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# -*- coding: utf-8 -*- 

 

u'''Geocentric conversions transcribed from I{Charles Karney}'s C++ classes U{Geocentric 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Geocentric.html>} and 

U{LocalCartesian<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1LocalCartesian.html>} 

into pure Python classes L{EcefKarney} respectively L{EcefCartesian}, class L{EcefSudano} 

based on I{John Sudano}'s U{paper<https://www.ResearchGate.net/publication/ 

3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}, 

class L{EcefVeness} transcribed from I{Chris Veness}' JavaScript classes U{LatLonEllipsoidal, 

Cartesian<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>} 

and class L{EcefYou} implementing I{Rey-Jer You}'s U{transformations 

<https://www.ResearchGate.net/publication/240359424>}. 

 

Convert between geodetic coordinates C{lat}-, C{lon}gitude and height C{h} 

(measured vertically from the surface of the ellipsoid) to geocentric C{x}, 

C{y} and C{z} coordinates, also known as I{Earth-Centered, Earth-Fixed} 

(U{ECEF<https://WikiPedia.org/wiki/ECEF>}). 

 

The origin of geocentric coordinates is at the center of the earth. The C{z} 

axis goes thru the North pole, C{lat} = 90°. The C{x} axis goes thru C{lat} 

= 0°, C{lon} = 0°. 

 

The local cartesian origin is at (C{lat0}, C{lon0}, C{height0}). The C{z} 

axis is normal to the ellipsoid, the C{y} axis points due North. The plane 

C{z = -height0} is tangent to the ellipsoid. 

 

Forward conversion from geodetic to geocentric (ECEF) coordinates is straightforward. 

 

For the reverse transformation we use Hugues Vermeille's U{Direct transformation 

from geocentric coordinates to geodetic coordinates 

<https://DOI.org/10.1007/s00190-002-0273-6>}, J. Geodesy (2002) 76, 451-454. 

 

Several changes have been made to ensure that the method returns accurate 

results for all finite inputs (even if h is infinite). The changes are 

described in Appendix B of C. F. F. Karney U{Geodesics on an ellipsoid of 

revolution<https://ArXiv.org/abs/1102.1215v1>}, Feb. 2011, 85, 105-117 

(U{preprint<https://ArXiv.org/abs/1102.1215v1>}). Vermeille similarly updated 

his method in U{An analytical method to transform geocentric into geodetic 

coordinates<https://DOI.org/10.1007/s00190-010-0419-x>}, J. Geodesy (2011) 85, 

105-117. See U{Geocentric coordinates 

<https://GeographicLib.SourceForge.io/html/geocentric.html>} for more information. 

 

The errors in these routines are close to round-off. Specifically, for points 

within 5,000 km of the surface of the ellipsoid (either inside or outside the 

ellipsoid), the error is bounded by 7 nm (7 nanometers) for the WGS84 ellipsoid. 

See U{Geocentric coordinates<https://GeographicLib.SourceForge.io/html/geocentric.html>} 

for further information on the errors. 

''' 

 

from pygeodesy.basics import isscalar, map1, property_RO, \ 

_xinstanceof, _xkwds, _xsubclassof 

from pygeodesy.datums import Datums, _ellipsoidal_datum 

from pygeodesy.ellipsoids import a_f2Tuple 

from pygeodesy.errors import _datum_datum, LenError, _ValueError 

from pygeodesy.fmath import cbrt, fdot, Fsum, fsum_, hypot1 

from pygeodesy.interns import EPS, EPS1, EPS_2, NN, PI, PI_2, _C_, _datum_, \ 

_ellipsoid_, _h_, _height_, _lat_, _lat0_, \ 

_lon_, _lon0_, _M_, _name_, _no_convergence_, \ 

_SPACE_, _x_, _y_, _z_, _0_, _0_0, _0_5, _1_0, \ 

_2_0, _3_0, _4_0, _6_0, _90_0 

from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY, _FOR_DOCS 

from pygeodesy.named import _NamedBase, _NamedTuple, notOverloaded, _Pass 

from pygeodesy.namedTuples import LatLon2Tuple, LatLon3Tuple, \ 

PhiLam2Tuple, Vector3Tuple 

from pygeodesy.streprs import unstr 

from pygeodesy.units import Height, Int, Lat, Lon, Meter, Scalar 

from pygeodesy.utily import atan2d, degrees90, sincos2, sincos2d 

from pygeodesy.vector3d import _xyzn4 

 

from math import asin, atan2, copysign, cos, degrees, hypot, radians, sqrt 

 

