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# http://pyrocko.org - GPLv3 # # The Pyrocko Developers, 21st Century # ---|P------/S----------~Lg----------
_testpath.contains_points([(0.5, 0.5), (1.5, 0.5)]) == [True, False]):
p = Path(verts, closed=True) return p.contains_points(points).astype(num.bool)
else: # work around missing contains_points and bug in matplotlib ~ v1.2.0
def path_contains_points(verts, points): p = Path(verts, closed=True) result = num.zeros(points.shape[0], dtype=num.bool) for i in range(result.size): result[i] = p.contains_point(points[i, :])
return result
except AttributeError: def cbrt(x): return x**(1./3.)
num.asarray(x, dtype=num.float) for x in args])
'''Simple location representation
:attrib lat: Latitude degree :attrib lon: Longitude degree ''' self.lat = lat self.lon = lon
''' Clipping data array ``x``
:param x: Continunous data to be clipped :param mi: Clip minimum :param ma: Clip maximum :type x: :py:class:`numpy.ndarray` :type mi: float :type ma: float
:return: Clipped data :rtype: :py:class:`numpy.ndarray` ''' return num.minimum(num.maximum(mi, x), ma)
'''Wrapping continuous data to fundamental phase values.
.. math:: x_{\\mathrm{wrapped}} = x_{\\mathrm{cont},i} - \\frac{ x_{\\mathrm{cont},i} - r_{\\mathrm{min}} } { r_{\\mathrm{max}} - r_{\\mathrm{min}}} \\cdot ( r_{\\mathrm{max}} - r_{\\mathrm{min}}),\\quad x_{\\mathrm{wrapped}}\\; \\in \\;[ r_{\\mathrm{min}},\\, r_{\\mathrm{max}}].
:param x: Continunous data to be wrapped :param mi: Minimum value of wrapped data :param ma: Maximum value of wrapped data :type x: :py:class:`numpy.ndarray` :type mi: float :type ma: float
:return: Wrapped data :rtype: :py:class:`numpy.ndarray` ''' return x - num.floor((x-mi)/(ma-mi)) * (ma-mi)
if len(args) == 2: a, b = args return a.lat, a.lon, b.lat, b.lon
elif len(args) == 4: return args
'''Cosine of the angular distance between two points ``a`` and ``b`` on a sphere.
This function (find implementation below) returns the cosine of the distance angle 'delta' between two points ``a`` and ``b``, coordinates of which are expected to be given in geographical coordinates and in degrees. For numerical stability a maximum of 1.0 is enforced.
.. math::
A_{\\mathrm{lat'}} = \\frac{ \\pi}{180} \\cdot A_{lat}, \\quad A_{\\mathrm{lon'}} = \\frac{ \\pi}{180} \\cdot A_{lon}, \\quad B_{\\mathrm{lat'}} = \\frac{ \\pi}{180} \\cdot B_{lat}, \\quad B_{\\mathrm{lon'}} = \\frac{ \\pi}{180} \\cdot B_{lon}\\\\[0.5cm]
\\cos(\\Delta) = \\min( 1.0, \\quad \\sin( A_{\\mathrm{lat'}}) \\sin( B_{\\mathrm{lat'}} ) + \\cos(A_{\\mathrm{lat'}}) \\cos( B_{\\mathrm{lat'}} ) \\cos( B_{\\mathrm{lon'}} - A_{\\mathrm{lon'}} )
:param a: Location point A :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B :type b: :py:class:`pyrocko.orthodrome.Loc`
:return: cosdelta :rtype: float '''
alat, alon, blat, blon = _latlon_pair(args)
return min( 1.0, math.sin(alat*d2r) * math.sin(blat*d2r) + math.cos(alat*d2r) * math.cos(blat*d2r) * math.cos(d2r*(blon-alon)))
'''Cosine of the angular distance between two points ``a`` and ``b`` on a sphere.
This function returns the cosines of the distance angles *delta* between two points ``a`` and ``b`` given as :py:class:`numpy.ndarray`. The coordinates are expected to be given in geographical coordinates and in degrees. For numerical stability a maximum of ``1.0`` is enforced.
Please find the details of the implementation in the documentation of the function :py:func:`pyrocko.orthodrome.cosdelta` above.
:param a_lats: Latitudes (degree) point A :param a_lons: Longitudes (degree) point A :param b_lats: Latitudes (degree) point B :param b_lons: Longitudes (degree) point B :type a_lats: :py:class:`numpy.ndarray` :type a_lons: :py:class:`numpy.ndarray` :type b_lats: :py:class:`numpy.ndarray` :type b_lons: :py:class:`numpy.ndarray`
:return: cosdelta :type b_lons: :py:class:`numpy.ndarray`, ``(N)`` ''' return num.minimum( 1.0, num.sin(a_lats*d2r) * num.sin(b_lats*d2r) + num.cos(a_lats*d2r) * num.cos(b_lats*d2r) * num.cos(d2r*(b_lons-a_lons)))
'''Azimuth calculation
This function (find implementation below) returns azimuth ... between points ``a`` and ``b``, coordinates of which are expected to be given in geographical coordinates and in degrees.
