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# -*- coding: utf-8 -*- 

 

u'''(INTERNAL) Spherical geodesy bases. 

 

Base classes C{CartesianSphericalBase} and C{LatLonSphericalBase} 

imported by L{sphericalNvector} or L{sphericalTrigonometry}. 

 

Pure Python implementation of geodetic (lat-/longitude) functions, 

transcoded in part from JavaScript originals by I{(C) Chris Veness 2011-2016} 

and published under the same MIT Licence**, see 

U{Latitude/Longitude<https://www.Movable-Type.co.UK/scripts/latlong.html>}. 

''' 

# make sure int/int division yields float quotient, see .basics 

from __future__ import division 

 

from pygeodesy.basics import isnear0, isnon0, map1 

from pygeodesy.cartesianBase import CartesianBase 

from pygeodesy.datums import Datums, _spherical_datum 

from pygeodesy.ellipsoids import R_M, R_MA 

from pygeodesy.errors import IntersectionError 

from pygeodesy.fmath import favg, fdot, hypot 

from pygeodesy.interns import EPS, NN, PI, PI2, PI_2, _COMMA_, \ 

_datum_, _distant_, _exceed_PI_radians_, \ 

_name_, _near_concentric_, _too_, \ 

_1_0, _180_0, _360_0 

from pygeodesy.latlonBase import LatLonBase, _trilaterate5 # PYCHOK passed 

from pygeodesy.lazily import _ALL_DOCS 

from pygeodesy.namedTuples import Bearing2Tuple 

from pygeodesy.nvectorBase import NvectorBase, _xattrs # streprs 

from pygeodesy.props import property_doc_ 

from pygeodesy.units import Bearing_, Height, Radians_, Radius, Radius_ 

from pygeodesy.utily import acos1, atan2b, degrees90, degrees180, \ 

sincos2, tanPI_2_2, wrapPI 

 

from math import cos, log, sin, sqrt 

 

__all__ = () 

__version__ = '21.06.30' 

 

 

def _angular(distance, radius): # PYCHOK for export 

'''(INTERNAL) Return the angular distance in C{radians}. 

 

@raise UnitError: Invalid B{C{distance}} or B{C{radius}}. 

''' 

return Radians_(distance / Radius_(radius=radius), low=EPS) 

 

 

def _rads3(rad1, rad2, radius): # in .sphericalTrigonometry 

'''(INTERNAL) Convert radii to radians. 

''' 

r1 = Radius_(rad1=rad1) 

r2 = Radius_(rad2=rad2) 

if radius is not None: # convert radii to radians 

r1 = _angular(r1, radius) 

r2 = _angular(r2, radius) 

 

x = r1 < r2 

if x: 

r1, r2 = r2, r1 

if r1 > PI: 

raise IntersectionError(rad1=rad1, rad2=rad2, 

txt=_exceed_PI_radians_) 

return r1, r2, x 

 

 

class CartesianSphericalBase(CartesianBase): 

'''(INTERNAL) Base class for spherical C{Cartesian}s. 

''' 

_datum = Datums.Sphere # L{Datum} 

 

def intersections2(self, rad1, other, rad2, radius=R_M): 

'''Compute the intersection points of two circles each defined 

by a center point and a radius. 

 

@arg rad1: Radius of the this circle (C{meter} or C{radians}, 

see B{C{radius}}). 

@arg other: Center of the other circle (C{Cartesian}). 

@arg rad2: Radius of the other circle (C{meter} or C{radians}, 

see B{C{radius}}). 

@kwarg radius: Mean earth radius (C{meter} or C{None} if both 

B{C{rad1}} and B{C{rad2}} are given in C{radians}). 

 

@return: 2-Tuple of the intersection points, each C{Cartesian}. 

The intersection points are the same C{Cartesian} 

instance for abutting circles. 

 

@raise IntersectionError: Concentric, antipodal, invalid or 

non-intersecting circles. 

 

@raise TypeError: If B{C{other}} is not C{Cartesian}. 

 

@raise ValueError: Invalid B{C{rad1}}, B{C{rad2}} or B{C{radius}}. 

 

@see: U{Calculating intersection of two Circles 

<https://GIS.StackExchange.com/questions/48937/ 

calculating-intersection-of-two-circles>} and method 

or function C{trilaterate3d2}. 

