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# -*- coding: utf-8 -*- 

 

u'''Trigonometric classes geodetic (lat-/longitude) L{LatLon} and 

geocentric (ECEF) L{Cartesian} and functions L{areaOf}, L{intersection}, 

L{isPoleEnclosedBy}, L{meanOf}, L{nearestOn2} and L{perimeterOf}, 

I{all spherical}. 

 

Pure Python implementation of geodetic (lat-/longitude) methods using 

spherical trigonometry, transcribed from JavaScript originals by 

I{(C) Chris Veness 2011-2016} published under the same MIT Licence**, see 

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

 

@newfield example: Example, Examples 

''' 

 

from pygeodesy.basics import EPS, PI2, PI_2, PI_4, R_M, \ 

isscalar, map1, _xkwds 

from pygeodesy.errors import CrossError, crosserrors, \ 

_item_, IntersectionError, \ 

_Not_convex, _ValueError, _xkwds_get 

from pygeodesy.fmath import acos1, favg, fdot, fmean, fsum, fsum_ 

from pygeodesy.formy import antipode_, bearing_, vincentys_ 

from pygeodesy.lazily import _ALL_LAZY 

from pygeodesy.named import LatLon2Tuple, LatLon3Tuple, NearestOn3Tuple, \ 

_xnamed 

from pygeodesy.nvectorBase import NvectorBase as _Nvector 

from pygeodesy.points import _imdex2, ispolar, nearestOn5 as _nearestOn5 

from pygeodesy.sphericalBase import _angular, CartesianSphericalBase, \ 

LatLonSphericalBase, _rads3 

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

from pygeodesy.utily import degrees90, degrees180, degrees2m, iterNumpy2, \ 

radiansPI2, sincos2, tan_2, unrollPI, wrapPI 

from pygeodesy.vector3d import Vector3d, sumOf 

 

from math import asin, atan2, copysign, cos, degrees, hypot, \ 

radians, sin 

 

# all public contants, classes and functions 

__all__ = _ALL_LAZY.sphericalTrigonometry + ( 

'Cartesian', 'LatLon', # classes 

'areaOf', # functions 

'intersection', 'intersections2', 'ispolar', 

'isPoleEnclosedBy', # DEPRECATED, use ispolar 

'meanOf', 

'nearestOn2', 'nearestOn3', 

'perimeterOf', 

'sumOf') # == vector3d.sumOf 

__version__ = '20.06.16' 

 

 

def _destination2(a, b, r, t): 

'''(INTERNAL) Destination phi- and longitude in C{radians}. 

 

@arg a: Latitude (C{radians}). 

@arg b: Longitude (C{radians}). 

@arg r: Angular distance (C{radians}). 

@arg t: Bearing (compass C{radians}). 

 

@return: 2-Tuple (phi, lam) of (C{radians}, C{radiansPI}). 

''' 

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

sa, ca, sr, cr, st, ct = sincos2(a, r, t) 

 

a = asin(ct * sr * ca + cr * sa) 

d = atan2(st * sr * ca, cr - sa * sin(a)) 

# note, in EdWilliams.org/avform.htm W is + and E is - 

return a, b + d 

 

 

class Cartesian(CartesianSphericalBase): 

'''Extended to convert geocentric, L{Cartesian} points to 

spherical, geodetic L{LatLon}. 

''' 

 

def toLatLon(self, **LatLon_datum_kwds): # PYCHOK LatLon=LatLon 

'''Convert this cartesian point to an C{Nvector}-based 

geodetic point. 

 

@kwarg LatLon_datum_kwds: Optional L{LatLon}, B{C{datum}} and 

other keyword arguments, ignored if 

B{C{LatLon=None}}. Use 

B{C{LatLon=...}} to override this 

L{LatLon} class or specify 

B{C{LatLon=None}}. 

 

@return: The geodetic point (L{LatLon}) or if B{C{LatLon}} 

is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, 

height, C, M, datum)} with C{C} and C{M} if available. 

 

@raise TypeError: Invalid B{C{LatLon_datum_kwds}}. 

''' 

kwds = _xkwds(LatLon_datum_kwds, LatLon=LatLon, datum=self.datum) 

return CartesianSphericalBase.toLatLon(self, **kwds) 

 

 

class LatLon(LatLonSphericalBase): 

'''New point on spherical model earth model. 

 

@example: 

 

>>> p = LatLon(52.205, 0.119) # height=0 

''' 

 

def _trackDistanceTo3(self, start, end, radius, wrap): 

'''(INTERNAL) Helper for along-/crossTrackDistanceTo. 

''' 

self.others(start, name='start') 

self.others(end, name='end') 

 

r = Radius_(radius) 

r = start.distanceTo(self, r, wrap=wrap) / r 

 

b = radians(start.initialBearingTo(self, wrap=wrap)) 

e = radians(start.initialBearingTo(end, wrap=wrap)) 

x = asin(sin(r) * sin(b - e)) 

return r, x, (e - b) 

 

def alongTrackDistanceTo(self, start, end, radius=R_M, wrap=False): 

'''Compute the (signed) distance from the start to the closest 

point on the great circle path defined by a start and an 

end point. 