.. math::
A_{\\mathrm{lat'}} = \\frac{ \\pi}{180} \\cdot A_{lat}, \\quad A_{\\mathrm{lon'}} = \\frac{ \\pi}{180} \\cdot A_{lon}, \\quad B_{\\mathrm{lat'}} = \\frac{ \\pi}{180} \\cdot B_{lat}, \\quad B_{\\mathrm{lon'}} = \\frac{ \\pi}{180} \\cdot B_{lon}\\\\
\\varphi_{\\mathrm{azi},AB} = \\frac{180}{\\pi} \\arctan \\left[ \\frac{ \\cos( A_{\\mathrm{lat'}}) \\cos( B_{\\mathrm{lat'}} ) \\sin(B_{\\mathrm{lon'}} - A_{\\mathrm{lon'}} )} {\\sin ( B_{\\mathrm{lat'}} ) - \\sin( A_{\\mathrm{lat'}} cosdelta) } \\right]
:param a: Location point A :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B :type b: :py:class:`pyrocko.orthodrome.Loc`
:return: Azimuth in degree '''
alat, alon, blat, blon = _latlon_pair(args)
return r2d*math.atan2( math.cos(alat*d2r) * math.cos(blat*d2r) * math.sin(d2r*(blon-alon)), math.sin(d2r*blat) - math.sin(d2r*alat) * cosdelta( alat, alon, blat, blon))
'''Calculation of the azimuth (*track angle*) from a location A towards B.
This function returns azimuths (*track angles*) from locations A towards B given in :py:class:`numpy.ndarray`. Coordinates are expected to be given in geographical coordinates and in degrees.
Please find the details of the implementation in the documentation of the function :py:func:`pyrocko.orthodrome.azimuth`.
:param a_lats: Latitudes (degree) point A :param a_lons: Longitudes (degree) point A :param b_lats: Latitudes (degree) point B :param b_lons: Longitudes (degree) point B :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)``
:return: Azimuths in degrees :rtype: :py:class:`numpy.ndarray`, ``(N)`` ''' if _cosdelta is None: _cosdelta = cosdelta_numpy(a_lats, a_lons, b_lats, b_lons)
return r2d*num.arctan2( num.cos(a_lats*d2r) * num.cos(b_lats*d2r) * num.sin(d2r*(b_lons-a_lons)), num.sin(d2r*b_lats) - num.sin(d2r*a_lats) * _cosdelta)
alat, alon, blat, blon = _latlon_pair(args) if alat == blat and alon == blon: return 0., 180.
implementation = kwargs.get('implementation', 'c') assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.azibazi(alat, alon, blat, blon)
cd = cosdelta(alat, alon, blat, blon) azi = r2d*math.atan2( math.cos(alat*d2r) * math.cos(blat*d2r) * math.sin(d2r*(blon-alon)), math.sin(d2r*blat) - math.sin(d2r*alat) * cd) bazi = r2d*math.atan2( math.cos(blat*d2r) * math.cos(alat*d2r) * math.sin(d2r*(alon-blon)), math.sin(d2r*alat) - math.sin(d2r*blat) * cd)
return azi, bazi
a_lats, a_lons, b_lats, b_lons)
_cosdelta = cosdelta_numpy(a_lats, a_lons, b_lats, b_lons) azis = azimuth_numpy(a_lats, a_lons, b_lats, b_lons, _cosdelta) bazis = azimuth_numpy(b_lats, b_lons, a_lats, a_lons, _cosdelta)
eq = num.logical_and(a_lats == b_lats, a_lons == b_lons) ii_eq = num.where(eq)[0] azis[ii_eq] = 0.0 bazis[ii_eq] = 180.0 return azis, bazis
'''Calculation of the azimuth (*track angle*) and the distance from locations A towards B on a sphere.
The assisting functions used are :py:func:`pyrocko.orthodrome.cosdelta` and :py:func:`pyrocko.orthodrome.azimuth`
:param a_lats: Latitudes (degree) point A :param a_lons: Longitudes (degree) point A :param b_lats: Latitudes (degree) point B :param b_lons: Longitudes (degree) point B :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)``
:return: Azimuths in degrees, distances in degrees :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' _cosdelta = cosdelta_numpy(*args) _azimuths = azimuth_numpy(_cosdelta=_cosdelta, *args) return _azimuths, r2d*num.arccos(_cosdelta)
''' Accurate distance calculation based on a spheroid of rotation.
Function returns distance in meter between points A and B, coordinates of which must be given in geographical coordinates and in degrees. The returned distance should be accurate to 50 m using WGS84. Values for the Earth's equator radius and the Earth's oblateness (``f_oblate``) are defined in the pyrocko configuration file :py:class:`pyrocko.config`.