''' 

x1, x2 = self, self.others(other) 

r1, r2, x = _rads3(rad1, rad2, radius) 

if x: 

x1, x2 = x2, x1 

try: 

n, q = x1.cross(x2), x1.dot(x2) 

n2, q1 = n.length2, (_1_0 - q**2) 

if n2 < EPS or isnear0(q1): 

raise ValueError(_near_concentric_) 

c1, c2 = cos(r1), cos(r2) 

x0 = x1.times((c1 - q * c2) / q1).plus( 

x2.times((c2 - q * c1) / q1)) 

n1 = _1_0 - x0.length2 

if n1 < EPS: 

raise ValueError(_too_(_distant_)) 

except ValueError as x: 

raise IntersectionError(center=self, rad1=rad1, 

other=other, rad2=rad2, txt=str(x)) 

n = n.times(sqrt(n1 / n2)) 

if n.length > EPS: 

x1 = x0.plus(n) 

x2 = x0.minus(n) 

else: # abutting circles 

x1 = x2 = x0 

 

return (_xattrs(x1, self, _datum_, _name_), 

_xattrs(x2, self, _datum_, _name_)) 

 

 

class LatLonSphericalBase(LatLonBase): 

'''(INTERNAL) Base class for spherical C{LatLon}s. 

''' 

_datum = Datums.Sphere # spherical L{Datum} 

 

def __init__(self, lat, lon, height=0, datum=None, name=NN): 

'''Create a spherical C{LatLon} point frome the given 

lat-, longitude and height on the given datum. 

 

@arg lat: Latitude (C{degrees} or DMS C{[N|S]}). 

@arg lon: Longitude (C{degrees} or DMS C{str[E|W]}). 

@kwarg height: Optional elevation (C{meter}, the same units 

as the datum's half-axes). 

@kwarg datum: Optional, spherical datum to use (L{Datum}, 

L{Ellipsoid}, L{Ellipsoid2}, L{a_f2Tuple}) 

or C{scalar} earth radius). 

@kwarg name: Optional name (string). 

 

@raise TypeError: If B{C{datum}} invalid or not 

not spherical. 

 

@example: 

 

>>> p = LatLon(51.4778, -0.0016) # height=0, datum=Datums.WGS84 

''' 

LatLonBase.__init__(self, lat, lon, height=height, name=name) 

if datum not in (None, self.datum): 

self._datum = _spherical_datum(datum, name=self.name, raiser=True) 

 

def bearingTo2(self, other, wrap=False, raiser=False): 

'''Return the initial and final bearing (forward and reverse 

azimuth) from this to an other point. 

 

@arg other: The other point (C{LatLon}). 

@kwarg wrap: Wrap and unroll longitudes (C{bool}). 

@kwarg raiser: Optionally, raise L{CrossError} (C{bool}). 

 

@return: A L{Bearing2Tuple}C{(initial, final)}. 

 

@raise TypeError: The B{C{other}} point is not spherical. 

 

@see: Methods C{initialBearingTo} and C{finalBearingTo}. 

''' 

# .initialBearingTo is inside .-Nvector and .-Trigonometry 

i = self.initialBearingTo(other, wrap=wrap, raiser=raiser) # PYCHOK .initialBearingTo 

f = self.finalBearingTo( other, wrap=wrap, raiser=raiser) 

return Bearing2Tuple(i, f, name=self.name) 

 

@property_doc_(''' this point's datum (L{Datum}).''') 

def datum(self): 

'''Get this point's datum (L{Datum}). 

''' 

return self._datum 

 

@datum.setter # PYCHOK setter! 

def datum(self, datum): 

'''Set this point's datum I{without conversion}. 

 

@arg datum: New spherical datum (L{Datum}, L{Ellipsoid}, 

L{Ellipsoid2}, L{a_f2Tuple}) or C{scalar} 

earth radius). 

 

@raise TypeError: If B{C{datum}} invalid or not 

not spherical. 

''' 

d = _spherical_datum(datum, name=self.name, raiser=True) 

self._update(d != self._datum) 

self._datum = d 

 

def finalBearingTo(self, other, wrap=False, raiser=False): 

'''Return the final bearing (reverse azimuth) from this to 

an other point. 

 

@arg other: The other point (spherical C{LatLon}). 

@kwarg wrap: Wrap and unroll longitudes (C{bool}). 

@kwarg raiser: Optionally, raise L{CrossError} (C{bool}). 

 

@return: Final bearing (compass C{degrees360}). 

 

@raise TypeError: The B{C{other}} point is not spherical. 

 

@example: 

 

>>> p = LatLon(52.205, 0.119) 

>>> q = LatLon(48.857, 2.351) 

>>> b = p.finalBearingTo(q) # 157.9 

''' 

self.others(other) 

 

# final bearing is the reverse of the other, initial one; 

# .initialBearingTo is inside .-Nvector and .-Trigonometry 

b = other.initialBearingTo(self, wrap=wrap, raiser=raiser) 

return (b + _180_0) % _360_0 # == wrap360 since b >= 0 

 

def maxLat(self, bearing): 

'''Return the maximum latitude reached when travelling 

on a great circle on given bearing from this point 

based on Clairaut's formula. 