 

That is, if a perpendicular is drawn from this point to the 

great circle path, the along-track distance is the distance 

from the start point to the point where the perpendicular 

crosses the path. 

 

@arg start: Start point of great circle path (L{LatLon}). 

@arg end: End point of great circle path (L{LatLon}). 

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

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

 

@return: Distance along the great circle path (C{meter}, 

same units as B{C{radius}}), positive if after the 

B{C{start}} toward the B{C{end}} point of the path or 

negative if before the B{C{start}} point. 

 

@raise TypeError: The B{C{start}} or B{C{end}} point is not L{LatLon}. 

 

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

 

@example: 

 

>>> p = LatLon(53.2611, -0.7972) 

 

>>> s = LatLon(53.3206, -1.7297) 

>>> e = LatLon(53.1887, 0.1334) 

>>> d = p.alongTrackDistanceTo(s, e) # 62331.58 

''' 

r, x, b = self._trackDistanceTo3(start, end, radius, wrap) 

cx = cos(x) 

if abs(cx) > EPS: 

return copysign(acos1(cos(r) / cx), cos(b)) * radius 

else: 

return 0.0 

 

def bearingTo(self, other, wrap=False, raiser=False): # PYCHOK no cover 

'''DEPRECATED, use method C{initialBearingTo}. 

''' 

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

 

def crossingParallels(self, other, lat, wrap=False): 

'''Return the pair of meridians at which a great circle defined 

by this and an other point crosses the given latitude. 

 

@arg other: The other point defining great circle (L{LatLon}). 

@arg lat: Latitude at the crossing (C{degrees}). 

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

 

@return: 2-Tuple C{(lon1, lon2)}, both in C{degrees180} or 

C{None} if the great circle doesn't reach B{C{lat}}. 

''' 

self.others(other) 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

 

a = radians(lat) 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

 

sa, ca, sa1, ca1, \ 

sa2, ca2, sdb, cdb = sincos2(a, a1, a2, db) 

 

x = sa1 * ca2 * ca * sdb 

y = sa1 * ca2 * ca * cdb - ca1 * sa2 * ca 

z = ca1 * ca2 * sa * sdb 

 

h = hypot(x, y) 

if h < EPS or abs(z) > h: 

return None # great circle doesn't reach latitude 

 

m = atan2(-y, x) + b1 # longitude at max latitude 

d = acos1(z / h) # delta longitude to intersections 

return degrees180(m - d), degrees180(m + d) 

 

def crossTrackDistanceTo(self, start, end, radius=R_M, wrap=False): 

'''Compute the (signed) distance from this point to the great 

circle defined by a start and an end point. 

 

@arg start: Start point of great circle path (L{LatLon}). 

@arg end: End point of great circle path (L{LatLon}). 

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

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

 

@return: Distance to great circle (negative if to the 

left or positive if to the right of the path). 

 

@raise TypeError: The B{C{start}} or B{C{end}} point is not L{LatLon}. 

 

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

 

@example: 

 

>>> p = LatLon(53.2611, -0.7972) 

 

>>> s = LatLon(53.3206, -1.7297) 

>>> e = LatLon(53.1887, 0.1334) 

>>> d = p.crossTrackDistanceTo(s, e) # -307.5 

''' 

_, x, _ = self._trackDistanceTo3(start, end, radius, wrap) 

return x * radius 

 

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

'''Locate the destination from this point after having 

travelled the given distance on the given initial 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 at destination (C{meter}, same 

units a B{C{radius}}). 

 

@return: Destination point (L{LatLon}). 

 

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

B{C{radius}} or B{C{height}}. 

 

@example: 

 

>>> p1 = LatLon(51.4778, -0.0015) 

>>> p2 = p1.destination(7794, 300.7) 

>>> p2.toStr() # '51.5135°N, 000.0983°W' 

 

@JSname: I{destinationPoint}. 

''' 

a, b = self.philam 

 

r, t = _angular(distance, radius), Bearing_(bearing) 

 

a, b = _destination2(a, b, r, t) 

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

return self.classof(degrees90(a), degrees180(b), height=h) 

 

def distanceTo(self, other, radius=R_M, wrap=False): 

'''Compute the distance from this to an other point. 

 

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

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

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

 

@return: Distance between this and the B{C{other}} point 

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

 

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

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351); 

>>> d = p1.distanceTo(p2) # 404300 

''' 

self.others(other) 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

 

db, _ = unrollPI(b1, b2, wrap=wrap) 

r = vincentys_(a2, a1, db) 

return r * Radius(radius) 

 

def greatCircle(self, bearing): 

'''Compute the vector normal to great circle obtained by heading 

on the given initial bearing from this point. 

 

Direction of vector is such that initial bearing vector 

b = c × n, where n is an n-vector representing this point. 

 

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

 

@return: Vector representing great circle (L{Vector3d}). 