__all__ = _ALL_LAZY.ecef 

__version__ = '20.10.11' 

 

_TRIPS = 17 # 8..9 sufficient, EcefSudano.reverse 

 

 

def _llhn4(latlonh, lon, height, suffix=NN): 

'''(INTERNAL) Get C{lat, lon, h, name} as C{4-tuple}. 

''' 

try: 

llh = latlonh.lat, latlonh.lon, getattr(latlonh, _height_, 

getattr(latlonh, _h_, height)) 

except AttributeError: 

llh = latlonh, lon, height 

try: 

lat, lon, h = map1(float, *llh) 

except (TypeError, ValueError) as x: 

t = _lat_, _lon_, _height_ 

if suffix: 

t = (_ + suffix for _ in t) 

raise EcefError(txt=str(x), **dict(zip(t, llh))) 

 

if abs(lat) > _90_0: # XXX RangeError 

raise EcefError(_lat_ + suffix, lat) 

 

return lat, lon, h, getattr(latlonh, _name_, NN) 

 

 

def _sch3(y, x): 

'''(INTERNAL) Compute sin, cos and hypotenuse. 

''' 

h = hypot(y, x) 

if h > 0: # EPS_2 

s, c = y / h, x / h 

else: 

s, c = _0_0, _1_0 

return s, c, h 

 

 

class EcefError(_ValueError): 

'''An ECEF or C{Ecef*} related issue. 

''' 

pass 

 

 

class _EcefBase(_NamedBase): 

'''(INTERNAL) Base class for L{EcefKarney}, L{EcefVeness} and L{EcefYou}. 

''' 

_datum = None 

_E = None 

 

def __init__(self, a_ellipsoid, f, name): 

'''(INTERNAL) New C{Ecef...}. 

''' 

try: 

E = a_ellipsoid 

if f is None: 

pass 

elif isscalar(E) and isscalar(f): 

E = a_f2Tuple(E, f) 

else: 

raise ValueError 

 

d = _ellipsoidal_datum(E, name=name) 

E = d.ellipsoid 

if E.a < EPS or E.f > EPS1: 

raise ValueError 

 

except (TypeError, ValueError) as x: 

t = unstr(self.classname, a=a_ellipsoid, f=f) 

raise EcefError(t + _SPACE_ + _ellipsoid_, txt=str(x)) 

 

self._datum = d 

self._E = E 

if name: 

self.name = name 

 

@property_RO 

def a(self): 

'''Get C{E.a}, the major, equatorial radius (C{meter}). 

''' 

return self._E.a 

 

equatorialRadius = a # Karney property 

 

@property_RO 

def datum(self): 

'''Get the datum (L{Datum} or C{None} if not available). 

''' 

return self._datum 

 

@property_RO 

def ellipsoid(self): 

'''Get C{E}, the ellipsoid (L{Ellipsoid} or L{Ellipsoid2}). 

''' 

return self._E 

 

@property_RO 

def f(self): 

'''Get C{E.f}, the flattening (C{float}), zero for sphere. 

''' 

return self._E.f 

 

flattening = f # Karney property 

 

def forward(self, latlonh, lon=None, height=0, M=False): # PYCHOK no cover 

'''(INTERNAL) I{Must be overloaded}. 

''' 

notOverloaded(self, self.forward, latlonh, lon=lon, 

height=height, M=M) 

 

def _Matrix(self, sa, ca, sb, cb): 

'''Creation a rotation matrix. 

 

@arg sa: C{sin(phi)} (C{float}). 

@arg ca: C{cos(phi)} (C{float}). 

@arg sb: C{sin(lambda)} (C{float}). 

@arg cb: C{cos(lambda)} (C{float}). 

 

@return: A L{EcefMatrix}. 