From wikipedia (http://de.wikipedia.org/wiki/Orthodrome), based on:
``Meeus, J.: Astronomical Algorithms, S 85, Willmann-Bell, Richmond 2000 (2nd ed., 2nd printing), ISBN 0-943396-61-1``
.. math::
F = \\frac{\\pi}{180} \\frac{(A_{lat} + B_{lat})}{2}, \\quad G = \\frac{\\pi}{180} \\frac{(A_{lat} - B_{lat})}{2}, \\quad l = \\frac{\\pi}{180} \\frac{(A_{lon} - B_{lon})}{2} \\quad \\\\[0.5cm] S = \\sin^2(G) \\cdot \\cos^2(l) + \\cos^2(F) \\cdot \\sin^2(l), \\quad \\quad C = \\cos^2(G) \\cdot \\cos^2(l) + \\sin^2(F) \\cdot \\sin^2(l)
.. math::
w = \\arctan \\left( \\sqrt{ \\frac{S}{C}} \\right) , \\quad r = \\sqrt{\\frac{S}{C} }
The spherical-earth distance D between A and B, can be given with:
.. math::
D_{sphere} = 2w \\cdot R_{equator}
The oblateness of the Earth requires some correction with correction factors h1 and h2:
.. math::
h_1 = \\frac{3r - 1}{2C}, \\quad h_2 = \\frac{3r +1 }{2S}\\\\[0.5cm]
D = D_{\\mathrm{sphere}} \\cdot [ 1 + h_1 \\,f_{\\mathrm{oblate}} \\cdot \\sin^2(F) \\cos^2(G) - h_2\\, f_{\\mathrm{oblate}} \\cdot \\cos^2(F) \\sin^2(G)]
:param a: Location point A :type a: :py:class:`pyrocko.orthodrome.Loc` :param b: Location point B :type b: :py:class:`pyrocko.orthodrome.Loc`
:return: Distance in meter :rtype: float '''
alat, alon, blat, blon = _latlon_pair(args)
implementation = kwargs.get('implementation', 'c') assert implementation in ('c', 'python') if implementation == 'c': from pyrocko import orthodrome_ext return orthodrome_ext.distance_accurate50m(alat, alon, blat, blon)
f = (alat + blat)*d2r / 2. g = (alat - blat)*d2r / 2. h = (alon - blon)*d2r / 2.
s = math.sin(g)**2 * math.cos(h)**2 + math.cos(f)**2 * math.sin(h)**2 c = math.cos(g)**2 * math.cos(h)**2 + math.sin(f)**2 * math.sin(h)**2
w = math.atan(math.sqrt(s/c))
if w == 0.0: return 0.0
r = math.sqrt(s*c)/w d = 2.*w*earthradius_equator h1 = (3.*r-1.)/(2.*c) h2 = (3.*r+1.)/(2.*s)
return d * (1. + earth_oblateness * h1 * math.sin(f)**2 * math.cos(g)**2 - earth_oblateness * h2 * math.cos(f)**2 * math.sin(g)**2)
a_lats, a_lons, b_lats, b_lons, implementation='c'):
''' Accurate distance calculation based on a spheroid of rotation.
Function returns distance in meter between points ``a`` and ``b``, coordinates of which must be given in geographical coordinates and in degrees. The returned distance should be accurate to 50 m using WGS84. Values for the Earth's equator radius and the Earth's oblateness (``f_oblate``) are defined in the pyrocko configuration file :py:class:`pyrocko.config`.
From wikipedia (http://de.wikipedia.org/wiki/Orthodrome), based on:
``Meeus, J.: Astronomical Algorithms, S 85, Willmann-Bell, Richmond 2000 (2nd ed., 2nd printing), ISBN 0-943396-61-1``
.. math::
F_i = \\frac{\\pi}{180} \\frac{(a_{lat,i} + a_{lat,i})}{2}, \\quad G_i = \\frac{\\pi}{180} \\frac{(a_{lat,i} - b_{lat,i})}{2}, \\quad l_i= \\frac{\\pi}{180} \\frac{(a_{lon,i} - b_{lon,i})}{2} \\\\[0.5cm] S_i = \\sin^2(G_i) \\cdot \\cos^2(l_i) + \\cos^2(F_i) \\cdot \\sin^2(l_i), \\quad \\quad C_i = \\cos^2(G_i) \\cdot \\cos^2(l_i) + \\sin^2(F_i) \\cdot \\sin^2(l_i)
.. math::
w_i = \\arctan \\left( \\sqrt{\\frac{S_i}{C_i}} \\right), \\quad r_i = \\sqrt{\\frac{S_i}{C_i} }
The spherical-earth distance ``D`` between ``a`` and ``b``, can be given with:
.. math::
D_{\\mathrm{sphere},i} = 2w_i \\cdot R_{\\mathrm{equator}}
The oblateness of the Earth requires some correction with correction factors ``h1`` and ``h2``:
.. math::
h_{1.i} = \\frac{3r - 1}{2C_i}, \\quad h_{2,i} = \\frac{3r +1 }{2S_i}\\\\[0.5cm]
D_{AB,i} = D_{\\mathrm{sphere},i} \\cdot [1 + h_{1,i} \\,f_{\\mathrm{oblate}} \\cdot \\sin^2(F_i) \\cos^2(G_i) - h_{2,i}\\, f_{\\mathrm{oblate}} \\cdot \\cos^2(F_i) \\sin^2(G_i)]
:param a_lats: Latitudes (degree) point A :param a_lons: Longitudes (degree) point A :param b_lats: Latitudes (degree) point B :param b_lons: Longitudes (degree) point B :type a_lats: :py:class:`numpy.ndarray`, ``(N)`` :type a_lons: :py:class:`numpy.ndarray`, ``(N)`` :type b_lats: :py:class:`numpy.ndarray`, ``(N)`` :type b_lons: :py:class:`numpy.ndarray`, ``(N)``
:return: Distances in meter :rtype: :py:class:`numpy.ndarray`, ``(N)`` '''
a_lats, a_lons, b_lats, b_lons)
a_lats, a_lons, b_lats, b_lons)
eq = num.logical_and(a_lats == b_lats, a_lons == b_lons) ii_neq = num.where(num.logical_not(eq))[0]
if num.all(eq): return num.zeros_like(eq, dtype=num.float)
def extr(x): if isinstance(x, num.ndarray) and x.size > 1: return x[ii_neq] else: return x
a_lats = extr(a_lats) a_lons = extr(a_lons) b_lats = extr(b_lats) b_lons = extr(b_lons)
f = (a_lats + b_lats)*d2r / 2. g = (a_lats - b_lats)*d2r / 2. h = (a_lons - b_lons)*d2r / 2.