 

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

 

@example: 

 

>>> p = LatLon(53.3206, -1.7297) 

>>> g = p.greatCircle(96.0) 

>>> g.toStr() # (-0.794, 0.129, 0.594) 

''' 

a, b = self.philam 

 

t = Bearing_(bearing) 

 

sa, ca, sb, cb, st, ct = sincos2(a, b, t) 

 

return Vector3d(sb * ct - cb * sa * st, 

-cb * ct - sb * sa * st, 

ca * st) # XXX .unit()? 

 

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

'''Compute the initial bearing (forward azimuth) from this 

to an other point. 

 

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

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

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

use B{C{raiser}}=C{True} for behavior like 

C{sphericalNvector.LatLon.initialBearingTo}. 

 

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

 

@raise CrossError: If this and the B{C{other}} point coincide, 

provided B{C{raiser}} is C{True} and 

L{crosserrors} is C{True}. 

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351) 

>>> b = p1.initialBearingTo(p2) # 156.2 

 

@JSname: I{bearingTo}. 

''' 

self.others(other) 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

 

# XXX behavior like sphericalNvector.LatLon.initialBearingTo 

if raiser and crosserrors() and max(abs(a2 - a1), abs(b2 - b1)) < EPS: 

raise CrossError('points', self, txt='coincident') 

 

return degrees(bearing_(a1, b1, a2, b2, final=False, wrap=wrap)) 

 

def intermediateTo(self, other, fraction, height=None, wrap=False): 

'''Locate the point at given fraction between this and an 

other point. 

 

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

@arg fraction: Fraction between both points (float, between 

0.0 for this and 1.0 for the other point). 

@kwarg height: Optional height, overriding the fractional 

height (C{meter}). 

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

 

@return: Intermediate point (L{LatLon}). 

 

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

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351) 

>>> p = p1.intermediateTo(p2, 0.25) # 51.3721°N, 000.7073°E 

 

@JSname: I{intermediatePointTo}. 

''' 

self.others(other) 

 

f = Scalar(fraction, name='fraction') 

 

a1, b1 = self.philam 

a2, b2 = other.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

r = vincentys_(a2, a1, db) 

sr = sin(r) 

if abs(sr) > EPS: 

sa1, ca1, sa2, ca2, \ 

sb1, cb1, sb2, cb2 = sincos2(a1, a2, b1, b2) 

 

A = sin((1 - f) * r) / sr 

B = sin( f * r) / sr 

 

x = A * ca1 * cb1 + B * ca2 * cb2 

y = A * ca1 * sb1 + B * ca2 * sb2 

z = A * sa1 + B * sa2 

 

a = atan2(z, hypot(x, y)) 

b = atan2(y, x) 

 

else: # points too close 

a = favg(a1, a2, f=f) 

b = favg(b1, b2, f=f) 

 

if height is None: 

h = self._havg(other, f=f) 

else: 

h = Height(height) 

return self.classof(degrees90(a), degrees180(b), height=h) 

 

def intersection(self, end1, other, end2, height=None, wrap=False): 

'''Locate the intersection point of two paths both defined 

by two points or a start point and bearing from North. 

 

@arg end1: End point of this path (L{LatLon}) or the 

initial bearing at this point (compass 

C{degrees360}). 

@arg other: Start point of the other path (L{LatLon}). 

@arg end2: End point of the other path (L{LatLon}) or 

the initial bearing at the other start point 

(compass C{degrees360}). 

@kwarg height: Optional height for intersection point, 

overriding the mean height (C{meter}). 

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

 

@return: The intersection point (L{LatLon}). An alternate 

intersection point might be the L{antipode} to 

the returned result. 

 

@raise IntersectionError: Intersection is ambiguous or infinite 

or the paths are coincident, colinear 

or parallel. 

 

@raise TypeError: If B{C{end1}}, B{C{other}} or B{C{end2}} is 

not L{LatLon}. 

 

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

 

@example: 

 

>>> p = LatLon(51.8853, 0.2545) 

>>> s = LatLon(49.0034, 2.5735) 

>>> i = p.intersection(108.547, s, 32.435) # '50.9078°N, 004.5084°E' 

''' 

return intersection(self, end1, other, end2, 

height=height, wrap=wrap, 

LatLon=self.classof) 

 

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

height=None, wrap=False): 

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

by a center point and radius. 

 

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

see B{C{radius}}). 

@arg other: Center of the other circle (L{LatLon}). 

@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}). 

@kwarg height: Optional height for the intersection point, 

overriding the mean height (C{meter}). 

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

 

@return: 2-Tuple of the intersection points, each a L{LatLon} 

instance. The intersection points are the same 

instance for abutting circles. 

 

@raise IntersectionError: Concentric, antipodal, invalid or 

non-intersecting circles. 

 

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

 

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

or B{C{height}}. 

''' 

return intersections2(self, rad1, other, rad2, radius=radius, 

height=height, wrap=wrap, 

LatLon=self.classof) 

 

def isenclosedBy(self, points): 

'''Check whether a (convex) polygon encloses this point. 

 

@arg points: The polygon points (L{LatLon}[]). 

 

@return: C{True} if the polygon encloses this point, 

C{False} otherwise. 