 

The maximum latitude is independent of longitude 

and the same for all points on a given latitude. 

 

Negate the result for the minimum latitude (on the 

Southern hemisphere). 

 

@arg bearing: Initial bearing (compass C{degrees360}). 

 

@return: Maximum latitude (C{degrees90}). 

 

@raise ValueError: Invalid B{C{bearing}}. 

 

@JSname: I{maxLatitude}. 

''' 

m = acos1(abs(sin(Bearing_(bearing)) * cos(self.phi))) 

return degrees90(m) 

 

def minLat(self, bearing): 

'''Return the minimum latitude reached when travelling 

on a great circle on given bearing from this point. 

 

@arg bearing: Initial bearing (compass C{degrees360}). 

 

@return: Minimum latitude (C{degrees90}). 

 

@see: Method L{maxLat} for more details. 

 

@raise ValueError: Invalid B{C{bearing}}. 

 

@JSname: I{minLatitude}. 

''' 

return -self.maxLat(bearing) 

 

def parse(self, strllh, height=0, sep=_COMMA_, name=NN): 

'''Parse a string representing a similar, spherical C{LatLon} 

point, consisting of C{"lat, lon[, height]"}. 

 

@arg strllh: Lat, lon and optional height (C{str}), 

see function L{parse3llh}. 

@kwarg height: Optional, default height (C{meter}). 

@kwarg sep: Optional separator (C{str}). 

@kwarg name: Optional instance name (C{str}), 

overriding this name. 

 

@return: The similar point (spherical C{LatLon}). 

 

@raise ParseError: Invalid B{C{strllh}}. 

''' 

from pygeodesy.dms import parse3llh 

r = self.classof(*parse3llh(strllh, height=height, sep=sep)) 

if name: 

r.rename(name) 

return r 

 

def _rhumb3(self, other): 

'''(INTERNAL) Rhumb_ helper function. 

 

@arg other: The other point (spherical C{LatLon}). 

''' 

self.others(other) 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

# if |db| > 180 take shorter rhumb 

# line across the anti-meridian 

db = wrapPI(b2 - b1) 

dp = log(tanPI_2_2(a2) / tanPI_2_2(a1)) 

return (a2 - a1), db, dp 

 

def rhumbBearingTo(self, other): 

'''Return the initial bearing (forward azimuth) from this to 

an other point along a rhumb (loxodrome) line. 

 

@arg other: The other point (spherical C{LatLon}). 

 

@return: Initial bearing (compass C{degrees360}). 

 

@raise TypeError: The B{C{other}} point is not spherical. 

 

@example: 

 

>>> p = LatLon(51.127, 1.338) 

>>> q = LatLon(50.964, 1.853) 

>>> b = p.rhumbBearingTo(q) # 116.7 

''' 

_, db, dp = self._rhumb3(other) 

return atan2b(db, dp) 

 

def rhumbDestination(self, distance, bearing, radius=R_M, height=None): 

'''Return the destination point having travelled along a rhumb 

(loxodrome) line from this point the given distance on the 

given bearing. 

 

@arg distance: Distance travelled (C{meter}, same units as 

B{C{radius}}). 

@arg bearing: Bearing from this point (compass C{degrees360}). 

@kwarg radius: Mean earth radius (C{meter}). 

@kwarg height: Optional height, overriding the default height 

(C{meter}, same unit as B{C{radius}}). 

 

@return: The destination point (spherical C{LatLon}). 

 

@raise ValueError: Invalid B{C{distance}}, B{C{bearing}}, 

B{C{radius}} or B{C{height}}. 

 

@example: 

 

>>> p = LatLon(51.127, 1.338) 

>>> q = p.rhumbDestination(40300, 116.7) # 50.9642°N, 001.8530°E 

 

@JSname: I{rhumbDestinationPoint} 

''' 

r = _angular(distance, radius) 

 

a1, b1 = self.philam 

sb, cb = sincos2(Bearing_(bearing)) 

 

da = r * cb 

a2 = a1 + da 

# normalize latitude if past pole 

if a2 > PI_2: 

a2 = PI - a2 

elif a2 < -PI_2: 

a2 = -PI - a2 

 

dp = log(tanPI_2_2(a2) / tanPI_2_2(a1)) 

# E-W course becomes ill-conditioned with 0/0 

q = (da / dp) if abs(dp) > EPS else cos(a1) 

b2 = (b1 + r * sb / q) if abs(q) > EPS else b1 

 

h = self.height if height is None else Height(height) 

return self.classof(degrees90(a2), degrees180(b2), height=h) 

 

def rhumbDistanceTo(self, other, radius=R_M): 

'''Return the distance from this to an other point along a rhumb 

(loxodrome) line. 