s = num.sin(g)**2 * num.cos(h)**2 + num.cos(f)**2 * num.sin(h)**2 c = num.cos(g)**2 * num.cos(h)**2 + num.sin(f)**2 * num.sin(h)**2
w = num.arctan(num.sqrt(s/c))
r = num.sqrt(s*c)/w
d = 2.*w*earthradius_equator h1 = (3.*r-1.)/(2.*c) h2 = (3.*r+1.)/(2.*s)
dists = num.zeros(eq.size, dtype=num.float) dists[ii_neq] = d * ( 1. + earth_oblateness * h1 * num.sin(f)**2 * num.cos(g)**2 - earth_oblateness * h2 * num.cos(f)**2 * num.sin(g)**2)
return dists
'''Transform local cartesian coordinates to latitude and longitude.
From east and north coordinates (``x`` and ``y`` coordinate :py:class:`numpy.ndarray`) relative to a reference differences in longitude and latitude are calculated, which are effectively changes in azimuth and distance, respectively:
.. math::
\\text{distance change:}\\; \\Delta {\\bf{a}} &= \\sqrt{{\\bf{y}}^2 + {\\bf{x}}^2 }/ \\mathrm{R_E},
\\text{azimuth change:}\\; \\Delta \\bf{\\gamma} &= \\arctan( \\bf{x} / \\bf{y}).
The projection used preserves the azimuths of the input points.
:param lat0: Latitude origin of the cartesian coordinate system. :param lon0: Longitude origin of the cartesian coordinate system. :param north_m: Northing distances from origin in meters. :param east_m: Easting distances from origin in meters. :type north_m: :py:class:`numpy.ndarray`, ``(N)`` :type east_m: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float
:return: Array with latitudes and longitudes :rtype: :py:class:`numpy.ndarray`, ``(2xN)``
'''
a = num.sqrt(north_m**2+east_m**2)/earthradius gamma = num.arctan2(east_m, north_m)
return azidist_to_latlon_rad(lat0, lon0, gamma, a)
'''(Durchreichen??).
'''
return azidist_to_latlon_rad( lat0, lon0, azimuth_deg/180.*num.pi, distance_deg/180.*num.pi)
''' Absolute latitudes and longitudes are calculated from relative changes.
For numerical stability a range between of ``-1.0`` and ``1.0`` is enforced for ``c`` and ``alpha``.
.. math::
\\Delta {\\bf a}_i \\; \\text{and} \\; \\Delta \\gamma_i \\; \\text{are relative distances and azimuths from lat0 and lon0 for \\textit{i} source points of a finite source.}
.. math::
\\mathrm{b} &= \\frac{\\pi}{2} -\\frac{\\pi}{180}\\;\\mathrm{lat_0}\\\\ {\\bf c}_i &=\\arccos[\\; \\cos(\\Delta {\\bf{a}}_i) \\cos(\\mathrm{b}) + |\\Delta \\gamma_i| \\, \\sin(\\Delta {\\bf a}_i) \\sin(\\mathrm{b})\\; ] \\\\ \\mathrm{lat}_i &= \\frac{180}{\\pi} \\left(\\frac{\\pi}{2} - {\\bf c}_i \\right)
.. math::
\\alpha_i &= \\arcsin \\left[ \\; \\frac{ \\sin(\\Delta {\\bf a}_i ) \\sin(|\\Delta \\gamma_i|)}{\\sin({\\bf c}_i)}\\; \\right] \\\\ \\alpha_i &= \\begin{cases} \\alpha_i, &\\text{if} \\; \\cos(\\Delta {\\bf a}_i) - \\cos(\\mathrm{b}) \\cos({\\bf{c}}_i) > 0, \\; \\text{else} \\\\ \\pi - \\alpha_i, & \\text{if} \\; \\alpha_i > 0,\\; \\text{else}\\\\ -\\pi - \\alpha_i, & \\text{if} \\; \\alpha_i < 0. \\end{cases} \\\\ \\mathrm{lon}_i &= \\mathrm{lon_0} + \\frac{180}{\\pi} \\, \\frac{\\Delta \\gamma_i }{|\\Delta \\gamma_i|} \\cdot \\alpha_i \\text{, with $\\alpha_i \\in [-\\pi,\\pi]$}
:param lat0: Latitude origin of the cartesian coordinate system. :param lon0: Longitude origin of the cartesian coordinate system. :param distance_rad: Distances from origin in radians. :param azimuth_rad: Azimuth from radians. :type distance_rad: :py:class:`numpy.ndarray`, ``(N)`` :type azimuth_rad: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float
:return: Array with latitudes and longitudes :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` '''
a = distance_rad gamma = azimuth_rad
b = math.pi/2.-lat0*d2r
alphasign = 1. alphasign = num.where(gamma < 0, -1., 1.) gamma = num.abs(gamma)
c = num.arccos(clip( num.cos(a)*num.cos(b)+num.sin(a)*num.sin(b)*num.cos(gamma), -1., 1.))
alpha = num.arcsin(clip( num.sin(a)*num.sin(gamma)/num.sin(c), -1., 1.))
alpha = num.where( num.cos(a)-num.cos(b)*num.cos(c) < 0, num.where(alpha > 0, math.pi-alpha, -math.pi-alpha), alpha)
lat = r2d * (math.pi/2. - c) lon = wrap(lon0 + r2d*alpha*alphasign, -180., 180.)
return lat, lon
'''Transform local cartesian coordinates to latitude and longitude.