 

@raise PointsError: Insufficient number of B{C{points}}. 

 

@raise TypeError: Some B{C{points}} are not L{LatLon}. 

 

@raise ValueError: Invalid B{C{points}}, non-convex polygon. 

 

@example: 

 

>>> b = LatLon(45,1), LatLon(45,2), LatLon(46,2), LatLon(46,1) 

>>> p = LatLon(45,1, 1.1) 

>>> inside = p.isEnclosedBy(b) # True 

''' 

n, points = self.points2(points, closed=True) 

 

n0 = self._N_vector 

 

if iterNumpy2(points): 

 

v1 = points[n-1]._N_vector 

v2 = points[n-2]._N_vector 

gc1 = v2.cross(v1) 

t0 = gc1.angleTo(n0) > PI_2 

for i in range(n): 

v2 = points[i]._N_vector 

gc = v1.cross(v2) 

v1 = v2 

 

ti = gc.angleTo(n0) > PI_2 

if ti != t0: 

return False # outside 

 

if gc1.angleTo(gc, vSign=n0) < 0: 

raise _ValueError(_item_(points=i), points[i], txt=_Not_convex) 

gc1 = gc 

 

else: 

# get great-circle vector for each edge 

gc, v1 = [], points[n-1]._N_vector 

for i in range(n): 

v2 = points[i]._N_vector 

gc.append(v1.cross(v2)) 

v1 = v2 

 

# check whether this point on same side of all 

# polygon edges (to the left or right depending 

# on anti-/clockwise polygon direction) 

t0 = gc[0].angleTo(n0) > PI_2 # True if on the right 

for i in range(1, n): 

ti = gc[i].angleTo(n0) > PI_2 

if ti != t0: # different sides of edge i 

return False # outside 

 

# check for convex polygon (otherwise 

# the test above is not reliable) 

gc1 = gc[n-1] 

for i, gc2 in enumerate(gc): 

# angle between gc vectors, signed by direction of n0 

if gc1.angleTo(gc2, vSign=n0) < 0: 

raise _ValueError(_item_(points=i), points[i], txt=_Not_convex) 

gc1 = gc2 

 

return True # inside 

 

def isEnclosedBy(self, points): # PYCHOK no cover 

'''DEPRECATED, use method C{isenclosedBy}. 

''' 

return self.isenclosedBy(points) 

 

def midpointTo(self, other, height=None, wrap=False): 

'''Find the midpoint between this and an other point. 

 

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

@kwarg height: Optional height for midpoint, overriding 

the mean height (C{meter}). 

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

 

@return: Midpoint (L{LatLon}). 

 

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

 

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

 

@example: 

 

>>> p1 = LatLon(52.205, 0.119) 

>>> p2 = LatLon(48.857, 2.351) 

>>> m = p1.midpointTo(p2) # '50.5363°N, 001.2746°E' 

''' 

self.others(other) 

 

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

a1, b1 = self.philam 

a2, b2 = other.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

 

sa1, ca1, sa2, ca2, sdb, cdb = sincos2(a1, a2, db) 

 

x = ca2 * cdb + ca1 

y = ca2 * sdb 

 

a = atan2(sa1 + sa2, hypot(x, y)) 

b = atan2(y, x) + b1 

 

if height is None: 

h = self._havg(other) 

else: 

h = Height(height) 

return self.classof(degrees90(a), degrees180(b), height=h) 

 

def nearestOn(self, point1, point2, radius=R_M, **options): 

'''Locate the point between two points closest and this point. 

 

Distances are approximated by function L{equirectangular_}, 

subject to the supplied B{C{options}}. 

 

@arg point1: Start point (L{LatLon}). 

@arg point2: End point (L{LatLon}). 

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

@kwarg options: Optional keyword arguments for function 

L{equirectangular_}. 

 

@return: Closest point on the arc (L{LatLon}). 

 

@raise LimitError: Lat- and/or longitudinal delta exceeds 

B{C{limit}}, see function L{equirectangular_}. 

 

@raise TypeError: If B{C{point1}} or B{C{point2}} is not L{LatLon}. 

 

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

 

@see: Functions L{equirectangular_} and L{nearestOn5} and 

method L{sphericalTrigonometry.LatLon.nearestOn3}. 

''' 

return self.nearestOn3([point1, point2], closed=False, radius=radius, 

**options)[0] 

 

def nearestOn2(self, points, closed=False, radius=R_M, **options): # PYCHOK no cover 

'''DEPRECATED, use method L{sphericalTrigonometry.LatLon.nearestOn3}. 

 

@return: ... 2-Tuple C{(closest, distance)} of the closest 

point (L{LatLon}) on the polygon and the distance 

to that point from this point in C{meter}, same 

units of B{C{radius}}. 

''' 

r = self.nearestOn3(points, closed=closed, radius=radius, **options) 

return tuple(r[:2]) 

 

def nearestOn3(self, points, closed=False, radius=R_M, **options): 

'''Locate the point on a polygon closest to this point. 

 

Distances are approximated by function L{equirectangular_}, 

subject to the supplied B{C{options}}. 