''' 

return self._xnamed(EcefMatrix(sa, ca, sb, cb)) 

 

def reverse(self, xyz, y=None, z=None, M=False): # PYCHOK no cover 

'''(INTERNAL) I{Must be overloaded}. 

''' 

notOverloaded(self, self.reverse, xyz, y=y, z=z, M=M) 

 

def toStr(self, prec=9): # PYCHOK signature 

'''Return this C{Ecef*} as a string. 

 

@kwarg prec: Optional precision, number of decimal digits (0..9). 

 

@return: This C{Ecef*} representation (C{str}). 

''' 

return self.attrs('a', 'f', _datum_, _ellipsoid_, _name_, prec=prec) 

 

# @property_RO 

# def xyz(self): 

# '''Get the geocentric C{(x, y, z)} coordinates (L{Vector3Tuple}C{(x, y, z)}). 

# ''' 

# return self._xnamed(Vector3Tuple(self.x, self.y, self.z)) 

 

 

class EcefKarney(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates based on I{Karney}'s U{Geocentric 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1Geocentric.html>} 

methods. 

''' 

_hmax = 0 # 12M light years 

 

def __init__(self, a_ellipsoid, f=None, name=NN): 

'''New L{EcefKarney} converter. 

 

@arg a_ellipsoid: An ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum} 

or L{a_f2Tuple}) or C{scalar} for the equatorial 

(major) radius of the ellipsoid (C{meter}). 

@kwarg f: C{None} or the ellipsoid flattening (C{scalar}), required 

for C{scalar} B{C{a_ellipsoid}}, B{C{f=0}} represents a 

sphere, negative B{C{f}} a prolate ellipsoid. 

@kwarg name: Optional name (C{str}). 

 

@raise EcefError: If B{C{a_ellipsoid}} not L{Ellipsoid}, L{Ellipsoid2}, 

L{Datum} or L{a_f2Tuple} or C{scalar} or B{C{f}} not 

C{scalar} or if C{scalar} B{C{a_ellipsoid}} not positive 

or B{C{f}} not less than 1.0. 

''' 

_EcefBase.__init__(self, a_ellipsoid, f, name) 

self._hmax = self.a / EPS_2 

 

@property_RO 

def e2(self): 

'''Get C{E.f * (2 - E.f)}, 1st eccentricty squared (C{float}). 

''' 

return self._E.e2 

 

@property_RO 

def e2a(self): 

'''Get C{abs(E.e2)} (C{float}). 

''' 

return abs(self._E.e2) 

 

@property_RO 

def e2m(self): 

'''Get C{1 - E.e2a} == C{(1 - E.f)**2} (C{float}). 

''' 

return self._E.e12 # == (1 - E.f)**2 

 

@property_RO 

def e4a(self): 

'''Get C{E.e2a**2} (C{float}). 

''' 

return self._E.e4 

 

def forward(self, latlonh, lon=None, height=0, M=False): 

'''Convert from geodetic C{(lat, lon, height)} to geocentric C{(x, y, z)}. 

 

@arg latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} 

latitude in C{degrees}. 

@kwarg lon: Optional C{scalar} longitude in C{degrees} for C{scalar} 

B{C{latlonh}}. 

@kwarg height: Optional height in C{meter}, vertically above (or 

below) the surface of the ellipsoid. 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geocentric C{(x, y, z)} coordinates for the given 

geodetic ones C{(lat, lon, height)}, case C{C} 0, optional 

L{EcefMatrix} C{M} and C{datum} if available. 

 

@raise EcefError: If B{C{latlonh}} not C{LatLon}, L{Ecef9Tuple} or 

C{scalar} or B{C{lon}} not C{scalar} for C{scalar} 

B{C{latlonh}} or C{abs(lat)} exceeds 90°. 

 

@note: Let C{v} be a unit vector located at C{(lat, lon, h)}. We can 

express C{v} as column vectors in one of two ways, C{v1} in east, 

north, up coordinates (where the components are relative 

to a local coordinate system at C{C(lat0, lon0, h0)}) or as 

C{v0} in geocentric C{x, y, z} coordinates. Then, M{v0 = 

M ⋅ v1} where C{M} is the rotation matrix. 