Like :py:func:`pyrocko.orthodrome.ne_to_latlon`, but this method (implementation below), although it should be numerically more stable, suffers problems at points which are *across the pole* as seen from the cartesian origin.
.. math::
\\text{distance change:}\\; \\Delta {{\\bf a}_i} &= \\sqrt{{\\bf{y}}^2_i + {\\bf{x}}^2_i }/ \\mathrm{R_E},\\\\ \\text{azimuth change:}\\; \\Delta {\\bf \\gamma}_i &= \\arctan( {\\bf x}_i {\\bf y}_i). \\\\ \\mathrm{b} &= \\frac{\\pi}{2} -\\frac{\\pi}{180} \\;\\mathrm{lat_0}\\\\
.. math::
{{\\bf z}_1}_i &= \\cos{\\left( \\frac{\\Delta {\\bf a}_i - \\mathrm{b}}{2} \\right)} \\cos {\\left( \\frac{|\\gamma_i|}{2} \\right) }\\\\ {{\\bf n}_1}_i &= \\cos{\\left( \\frac{\\Delta {\\bf a}_i + \\mathrm{b}}{2} \\right)} \\sin {\\left( \\frac{|\\gamma_i|}{2} \\right) }\\\\ {{\\bf z}_2}_i &= \\sin{\\left( \\frac{\\Delta {\\bf a}_i - \\mathrm{b}}{2} \\right)} \\cos {\\left( \\frac{|\\gamma_i|}{2} \\right) }\\\\ {{\\bf n}_2}_i &= \\sin{\\left( \\frac{\\Delta {\\bf a}_i + \\mathrm{b}}{2} \\right)} \\sin {\\left( \\frac{|\\gamma_i|}{2} \\right) }\\\\ {{\\bf t}_1}_i &= \\arctan{\\left( \\frac{{{\\bf z}_1}_i} {{{\\bf n}_1}_i} \\right) }\\\\ {{\\bf t}_2}_i &= \\arctan{\\left( \\frac{{{\\bf z}_2}_i} {{{\\bf n}_2}_i} \\right) } \\\\[0.5cm] c &= \\begin{cases} 2 \\cdot \\arccos \\left( {{\\bf z}_1}_i / \\sin({{\\bf t}_1}_i) \\right),\\; \\text{if } |\\sin({{\\bf t}_1}_i)| > |\\sin({{\\bf t}_2}_i)|,\\; \\text{else} \\\\ 2 \\cdot \\arcsin{\\left( {{\\bf z}_2}_i / \\sin({{\\bf t}_2}_i) \\right)}. \\end{cases}\\\\
.. math::
{\\bf {lat}}_i &= \\frac{180}{ \\pi } \\left( \\frac{\\pi}{2} - {\\bf {c}}_i \\right) \\\\ {\\bf {lon}}_i &= {\\bf {lon}}_0 + \\frac{180}{ \\pi } \\frac{\\gamma_i}{|\\gamma_i|}, \\text{ with}\\; \\gamma \\in [-\\pi,\\pi]
:param lat0: Latitude origin of the cartesian coordinate system. :param lon0: Longitude origin of the cartesian coordinate system. :param north_m: Northing distances from origin in meters. :param east_m: Easting distances from origin in meters. :type north_m: :py:class:`numpy.ndarray`, ``(N)`` :type east_m: :py:class:`numpy.ndarray`, ``(N)`` :type lat0: float :type lon0: float
:return: Array with latitudes and longitudes :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` '''
b = math.pi/2.-lat0*d2r a = num.sqrt(north_m**2+east_m**2)/earthradius
gamma = num.arctan2(east_m, north_m) alphasign = 1. alphasign = num.where(gamma < 0., -1., 1.) gamma = num.abs(gamma)
z1 = num.cos((a-b)/2.)*num.cos(gamma/2.) n1 = num.cos((a+b)/2.)*num.sin(gamma/2.) z2 = num.sin((a-b)/2.)*num.cos(gamma/2.) n2 = num.sin((a+b)/2.)*num.sin(gamma/2.) t1 = num.arctan2(z1, n1) t2 = num.arctan2(z2, n2)
alpha = t1 + t2
sin_t1 = num.sin(t1) sin_t2 = num.sin(t2) c = num.where( num.abs(sin_t1) > num.abs(sin_t2), num.arccos(z1/sin_t1)*2., num.arcsin(z2/sin_t2)*2.)
lat = r2d * (math.pi/2. - c) lon = wrap(lon0 + r2d*alpha*alphasign, -180., 180.) return lat, lon
'''Relative cartesian coordinates with respect to a reference location.