 

@arg points: The polygon points (L{LatLon}[]). 

@kwarg closed: Optionally, close the polygon (C{bool}). 

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

@kwarg options: Optional keyword arguments for function 

L{equirectangular_}. 

 

@return: A L{NearestOn3Tuple}C{(closest, distance, angle)} of 

the C{closest} point (L{LatLon}), the L{equirectangular_} 

C{distance} between this and the C{closest} point in 

C{meter}, same units as B{C{radius}}. The C{angle} 

from this to the C{closest} point is in compass 

C{degrees360}, like function L{compassAngle}. 

 

@raise LimitError: Lat- and/or longitudinal delta exceeds 

B{C{limit}}, see function L{equirectangular_}. 

 

@raise PointsError: Insufficient number of B{C{points}}. 

 

@raise TypeError: Some B{C{points}} are not C{LatLon}. 

 

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

 

@see: Functions L{compassAngle}, L{equirectangular_} and 

L{nearestOn5}. 

''' 

lat, lon, d, c, h = _nearestOn5(self, points, closed=closed, **options) 

return NearestOn3Tuple(self.classof(lat, lon, height=h), 

degrees2m(d, radius=radius), c) 

 

def toCartesian(self, **Cartesian_datum_kwds): # PYCHOK Cartesian=Cartesian, datum=None 

'''Convert this point to C{Karney}-based cartesian (ECEF) 

coordinates. 

 

@kwarg Cartesian_datum_kwds: Optional L{Cartesian}, B{C{datum}} 

and other keyword arguments, ignored 

if B{C{Cartesian=None}}. Use 

B{C{Cartesian=...}} to override 

this L{Cartesian} class or specify 

B{C{Cartesian=None}}. 

 

@return: The cartesian point (L{Cartesian}) or if B{C{Cartesian}} 

is C{None}, an L{Ecef9Tuple}C{(x, y, z, lat, lon, height, 

C, M, datum)} with C{C} and C{M} if available. 

 

@raise TypeError: Invalid B{C{Cartesian_datum_kwds}}. 

''' 

kwds = _xkwds(Cartesian_datum_kwds, Cartesian=Cartesian, datum=self.datum) 

return LatLonSphericalBase.toCartesian(self, **kwds) 

 

 

_Trll = LatLon(0, 0) #: (INTERNAL) Reference instance (L{LatLon}). 

 

 

def areaOf(points, radius=R_M, wrap=True): 

'''Calculate the area of a (spherical) polygon (with great circle 

arcs joining the points). 

 

@arg points: The polygon points (L{LatLon}[]). 

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

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

 

@return: Polygon area (C{meter}, same units as B{C{radius}}, squared). 

 

@raise PointsError: Insufficient number of B{C{points}}. 

 

@raise TypeError: Some B{C{points}} are not L{LatLon}. 

 

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

 

@note: The area is based on Karney's U{'Area of a spherical polygon' 

<https://OSGeo-org.1560.x6.nabble.com/ 

Area-of-a-spherical-polygon-td3841625.html>}. 

 

@see: L{pygeodesy.areaOf}, L{sphericalNvector.areaOf} and 

L{ellipsoidalKarney.areaOf}. 

 

@example: 

 

>>> b = LatLon(45, 1), LatLon(45, 2), LatLon(46, 2), LatLon(46, 1) 

>>> areaOf(b) # 8666058750.718977 

 

>>> c = LatLon(0, 0), LatLon(1, 0), LatLon(0, 1) 

>>> areaOf(c) # 6.18e9 

''' 

n, points = _Trll.points2(points, closed=True) 

 

# Area method due to Karney: for each edge of the polygon, 

# 

# tan(Δλ/2) · (tan(φ1/2) + tan(φ2/2)) 

# tan(E/2) = ------------------------------------ 

# 1 + tan(φ1/2) · tan(φ2/2) 

# 

# where E is the spherical excess of the trapezium obtained by 

# extending the edge to the equator-circle vector for each edge 

 

def _exs(n, points): # iterate over spherical edge excess 

a1, b1 = points[n-1].philam 

ta1 = tan_2(a1) 

for i in range(n): 

a2, b2 = points[i].philam 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

ta2, tdb = map1(tan_2, a2, db) 

yield atan2(tdb * (ta1 + ta2), 1 + ta1 * ta2) 

ta1, b1 = ta2, b2 

 

s = fsum(_exs(n, points)) * 2 

 

if isPoleEnclosedBy(points): 

s = abs(s) - PI2 

 

return abs(s * Radius(radius)**2) 

 

 

def _xb(a1, b1, end, a, b, wrap): 

# difference between the bearing to (a, b) and the given 

# bearing is negative if both are in opposite directions 

r = bearing_(a1, b1, a, b, wrap=wrap) 

return PI_2 - abs(wrapPI(r - radians(end))) 

 

 

def _xdot(d, a1, b1, a, b, wrap): 

# compute dot product d . (-b + b1, a - a1) 

db, _ = unrollPI(b1, b, wrap=wrap) 

return fdot(d, db, a - a1) 

 

 

def _x3d2(start, end, wrap, n, hs): 