 

@arg other: The other point (spherical C{LatLon}). 

@kwarg radius: Mean earth radius (C{meter}) or C{None}. 

 

@return: Distance (C{meter}, the same units as B{C{radius}} 

or C{radians} if B{C{radius}} is C{None}). 

 

@raise TypeError: The B{C{other}} point is not spherical. 

 

@raise ValueError: Invalid B{C{radius}}. 

 

@example: 

 

>>> p = LatLon(51.127, 1.338) 

>>> q = LatLon(50.964, 1.853) 

>>> d = p.rhumbDistanceTo(q) # 403100 

''' 

# see <https://www.EdWilliams.org/avform.htm#Rhumb> 

da, db, dp = self._rhumb3(other) 

 

# on Mercator projection, longitude distances shrink 

# by latitude; the 'stretch factor' q becomes ill- 

# conditioned along E-W line (0/0); use an empirical 

# tolerance to avoid it 

q = (da / dp) if abs(dp) > EPS else cos(self.phi) 

r = hypot(da, q * db) 

return r if radius is None else (Radius(radius) * r) 

 

def rhumbMidpointTo(self, other, height=None): 

'''Return the (loxodromic) midpoint between this and 

an other point. 

 

@arg other: The other point (spherical LatLon). 

@kwarg height: Optional height, overriding the mean height 

(C{meter}). 

 

@return: The midpoint (spherical C{LatLon}). 

 

@raise TypeError: The B{C{other}} point is not spherical. 

 

@raise ValueError: Invalid B{C{height}}. 

 

@example: 

 

>>> p = LatLon(51.127, 1.338) 

>>> q = LatLon(50.964, 1.853) 

>>> m = p.rhumb_midpointTo(q) 

>>> m.toStr() # '51.0455°N, 001.5957°E' 

''' 

self.others(other) 

 

# see <https://MathForum.org/library/drmath/view/51822.html> 

a1, b1 = self.philam 

a2, b2 = other.philam 

if abs(b2 - b1) > PI: 

b1 += PI2 # crossing anti-meridian 

 

a3 = favg(a1, a2) 

b3 = favg(b1, b2) 

 

f1 = tanPI_2_2(a1) 

if isnon0(f1): 

f2 = tanPI_2_2(a2) 

f = f2 / f1 

if isnon0(f): 

f = log(f) 

if isnon0(f): 

f3 = tanPI_2_2(a3) 

b3 = fdot(map1(log, f1, f2, f3), 

-b2, b1, b2 - b1) / f 

 

h = self._havg(other) if height is None else Height(height) 

return self.classof(degrees90(a3), degrees180(b3), height=h) 

 

def toNvector(self, Nvector=NvectorBase, **Nvector_kwds): # PYCHOK signature 

'''Convert this point to C{Nvector} components, I{including 

height}. 

 

@kwarg Nvector_kwds: Optional, additional B{C{Nvector}} 

keyword arguments, ignored if 

C{B{Nvector}=None}. 

 

@return: An B{C{Nvector}} or a L{Vector4Tuple}C{(x, y, z, h)} 

if B{C{Nvector}} is C{None}. 

 

@raise TypeError: Invalid B{C{Nvector}} or B{C{Nvector_kwds}}. 

''' 

return LatLonBase.toNvector(self, Nvector=Nvector, **Nvector_kwds) 

 

def toWm(self, radius=R_MA): 

'''Convert this point to a I{WM} coordinate. 

 

@kwarg radius: Optional earth radius (C{meter}). 

 

@return: The WM coordinate (L{Wm}). 

 

@see: Function L{toWm} in module L{webmercator} for details. 

''' 

from pygeodesy.webmercator import toWm 

return toWm(self, radius=radius) 

 

 

__all__ += _ALL_DOCS(CartesianSphericalBase, LatLonSphericalBase) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2021 -- mrJean1 at Gmail -- All Rights Reserved. 

# 

# Permission is hereby granted, free of charge, to any person obtaining a 

# copy of this software and associated documentation files (the "Software"), 

# to deal in the Software without restriction, including without limitation 

# the rights to use, copy, modify, merge, publish, distribute, sublicense, 

# and/or sell copies of the Software, and to permit persons to whom the 

# Software is furnished to do so, subject to the following conditions: 

# 

# The above copyright notice and this permission notice shall be included 

# in all copies or substantial portions of the Software. 

# 

# THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS 

# OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 

# FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL 

# THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR 

# OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, 

# ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR 

# OTHER DEALINGS IN THE SOFTWARE.