For two locations, a reference location A and another location B, given in geographical coordinates in degrees, the corresponding cartesian coordinates are calculated. Assisting functions are :py:func:`pyrocko.orthodrome.azimuth` and :py:func:`pyrocko.orthodrome.distance_accurate50m`.
.. math::
D_{AB} &= \\mathrm{distance\\_accurate50m(}A, B \\mathrm{)}, \\quad \\varphi_{\\mathrm{azi},AB} = \\mathrm{azimuth(}A,B \\mathrm{)}\\\\[0.3cm]
n &= D_{AB} \\cdot \\cos( \\frac{\\pi }{180} \\varphi_{\\mathrm{azi},AB} )\\\\ e &= D_{AB} \\cdot \\sin( \\frac{\\pi }{180} \\varphi_{\\mathrm{azi},AB})
:param refloc: Location reference point :type refloc: :py:class:`pyrocko.orthodrome.Loc` :param loc: Location of interest :type loc: :py:class:`pyrocko.orthodrome.Loc`
:return: Northing and easting from refloc to location :rtype: tuple, float
'''
azi = azimuth(*args) dist = distance_accurate50m(*args) n, e = math.cos(azi*d2r)*dist, math.sin(azi*d2r)*dist return n, e
'''Relative cartesian coordinates with respect to a reference location.
For two locations, a reference location (``lat0``, ``lon0``) and another location B, given in geographical coordinates in degrees, the corresponding cartesian coordinates are calculated. Assisting functions are :py:func:`azimuth` and :py:func:`distance_accurate50m`.
:param lat0: reference location latitude :param lon0: reference location longitude :param lat: absolute location latitude :param lon: absolute location longitude
:return: ``(n, e)``: relative north and east positions :rtype: :py:class:`numpy.ndarray`, ``(2xN)``
Implemented formulations:
.. math::
D_{AB} &= \\mathrm{distance\\_accurate50m(}A, B \\mathrm{)}, \\quad \\varphi_{\\mathrm{azi},AB} = \\mathrm{azimuth(}A,B \\mathrm{)}\\\\[0.3cm]
n &= D_{AB} \\cdot \\cos( \\frac{\\pi }{180} \\varphi_{ \\mathrm{azi},AB} )\\\\ e &= D_{AB} \\cdot \\sin( \\frac{\\pi }{180} \\varphi_{ \\mathrm{azi},AB} ) '''
azi = azimuth_numpy(lat0, lon0, lat, lon) dist = distance_accurate50m_numpy(lat0, lon0, lat, lon) n = num.cos(azi*d2r)*dist e = num.sin(azi*d2r)*dist return n, e
global _wgs84 if _wgs84 is None: from geographiclib.geodesic import Geodesic _wgs84 = Geodesic.WGS84
return _wgs84
it = num.nditer(args + (None,)) for ops in it: ops[-1][...] = f(*ops[:-1])
return it.operands[-1] it = num.nditer(args + (None, None)) for ops in it: ops[-2][...], ops[-1][...] = f(*ops[:-2])
return it.operands[-2], it.operands[-1]
def ne_to_latlon2(lat0, lon0, north_m, east_m): wgs84 = get_wgs84() az = num.arctan2(east_m, north_m)*r2d dist = num.sqrt(east_m**2 + north_m**2) x = wgs84.Direct(lat0, lon0, az, dist) return x['lat2'], x['lon2']
def latlon_to_ne2(lat0, lon0, lat1, lon1): wgs84 = get_wgs84() x = wgs84.Inverse(lat0, lon0, lat1, lon1) dist = x['s12'] az = x['azi1'] n = num.cos(az*d2r)*dist e = num.sin(az*d2r)*dist return n, e
def distance_accurate15nm(lat1, lon1, lat2, lon2): wgs84 = get_wgs84() return wgs84.Inverse(lat1, lon1, lat2, lon2)['s12']
'''Normalize parameterization of a rectangular geographical region.
:param region: ``(west, east, south, north)`` :returns: ``(west, east, south, north)``, where ``west <= east`` and where ``west`` and ``east`` are in the range ``[-180., 180.+360.[`` ''' west, east, south, north = [float(x) for x in region]
assert -180. - 360. <= west < 180. assert -180. < east <= 180. + 360. assert -90. <= south < 90. assert -90. < north <= 90.
if east < west: east += 360.
if west < -180.: west += 360. east += 360.
return (west, east, south, north)
''' Check what points are contained in a rectangular geographical region.
:param p: NumPy array of shape ``(N, 2)`` where each row is a ``(lat, lon)`` pair [deg] :param region: ``(west, east, south, north)`` [deg] :returns: NumPy array of shape ``(N)``, type ``bool`` '''
w, e, s, n = positive_region(region) return num.logical_and( num.logical_and(s <= p[:, 0], p[:, 0] <= n), num.logical_or( num.logical_and(w <= p[:, 1], p[:, 1] <= e), num.logical_and(w-360. <= p[:, 1], p[:, 1] <= e-360.)))
''' Check if a point is contained in a rectangular geographical region.
:param p: ``(lat, lon)`` [deg] :param region: ``(west, east, south, north)`` [deg] :returns: ``bool`` '''
w, e, s, n = positive_region(region) return num.logical_and( num.logical_and(s <= p[0], p[0] <= n), num.logical_or( num.logical_and(w <= p[1], p[1] <= e), num.logical_and(w-360. <= p[1], p[1] <= e-360.)))