# see <https://www.EdWilliams.org/intersect.htm> (5) ff 

a1, b1 = start.philam 

 

if isscalar(end): # bearing, make a point 

a2, b2 = _destination2(a1, b1, PI_4, radians(end)) 

else: # must be a point 

_Trll.others(end, name='end' + n) 

hs.append(end.height) 

a2, b2 = end.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

if max(abs(db), abs(a2 - a1)) < EPS: 

raise IntersectionError(start=start, end=end, txt='null path' + n) 

 

# note, in EdWilliams.org/avform.htm W is + and E is - 

b21, b12 = db * 0.5, -(b1 + b2) * 0.5 

 

sb21, cb21, sb12, cb12, \ 

sa21, _, sa12, _ = sincos2(b21, b12, a1 - a2, a1 + a2) 

 

x = _Nvector(sa21 * sb12 * cb21 - sa12 * cb12 * sb21, 

sa21 * cb12 * cb21 + sa12 * sb12 * sb21, 

cos(a1) * cos(a2) * sin(db)) # ll=start 

return x.unit(), (db, (a2 - a1)) # negated d 

 

 

def intersection(start1, end1, start2, end2, 

height=None, wrap=False, LatLon=LatLon, **LatLon_kwds): 

'''Compute the intersection point of two paths both defined 

by two points or a start point and bearing from North. 

 

@arg start1: Start point of the first path (L{LatLon}). 

@arg end1: End point ofthe first path (L{LatLon}) or 

the initial bearing at the first start point 

(compass C{degrees360}). 

@arg start2: Start point of the second path (L{LatLon}). 

@arg end2: End point of the second path (L{LatLon}) or 

the initial bearing at the second start point 

(compass C{degrees360}). 

@kwarg height: Optional height for the intersection point, 

overriding the mean height (C{meter}). 

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

@kwarg LatLon: Optional class to return the intersection 

point (L{LatLon}) or C{None}. 

@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

arguments, ignored if B{C{LatLon=None}}. 

 

@return: The intersection point (B{C{LatLon}}) or a 

L{LatLon3Tuple}C{(lat, lon, height)} if B{C{LatLon}} 

is C{None}. An alternate intersection point might 

be the L{antipode} to the returned result. 

 

@raise IntersectionError: Intersection is ambiguous or infinite 

or the paths are coincident, colinear 

or parallel. 

 

@raise TypeError: A B{C{start}} or B{C{end}} point not L{LatLon}. 

 

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

 

@example: 

 

>>> p = LatLon(51.8853, 0.2545) 

>>> s = LatLon(49.0034, 2.5735) 

>>> i = intersection(p, 108.547, s, 32.435) # '50.9078°N, 004.5084°E' 

''' 

_Trll.others(start1, name='start1') 

_Trll.others(start2, name='start2') 

 

hs = [start1.height, start2.height] 

 

a1, b1 = start1.philam 

a2, b2 = start2.philam 

 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

r12 = vincentys_(a2, a1, db) 

if abs(r12) < EPS: # [nearly] coincident points 

a, b = favg(a1, a2), favg(b1, b2) 

 

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

elif isscalar(end1) and isscalar(end2): # both bearings 

sa1, ca1, sa2, ca2, sr12, cr12 = sincos2(a1, a2, r12) 

 

x1, x2 = (sr12 * ca1), (sr12 * ca2) 

if abs(x1) < EPS or abs(x2) < EPS: 

raise IntersectionError(start1=start1, end1=end1, 

start2=start2, end2=end2, txt='parallel') 

 

# handle domain error for equivalent longitudes, 

# see also functions asin_safe and acos_safe at 

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

t1, t2 = map1(acos1, (sa2 - sa1 * cr12) / x1, 

(sa1 - sa2 * cr12) / x2) 

if sin(db) > 0: 

t12, t21 = t1, PI2 - t2 

else: 

t12, t21 = PI2 - t1, t2 

 

t13, t23 = map1(radiansPI2, end1, end2) 

x1, x2 = map1(wrapPI, t13 - t12, # angle 2-1-3 

t21 - t23) # angle 1-2-3 

sx1, cx1, sx2, cx2 = sincos2(x1, x2) 

if sx1 == 0 and sx2 == 0: # max(abs(sx1), abs(sx2)) < EPS 

raise IntersectionError(start1=start1, end1=end1, 

start2=start2, end2=end2, txt='infinite') 

sx3 = sx1 * sx2 

# if sx3 < 0: 

# raise IntersectionError(start1=start1, end1=end1, 

# start2=start2, end2=end2, txt='ambiguous') 

x3 = acos1(cr12 * sx3 - cx2 * cx1) 

r13 = atan2(sr12 * sx3, cx2 + cx1 * cos(x3)) 

 

a, b = _destination2(a1, b1, r13, t13) 

