coordinate 1, not 0.
#see http://docs.astropy.org/en/stable/io/fits/#working-with-image-data
assert self._map.shape[1] == self._nX, "NAXIS1 = %d in fits header, but %d in map" % (self._nX, self._map.shape[1])
assert self._map.shape[0] == self._nY, "NAXIS2 = %d in fits header, but %d in map" % (self._nY, self._map.shape[0])
#hash sum uniquely identifying the template function (defined by its 2D map array and coordinate system)
#this is needed so that the memoization won't confuse different SpatialTemplate_2D objects.
h = hashlib.sha224()
h.update( self._map)
h.update( repr(self._wcs).encode('utf-8') )
self.hash = int(h.hexdigest(), 16)
def to_dict(self, minimal=False):
data = super(Function2D, self).to_dict(minimal)
if not minimal:
data['extra_setup'] = {"_fitsfile": self._fitsfile, "_frame": self._frame }
return data
def set_frame(self, new_frame):
"""
Set a new frame for the coordinates (the default is ICRS J2000)
:param new_frame: a coordinate frame from astropy
:return: (none)
"""
assert new_frame.lower() in ['icrs', 'galactic', 'fk5', 'fk4', 'fk4_no_e' ]
self._frame = new_frame
def evaluate(self, x, y, K, hash):
if self._map is None:
self.load_file(self._fitsfile)
# We assume x and y are R.A. and Dec
coord = SkyCoord(ra=x, dec=y, frame=self._frame, unit="deg")
#transform input coordinates to pixel coordinates;
#SkyCoord takes care of necessary coordinate frame transformations.
Xpix, Ypix = coord.to_pixel(self._wcs)
Xpix = np.atleast_1d(Xpix.astype(int))
Ypix = np.atleast_1d(Ypix.astype(int))
# find pixels that are in the template ROI, otherwise return zero
#iz = np.where((Xpix<self._nX) & (Xpix>=0) & (Ypix<self._nY) & (Ypix>=0))[0]
iz = (Xpix<self._nX) & (Xpix>=0) & (Ypix<self._nY) & (Ypix>=0)
out = np.zeros_like(Xpix)
out[iz] = self._map[Ypix[iz], Xpix[iz]]
return np.multiply(K,out)
def get_boundaries(self):
if self._map is None:
self.load_file(self._fitsfile)
#We use the max/min RA/Dec of the image corners to define the boundaries.
#Use the 'outside' of the pixel corners, i.e. from pixel 0 to nX in 0-indexed accounting.
Xcorners = np.array( [0, 0, self._nX, self._nX] )
Ycorners = np.array( [0, self._nY, 0, self._nY] )
corners = SkyCoord.from_pixel( Xcorners, Ycorners, wcs=self._wcs, origin = 0).transform_to(self._frame)
min_lon = min(corners.ra.degree)
max_lon = max(corners.ra.degree)
min_lat = min(corners.dec.degree)
max_lat = max(corners.dec.degree)
return (min_lon, max_lon), (min_lat, max_lat)
def get_total_spatial_integral(self, z=None):
"""
Returns the total integral (for 2D functions) or the integral over the spatial components (for 3D functions).
needs to be implemented in subclasses.
:return: an array of values of the integral (same dimension as z).
"""
if isinstance( z, u.Quantity):
z = z.value
return np.multiply(self.K.value,np.ones_like( z ) )
class Power_law_on_sphere(with_metaclass(FunctionMeta, Function2D)):
r"""
description :
A power law function on a sphere (in spherical coordinates)
latex : $$ f(\vec{x}) = \left(\frac{180}{\pi}\right)^{-1.*index} \left\{\begin{matrix} 0.05^{index} & {\rm if} & |\vec{x}-\vec{x}_0| \le 0.05\\ |\vec{x}-\vec{x}_0|^{index} & {\rm if} & 0.05 < |\vec{x}-\vec{x}_0| \le maxr \\ 0 & {\rm if} & |\vec{x}-\vec{x}_0|>maxr\end{matrix}\right. $$
parameters :
lon0 :
desc : Longitude of the center of the source
initial value : 0.0
min : 0.0
max : 360.0
lat0 :
desc : Latitude of the center of the source
initial value : 0.0
min : -90.0
max : 90.0
index :
desc : power law index
initial value : -2.0
min : -5.0
max : -1.0
maxr :
desc : max radius
initial value : 5.