''' 

lat, lon, h, name = _llhn4(latlonh, lon, height) 

sa, ca, sb, cb = sincos2d(lat, lon) 

 

n = self.a / self.ellipsoid.e2s(sa) # ... / sqrt(1 - self.e2 * sa**2) 

z = (self.e2m * n + h) * sa 

x = (n + h) * ca 

 

m = self._Matrix(sa, ca, sb, cb) if M else None 

r = Ecef9Tuple(x * cb, x * sb, z, lat, lon, h, 0, m, self.datum) 

return self._xnamed(r, name=name) 

 

@property_RO 

def hmax(self): 

'''Get the distance limit (C{float}). 

''' 

return self._hmax 

 

def reverse(self, xyz, y=None, z=None, M=False): 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)}. 

 

@arg xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} 

ECEF C{x} coordinate in C{meter}. 

@kwarg y: ECEF C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} 

and B{C{z}}. 

@kwarg z: ECEF C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} 

and B{C{y}}. 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geodetic coordinates C{(lat, lon, height)} for the given 

geocentric ones C{(x, y, z)}, case C{C}, optional L{EcefMatrix} 

C{M} and C{datum} if available. 

 

@raise EcefError: If B{C{xyz}} not L{Ecef9Tuple} or C{scalar} C{x} 

or B{C{y}} and/or B{C{z}} not C{scalar} for C{scalar} 

B{C{xyz}}. 

 

@note: In general, there are multiple solutions and the result 

which minimizes C{height} is returned, i.e., C{(lat, lon)} 

corresponds to the closest point on the ellipsoid. If 

there are still multiple solutions with different latitudes 

(applies only if C{z} = 0), then the solution with C{lat} > 0 

is returned. If there are still multiple solutions with 

different longitudes (applies only if C{x} = C{y} = 0) then 

C{lon} = 0 is returned. The returned C{height} value is not 

below M{−E.a * (1 − E.e2) / sqrt(1 − E.e2 * sin(lat)**2)}. 

The returned C{lon} is in the range [−180°, 180°]. Like 

C{forward} above, M{v1 = Transpose(M) ⋅ v0}. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, Error=EcefError) 

 

sb, cb, d = _sch3(y, x) 

h = hypot(d, z) # distance to earth center 

if h > self._hmax: # PYCHOK no cover 

# We are really far away (> 12M light years). Treat the earth 

# as a point and h, above, is an acceptable approximation to the 

# height. This avoids overflow, e.g., in the computation of d 

# below. It's possible that h has overflowed to INF, that's OK. 

# Treat finite x, y, but r overflows to +INF by scaling by 2. 

sb, cb, r = _sch3(y * _0_5, x * _0_5) 

sa, ca, _ = _sch3(z * _0_5, r) 

C = 1 

 

elif self.e4a: 

# Treat prolate spheroids by swapping R and Z here and by switching 

# the arguments to phi = atan2(...) at the end. 

p = (d / self.a)**2 

q = self.e2m * (z / self.a)**2 

if self.f < 0: 

p, q = q, p 

r = p + q - self.e4a 

e = self.e4a * q 

if e or r > 0: 

# Avoid possible division by zero when r = 0 by multiplying 

# equations for s and t by r^3 and r, resp. 

s = e * p / _4_0 # s = r^3 * s 

u = r = r / _6_0 

r2 = r**2 

r3 = r * r2 

t3 = s + r3 

disc = s * (r3 + t3) 

if disc < 0: 

# t is complex, but the way u is defined the result is real. 

# There are three possible cube roots. We choose the root 

# which avoids cancellation. Note, disc < 0 implies r < 0. 

u += _2_0 * r * cos(atan2(sqrt(-disc), -t3) / _3_0) 

else: 