# choose antipode for opposing bearings 

if _xb(a1, b1, end1, a, b, wrap) < 0 or \ 

_xb(a2, b2, end2, a, b, wrap) < 0: 

a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple 

 

else: # end point(s) or bearing(s) 

x1, d1 = _x3d2(start1, end1, wrap, '1', hs) 

x2, d2 = _x3d2(start2, end2, wrap, '2', hs) 

x = x1.cross(x2) 

if x.length < EPS: # [nearly] colinear or parallel paths 

raise IntersectionError(start1=start1, end1=end1, 

start2=start2, end2=end2, txt='colinear') 

a, b = x.philam 

# choose intersection similar to sphericalNvector 

d1 = _xdot(d1, a1, b1, a, b, wrap) 

if d1: 

d2 = _xdot(d2, a2, b2, a, b, wrap) 

if (d2 < 0 and d1 > 0) or (d2 > 0 and d1 < 0): 

a, b = antipode_(a, b) # PYCHOK PhiLam2Tuple 

 

h = fmean(hs) if height is None else Height(height) 

return _latlon3(degrees90(a), degrees180(b), h, 

intersection, LatLon, **LatLon_kwds) 

 

 

def intersections2(center1, rad1, center2, rad2, radius=R_M, # MCCABE 13 

height=None, wrap=False, LatLon=LatLon, **LatLon_kwds): 

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

by a center point and radius. 

 

@arg center1: Center of the first circle (L{LatLon}). 

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

see B{C{radius}}). 

@arg center2: Center of the second circle (L{LatLon}). 

@arg rad2: Radius of the second 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}). 

@kwarg height: Optional height for the intersection point, 

overriding the mean height (C{meter}). 

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

@kwarg LatLon: Optional class to return the intersection 

points (L{LatLon}) or C{None}. 

@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} keyword 

arguments, ignored if B{C{LatLon=None}}. 

 

@return: 2-Tuple of the intersection points, each a B{C{LatLon}} 

instance or L{LatLon3Tuple}C{(lat, lon, height)} if 

B{C{LatLon}} is C{None}. The intersection points are 

the same instance for abutting circles. 

 

@raise IntersectionError: Concentric, antipodal, invalid or 

non-intersecting circles. 

 

@raise TypeError: If B{C{center1}} or B{C{center2}} not L{LatLon}. 

 

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

B{C{height}}. 

 

@note: Courtesy U{Samuel Čavoj<https://GitHub.com/mrJean1/PyGeodesy/issues/41>}. 

 

@see: This U{Answer<https://StackOverflow.com/questions/53324667/ 

find-intersection-coordinates-of-two-circles-on-earth/53331953>}. 

''' 

def _destination1(bearing): 

a, b = _destination2(a1, b1, r1, bearing) 

return _latlon3(degrees90(a), degrees180(b), h, 

intersections2, LatLon, **LatLon_kwds) 

 

_Trll.others(center1, name='center1') 

_Trll.others(center2, name='center2') 

 

a1, b1 = center1.philam 

a2, b2 = center2.philam 

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

if x: 

a1, b1, a2, b2 = a2, b2, a1, b1 

 

db, _ = unrollPI(b1, b2, wrap=wrap) 

d = vincentys_(a2, a1, db) # radians 

if d < max(r1 - r2, EPS): 

raise IntersectionError(center1=center1, rad1=rad1, 

center2=center2, rad2=rad2, txt='near-concentric') 

 

x = fsum_(r1, r2, -d) 

if x > EPS: 

try: 

sd, cd, s1, c1, _, c2 = sincos2(d, r1, r2) 

x = sd * s1 

if abs(x) < EPS: 

raise ValueError 

x = acos1((c2 - cd * c1) / x) 

except ValueError: 

raise IntersectionError(center1=center1, rad1=rad1, 

center2=center2, rad2=rad2) 

elif x < 0: 

raise IntersectionError(center1=center1, rad1=rad1, 

center2=center2, rad2=rad2, txt='too distant') 

 

b = bearing_(a1, b1, a2, b2, final=False, wrap=wrap) 

if height is None: 

h = fmean((center1.height, center2.height)) 

else: 

Height(height) 

if abs(x) > EPS: 

return _destination1(b + x), _destination1(b - x) 

else: # abutting circles 

x = _destination1(b) 

return x, x 

 

 

def isPoleEnclosedBy(points, wrap=False): # PYCHOK no cover 

'''DEPRECATED, use function L{ispolar}. 

''' 

return ispolar(points, wrap=wrap) 

 

 

def _latlon3(lat, lon, height, func, LatLon, **LatLon_kwds): 

'''(INTERNAL) Helper for L{intersection}, L{intersections2} and L{meanof}. 

''' 

if LatLon is None: 

r = LatLon3Tuple(lat, lon, height) 

else: 

kwds = _xkwds(LatLon_kwds, height=height) 

r = LatLon(lat, lon, **kwds) 

return _xnamed(r, func.__name__) 

 

 

def meanOf(points, height=None, LatLon=LatLon, **LatLon_kwds): 

'''Compute the geographic mean of several points. 

 

@arg points: Points to be averaged (L{LatLon}[]). 

@kwarg height: Optional height at mean point, overriding 

the mean height (C{meter}). 

@kwarg LatLon: Optional class to return the mean point 

(L{LatLon}) or C{None}. 