''' Get a rectangular region which fully contains a given circular region.
:param lat,lon: center of circular region [deg] :param radius: radius of circular region [m] :return: rectangular region as ``(east, west, south, north)`` [deg] ''' radius_deg = radius * m2d if radius_deg < 45.: lat_min = max(-90., lat - radius_deg) lat_max = min(90., lat + radius_deg) absmaxlat = max(abs(lat_min), abs(lat_max)) if absmaxlat > 89: lon_min = -180. lon_max = 180. else: lon_min = max( -180. - 360., lon - radius_deg / math.cos(absmaxlat*d2r)) lon_max = min( 180. + 360., lon + radius_deg / math.cos(absmaxlat*d2r))
lon_min, lon_max, lat_min, lat_max = positive_region( (lon_min, lon_max, lat_min, lat_max))
return lon_min, lon_max, lat_min, lat_max
else: return None
'''Calculate geographic midpoints by finding the center of gravity.
This method suffers from instabilities if points are centered around the poles.
:param lats: array of latitudes :param lons: array of longitudes :param weights: array weighting factors (optional) :type lats: :py:class:`numpy.ndarray`, ``(N)`` :type lons: :py:class:`numpy.ndarray`, ``(N)`` :type weights: :py:class:`numpy.ndarray`, ``(N)``
:return: Latitudes and longitudes of the modpoints :rtype: :py:class:`numpy.ndarray`, ``(2xN)`` ''' if not weights: weights = num.ones(len(lats))
total_weigth = num.sum(weights) weights /= total_weigth lats = lats * d2r lons = lons * d2r x = num.sum(num.cos(lats) * num.cos(lons) * weights) y = num.sum(num.cos(lats) * num.sin(lons) * weights) z = num.sum(num.sin(lats) * weights)
lon = num.arctan2(y, x) hyp = num.sqrt(x**2 + y**2) lat = num.arctan2(z, hyp)
return lat/d2r, lon/d2r
coords = num.array([loc.effective_latlon for loc in locations]) return geographic_midpoint(coords[:, 0], coords[:, 1], weights)
''' Convert geodetic coordinates to Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates. [#1]_ [#2]_
:param lat: Geodetic latitude in [deg]. :param lon: Geodetic longitude in [deg]. :param alt: Geodetic altitude (height) in [m] (positive for points outside the geoid). :type lat: float :type lon: float :type alt: float
:return: ECEF Cartesian coordinates (X, Y, Z) in [m]. :rtype: tuple, float
.. [#1] https://en.wikipedia.org/wiki/ECEF .. [#2] https://en.wikipedia.org/wiki/Geographic_coordinate_conversion #From_geodetic_to_ECEF_coordinates '''
f = earth_oblateness a = earthradius_equator e2 = 2*f - f**2
lat, lon = num.radians(lat), num.radians(lon) # Normal (plumb line) N = a / num.sqrt(1.0 - (e2 * num.sin(lat)**2))
X = (N+alt) * num.cos(lat) * num.cos(lon) Y = (N+alt) * num.cos(lat) * num.sin(lon) Z = (N*(1.0-e2) + alt) * num.sin(lat)
return (X, Y, Z)
''' Convert Earth-Centered, Earth-Fixed (ECEF) Cartesian coordinates to geodetic coordinates (Ferrari's solution).
:param X, Y, Z: Cartesian coordinates in ECEF system in [m]. :type X, Y, Z: float
:return: Geodetic coordinates (lat, lon, alt). Latitude and longitude are in [deg] and altitude is in [m] (positive for points outside the geoid). :rtype: tuple, float
.. seealso :: https://en.wikipedia.org/wiki/Geographic_coordinate_conversion #The_application_of_Ferrari.27s_solution ''' f = earth_oblateness a = earthradius_equator b = a * (1. - f) e2 = 2.*f - f**2
# usefull a2 = a**2 b2 = b**2 e4 = e2**2 X2 = X**2 Y2 = Y**2 Z2 = Z**2
r = num.sqrt(X2 + Y2) r2 = r**2
e_prime2 = (a2 - b2)/b2 E2 = a2 - b2 F = 54. * b2 * Z2 G = r2 + (1.-e2)*Z2 - (e2*E2) C = (e4 * F * r2) / (G**3) S = cbrt(1. + C + num.sqrt(C**2 + 2.*C)) P = F / (3. * (S + 1./S + 1.)**2 * G**2) Q = num.sqrt(1. + (2.*e4*P))
dum1 = -(P*e2*r) / (1.+Q) dum2 = 0.5 * a2 * (1. + 1./Q) dum3 = (P * (1.-e2) * Z2) / (Q * (1.+Q)) dum4 = 0.5 * P * r2 r0 = dum1 + num.sqrt(dum2 - dum3 - dum4)
U = num.sqrt((r - e2*r0)**2 + Z2) V = num.sqrt((r - e2*r0)**2 + (1.-e2)*Z2) Z0 = (b2*Z) / (a*V)
alt = U * (1. - (b2 / (a*V))) lat = num.arctan((Z + e_prime2 * Z0)/r) lon = num.arctan2(Y, X)
return (lat*r2d, lon*r2d, alt)
if latlons.ndim == 1: return latlon_to_xyz(latlons[num.newaxis, :])[0]
points = num.zeros((latlons.shape[0], 3)) lats = latlons[:, 0] lons = latlons[:, 1] points[:, 0] = num.cos(lats*d2r) * num.cos(lons*d2r) points[:, 1] = num.cos(lats*d2r) * num.sin(lons*d2r) points[:, 2] = num.sin(lats*d2r) return points