fix : yes
minr :
desc : radius below which the PL is approximated as a constant
initial value : 0.05
fix : yes
"""
def _set_units(self, x_unit, y_unit, z_unit):
# lon0 and lat0 and rdiff have most probably all units of degrees. However,
# let's set them up here just to save for the possibility of using the
# formula with other units (although it is probably never going to happen)
self.lon0.unit = x_unit
self.lat0.unit = y_unit
self.index.unit = u.dimensionless_unscaled
self.maxr.unit = x_unit
self.minr.unit = x_unit
def evaluate(self, x, y, lon0, lat0, index, maxr, minr):
lon, lat = x,y
angsep = angular_distance(lon0, lat0, lon, lat)
if maxr <= minr:
norm = np.power(np.pi / 180., 2.+index) * np.pi * maxr**2 * minr**index
elif self.index.value == -2.:
norm = np.pi * (1.0 + 2. * np.log(maxr / minr) )
else:
norm = np.power(minr * np.pi / 180., 2.+index) * np.pi + 2. * np.pi / (2.+index) * (np.power(maxr * np.pi / 180., index+2.) - np.power(minr * np.pi / 180., index+2.))
value = np.less_equal(angsep,maxr) * np.power(np.pi / 180., index) * np.power(np.add(np.multiply(angsep, np.greater(angsep, minr)), np.multiply(minr, np.less_equal(angsep, minr))), index)
return value / norm
def get_boundaries(self):
return ((self.lon0.value - self.maxr.value), (self.lon0.value + self.maxr.value)), ((self.lat0.value - self.maxr.value), (self.lat0.value + self.maxr.value))
def get_total_spatial_integral(self, z=None):
"""
Returns the total integral (for 2D functions) or the integral over the spatial components (for 3D functions).
needs to be implemented in subclasses.
:return: an array of values of the integral (same dimension as z).
"""
if isinstance( z, u.Quantity):
z = z.value
return np.ones_like( z )
# class FunctionIntegrator(Function2D):
# r"""
# description :
#
# Returns the average of the integrand function (a 1-d function) over the interval x-y. The integrand is set
# using the .integrand property, like in:
#
# > G = FunctionIntegrator()
# > G.integrand = Powerlaw()
#
# latex : $$ G(x,y) = \frac{\int_{x}^{y}~f(x)~dx}{y-x}$$
#
# parameters :
#
# s :
#
# desc : if s=0, then the integral will *not* be normalized by (y-x), otherwise (default) it will.
# initial value : 1
# fix : yes
# """
#
# __metaclass__ = FunctionMeta
#
# def _set_units(self, x_unit, y_unit, z_unit):
#
# # lon0 and lat0 and rdiff have most probably all units of degrees. However,
# # let's set them up here just to save for the possibility of using the
# # formula with other units (although it is probably never going to happen)
#
# self.s = u.dimensionless_unscaled
#
# def evaluate(self, x, y, s):
#
# assert y-x >= 0, "Cannot obtain the integral if the integration interval is zero or negative!"
#
# integral = self._integrand.integral(x, y)
#
# if s==0:
#
# return integral
#
# else:
#
# return integral / (y-x)
#
#
# def get_boundaries(self):
#
# return (-np.inf, +np.inf), (-np.inf, +np.inf)
#
# def _set_integrand(self, function):
#
# self._integrand = function
#
# def _get_integrand(self):
#
# return self._integrand
#
# integrand = property(_get_integrand, _set_integrand,
# doc="""Get/set the integrand""")
#
#
# def to_dict(self, minimal=False):
#
# data = super(Function2D, self).to_dict(minimal)
#
# if not minimal:
# data['extra_setup'] = {'integrand': self.integrand.path}
#
# return data