# Pick the sign on the sqrt to maximize abs(T3). This 

# minimizes loss of precision due to cancellation. The 

# result is unchanged because of the way the T is used 

# in definition of u. 

if disc > 0: 

t3 += copysign(sqrt(disc), t3) # t3 = (r * t)^3 

# N.B. cbrt always returns the real root, cbrt(-8) = -2. 

t = cbrt(t3) # t = r * t 

# t can be zero; but then r2 / t -> 0. 

if t: 

u += t + r2 / t 

v = sqrt(u**2 + e) # guaranteed positive 

# Avoid loss of accuracy when u < 0. Underflow doesn't occur in 

# E.e4 * q / (v - u) because u ~ e^4 when q is small and u < 0. 

uv = (e / (v - u)) if u < 0 else (u + v) # u+v, guaranteed positive 

# Need to guard against w going negative due to roundoff in uv - q. 

w = max(_0_0, self.e2a * (uv - q) / (2 * v)) 

# Rearrange expression for k to avoid loss of accuracy due to 

# subtraction. Division by 0 not possible because uv > 0, w >= 0. 

k1 = k2 = uv / (sqrt(uv + w**2) + w) 

if self.f < 0: 

k1 -= self.e2 

else: 

k2 += self.e2 

sa, ca, h = _sch3(z / k1, d / k2) 

h *= k1 - self.e2m 

C = 2 

 

else: # e = self.e4a * q == 0 and r <= 0 

# This leads to k = 0 (oblate, equatorial plane) and k + E.e^2 = 0 

# (prolate, rotation axis) and the generation of 0/0 in the general 

# formulas for phi and h, using the general formula and division 

# by 0 in formula for h. Handle this case by taking the limits: 

# f > 0: z -> 0, k -> E.e2 * sqrt(q) / sqrt(E.e4 - p) 

# f < 0: r -> 0, k + E.e2 -> -E.e2 * sqrt(q) / sqrt(E.e4 - p) 

q = self.e4a - p 

if self.f < 0: 

p, q, e = q, p, -_1_0 

else: 

e = -self.e2m 

sa, ca, h = _sch3(sqrt(q / self.e2m), sqrt(p)) 

h *= self.a * e / self.e2a 

if z < 0: 

sa = -sa # for tiny negative z, not for prolate 

C = 3 

 

else: # self.e4a == 0 

# Treat the spherical case. Dealing with underflow in the general 

# case with E.e2 = 0 is difficult. Origin maps to North pole, same 

# as with ellipsoid. 

sa, ca, _ = _sch3(z if h else _1_0, d) 

h -= self.a 

C = 4 

 

r = Ecef9Tuple(x, y, z, atan2d(sa, ca), 

atan2d(sb, cb), h, C, 

self._Matrix(sa, ca, sb, cb) if M else None, 

self.datum) 

return self._xnamed(r, name=name) 

 

 

class EcefCartesian(_NamedBase): 

'''Conversion between geodetic C{(lat, lon, height)} and local cartesian 

C{(x, y, z)} coordinates with a local cartesian origin at C{(lat0, lon0, 

height0)}, transcribed from I{Karney}'s C++ class U{LocalCartesian 

<https://GeographicLib.SourceForge.io/html/classGeographicLib_1_1LocalCartesian.html>}. 

 

The C{z} axis is normal to the ellipsoid, the C{y} axis points due 

North. The plane C{z = -heighth0} is tangent to the ellipsoid. 

 

The conversions all take place via geocentric coordinates using a 

geocentric L{EcefKarney}, by default the WGS84 datum/ellipsoid. 

''' 

_ecef = EcefKarney(Datums.WGS84) 

_t0 = None 

 

def __init__(self, latlonh0=0, lon0=0, height0=0, ecef=None, name=NN): 

'''New L{EcefCartesian} converter. 

 

@kwarg latlonh0: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} 

latitude in C{degrees} of the cartesian origin. 

@kwarg lon0: Optional C{scalar} longitude of the cartesian origin 

in C{degrees} for C{scalar} B{C{latlonh0}}. 

@kwarg height0: Optional height of the cartesian origin in C{meter}, 

vertically above (or below) the surface of the ellipsoid. 

@kwarg ecef: An ECEF converter (L{EcefKarney}). 

@kwarg name: Optional name (C{str}). 

 

@raise EcefError: If B{C{latlonh0}} not C{LatLon}, L{Ecef9Tuple} or 

C{scalar} or B{C{lon0}} not C{scalar} for C{scalar} 

B{C{latlonh0}} or C{abs(lat)} exceeds 90°. 