@kwarg LatLon_kwds: Optional, additional B{C{LatLon}} 

keyword arguments, ignored if 

B{C{LatLon=None}}. 

 

@return: Point at geographic mean and height (B{C{LatLon}}) 

or a L{LatLon3Tuple}C{(lat, lon, height)} if 

B{C{LatLon}} is C{None}. 

 

@raise TypeError: Some B{C{points}} are not L{LatLon}. 

 

@raise ValueError: No B{C{points}} or invalid B{C{height}}. 

''' 

# geographic mean 

n, points = _Trll.points2(points, closed=False) 

 

m = sumOf(points[i]._N_vector for i in range(n)) 

lat, lon = m._N_vector.latlon 

 

if height is None: 

h = fmean(points[i].height for i in range(n)) 

else: 

h = Height(height) 

return _latlon3(lat, lon, h, meanOf, LatLon, **LatLon_kwds) 

 

 

def nearestOn2(point, points, **closed_radius_LatLon_options): # PYCHOK no cover 

'''DEPRECATED, use function L{sphericalTrigonometry.nearestOn3}. 

 

@return: ... 2-tuple C{(closest, distance)} of the C{closest} 

point (L{LatLon}) on the polygon and the C{distance} 

between the C{closest} and the given B{C{point}}. The 

C{closest} is a B{C{LatLon}} or a L{LatLon2Tuple}C{(lat, 

lon)} if B{C{LatLon}} is C{None} ... 

''' 

ll, d, _ = nearestOn3(point, points, **closed_radius_LatLon_options) 

if _xkwds_get(closed_radius_LatLon_options, LatLon=LatLon) is None: 

ll = LatLon2Tuple(ll.lat, ll.lon) 

return ll, d 

 

 

def nearestOn3(point, points, closed=False, radius=R_M, 

LatLon=LatLon, **options): 

'''Locate the point on a polygon closest to an other, reference point. 

 

Distances are approximated by function L{equirectangular_}, 

subject to the supplied B{C{options}}. 

 

@arg point: The other, reference point (L{LatLon}). 

@arg points: The polygon points (L{LatLon}[]). 

@kwarg closed: Optionally, close the polygon (C{bool}). 

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

@kwarg LatLon: Optional class to return the closest point 

(L{LatLon}) or C{None}. 

@kwarg options: Optional keyword arguments for function 

L{equirectangular_}. 

 

@return: A L{NearestOn3Tuple}C{(closest, distance, angle)} with the 

C{closest} point as B{L{LatLon}} or L{LatLon3Tuple}C{(lat, 

lon, height)} if B{C{LatLon}} is C{None}. The C{distance} 

is the L{equirectangular_} distance between the C{closest} 

and the given B{C{point}} in C{meter}, same units as 

B{C{radius}}. The C{angle} from the given B{C{point}} 

to the C{closest} is in compass C{degrees360}, like function 

L{compassAngle}. The C{height} is the (interpolated) height 

at the C{closest} point. 

 

@raise LimitError: Lat- and/or longitudinal delta exceeds the 

B{C{limit}}, see function L{equirectangular_}. 

 

@raise PointsError: Insufficient number of B{C{points}}. 

 

@raise TypeError: Some B{C{points}} are not C{LatLon}. 

 

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

 

@see: Functions L{equirectangular_} and L{nearestOn5}. 

''' 

lat, lon, d, c, h = _nearestOn5(point, points, closed=closed, 

LatLon=None, **options) 

r = LatLon3Tuple(lat, lon, h) if LatLon is None else \ 

LatLon(lat, lon, height=h) 

r = NearestOn3Tuple(r, degrees2m(d, radius=radius), c) 

return _xnamed(r, nearestOn3.__name__) 

 

 

def perimeterOf(points, closed=False, radius=R_M, wrap=True): 

'''Compute the perimeter of a (spherical) polygon (with great circle 

arcs joining the points). 

 

@arg points: The polygon points (L{LatLon}[]). 

@kwarg closed: Optionally, close the polygon (C{bool}). 

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

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

 

@return: Polygon perimeter (C{meter}, same units as B{C{radius}}). 

 

@raise PointsError: Insufficient number of B{C{points}}. 

 

@raise TypeError: Some B{C{points}} are not L{LatLon}. 

 

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

 

@note: This perimeter is based on the L{haversine} formula. 

 

@see: L{pygeodesy.perimeterOf}, L{sphericalNvector.perimeterOf} 

and L{ellipsoidalKarney.perimeterOf}. 

''' 

n, points = _Trll.points2(points, closed=closed) 

 

def _rads(n, points, closed): # angular edge lengths in radians 

i, m = _imdex2(closed, n) 

a1, b1 = points[i].philam 

for i in range(m, n): 

a2, b2 = points[i].philam 

db, b2 = unrollPI(b1, b2, wrap=wrap) 

yield vincentys_(a2, a1, db) 

a1, b1 = a2, b2 

 

r = fsum(_rads(n, points, closed)) 

return r * Radius(radius) 

 

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

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# in all copies or substantial portions of the Software. 

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