if xyz.ndim == 1: return xyz_to_latlon(xyz[num.newaxis, :])[0]
latlons = num.zeros((xyz.shape[0], 2)) latlons[:, 0] = num.arctan2( xyz[:, 2], num.sqrt(xyz[:, 0]**2 + xyz[:, 1]**2)) * r2d latlons[:, 1] = num.arctan2( xyz[:, 1], xyz[:, 0]) * r2d return latlons
rot0 = euler_to_matrix(0., -90.*d2r, 0.).A rot1 = euler_to_matrix(-d2r*lat, 0., -d2r*lon).A return num.dot(rot0.T, num.dot(rot1, rot0)).T
return num.sqrt(num.sum((a-b)**2, axis=a.ndim-1))
return num.sum( (points2[1:, 0] - points2[:-1, 0]) * (points2[1:, 1] + points2[:-1, 1]))
dists = distances3d(points[1:, :], points[:-1, :]) if dists.size > 0: maxdist = num.max(dists) cutoff = maxdist**2 / 2. else: cutoff = 1.0e-5
points = points.copy() if num.any(points[:, 0] < -1. + cutoff): raise Farside()
points_out = points[:, 1:].copy() factor = 1.0 / (1.0 + points[:, 0]) points_out *= factor[:, num.newaxis]
return points_out
dists = distances3d(points[1:, :], points[:-1, :]) if dists.size > 0: maxdist = num.max(dists) cutoff = maxdist**2 / 2. else: cutoff = 1.0e-5
points = points.copy() if num.any(points[:, 0] < -1. + cutoff): raise Farside()
points_out = points[:, 1:].copy() factor = 1.0 / (1.0 + points[:, 0]) points_out *= factor[:, num.newaxis]
if circulation(points_out) >= 0: raise Farside()
return points_out
points_out = points[:, 1:].copy() if num.any(points[:, 0] < cutoff): raise Farside()
factor = 1.0 / points[:, 0] points_out *= factor[:, num.newaxis] return points_out
if i == 1: return x else: return num.logical_not(x)
''' Test which points are inside polygon on a sphere.
:param polygon: Point coordinates defining the polygon [deg]. :type polygon: :py:class:`numpy.ndarray` of shape (N, 2), second index 0=lat, 1=lon :param points: Coordinates of points to test [deg]. :type points: :py:class:`numpy.ndarray` of shape (N, 2), second index 0=lat, 1=lon
:returns: Boolean mask array. :rtype: :py:class:`numpy.ndarray` of shape (N,)
The inside of the polygon is defined as the area which is to the left hand side of an observer walking the polygon line, points in order, on the sphere. Lines between the polygon points are treated as great circle paths. The polygon may be arbitrarily complex, as long as it does not have any crossings or thin parts with zero width. The polygon may contain the poles and is allowed to wrap around the sphere multiple times.
The algorithm works by consecutive cutting of the polygon into (almost) hemispheres and subsequent Gnomonic projections to perform the point-in-polygon tests on a 2D plane. '''
and_ = num.logical_and
points_xyz = latlon_to_xyz(points) mask_x = 0. <= points_xyz[:, 0] mask_y = 0. <= points_xyz[:, 1] mask_z = 0. <= points_xyz[:, 2]
result = num.zeros(points.shape[0], dtype=num.int)
for ix in [-1, 1]: for iy in [-1, 1]: for iz in [-1, 1]: mask = and_( and_(cneg(ix, mask_x), cneg(iy, mask_y)), cneg(iz, mask_z))
center_xyz = num.array([ix, iy, iz], dtype=num.float)
lat, lon = xyz_to_latlon(center_xyz) rot = rot_to_00(lat, lon)
points_rot_xyz = num.dot(rot, points_xyz[mask, :].T).T points_rot_pro = gnomonic_x(points_rot_xyz)
offset = 0.01
poly_xyz = latlon_to_xyz(polygon) poly_rot_xyz = num.dot(rot, poly_xyz.T).T poly_rot_xyz[:, 0] -= offset groups = spoly_cut([poly_rot_xyz], axis=0)
for poly_rot_group_xyz in groups[1]: poly_rot_group_xyz[:, 0] += offset
poly_rot_group_pro = gnomonic_x( poly_rot_group_xyz)
if circulation(poly_rot_group_pro) > 0: result[mask] += path_contains_points( poly_rot_group_pro, points_rot_pro) else: result[mask] -= path_contains_points( poly_rot_group_pro, points_rot_pro)
return result.astype(num.bool)
''' Test if point is inside polygon on a sphere.
:param polygon: Point coordinates defining the polygon [deg]. :type polygon: :py:class:`numpy.ndarray` of shape (N, 2), second index 0=lat, 1=lon :param point: Coordinates ``(lat, lon)`` of point to test [deg].
Convenience wrapper to :py:func:`contains_points` to test a single point. '''
return bool( contains_points(polygon, num.asarray(point)[num.newaxis, :])[0]) |