 

@raise TypeError: Invalid B{C{ecef}}, not L{EcefKarney}. 

''' 

if ecef: 

_xinstanceof(EcefKarney, ecef=ecef) 

self._ecef = ecef 

self.reset(latlonh0, lon0, height0, name=name) 

 

@property_RO 

def ecef(self): 

'''Get the ECEF converter (L{EcefKarney}). 

''' 

return self._ecef 

 

def forward(self, latlonh, lon=None, height=0, M=False): 

'''Convert from geodetic C{(lat, lon, height)} to local cartesian 

C{(x, y, z)}. 

 

@arg latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} 

latitude in C{degrees}. 

@kwarg lon: Optional C{scalar} longitude in C{degrees} for C{scalar} 

B{C{latlonh}}. 

@kwarg height: Optional height in C{meter}, vertically above (or 

below) the surface of the ellipsoid. 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geocentric C{(x, y, z)} coordinates for the given 

geodetic ones C{(lat, lon, height)}, case C{C} 0, optional 

L{EcefMatrix} C{M} and C{datum} if available. 

 

@raise EcefError: If B{C{latlonh}} not C{LatLon}, L{Ecef9Tuple} or 

C{scalar} or B{C{lon}} not C{scalar} for C{scalar} 

B{C{latlonh}} or C{abs(lat)} exceeds 90°. 

 

@see: Note at method L{EcefKarney.forward}. 

''' 

lat, lon, h, name = _llhn4(latlonh, lon, height) 

t = self.ecef.forward(lat, lon, h, M=M) 

x, y, z = self.M.rotate(t[:3], *self._t0[:3]) # .x, .y, .z 

 

m = self.M.multiply(t.M) if M else None 

r = Ecef9Tuple(x, y, z, t.lat, t.lon, t.height, 0, m, self.ecef.datum) 

return self._xnamed(r, name=name) 

 

@property_RO 

def height0(self): 

'''Get origin's height (C{meter}). 

''' 

return self._t0.height 

 

@property_RO 

def lat0(self): 

'''Get origin's latitude (C{degrees}). 

''' 

return self._t0.lat 

 

@property_RO 

def lon0(self): 

'''Get origin's longitude (C{degrees}). 

''' 

return self._t0.lon 

 

@property_RO 

def M(self): 

'''Get the rotation matrix (C{EcefMatrix}). 

''' 

return self._t0.M 

 

def reset(self, latlonh0=0, lon0=0, height0=0, name=NN): 

'''Reset the local cartesian origin. 

 

@kwarg latlonh0: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} 

latitude in C{degrees} of the cartesian origin. 

@kwarg lon0: Optional, C{scalar} longitude of the cartesian origin 

in C{degrees} for C{scalar} B{C{latlonh0}}. 

@kwarg height0: Optional, height of the cartesian origin in C{meter}, 

vertically above (or below) the surface of the ellipsoid. 

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

 

@raise EcefError: If B{C{latlonh0}} not C{LatLon}, L{Ecef9Tuple} or 

C{scalar} or B{C{lon0}} not C{scalar} for C{scalar} 

B{C{latlonh0}} or C{abs(lat)} exceeds 90°. 

''' 

lat0, lon0, height0, n = _llhn4(latlonh0, lon0, height0, suffix=_0_) 

if name or n: 

self.ecef.name = self.name = name or n 

self._t0 = self.ecef.forward(lat0, lon0, height0, M=True) 

 

def reverse(self, xyz, y=None, z=None, M=False): 

'''Convert from local cartesian C{(x, y, z)} to geodetic C{(lat, lon, height)}. 

 

@arg xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} 

local cartesian C{x} coordinate in C{meter}. 

@kwarg y: Local cartesian C{y} coordinate in C{meter} for C{scalar} 

B{C{xyz}} and B{C{z}}. 

@kwarg z: Local cartesian C{z} coordinate in C{meter} for C{scalar} 

B{C{xyz}} and B{C{y}}. 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geodetic coordinates C{(lat, lon, height)} for the given 

geocentric ones C{(x, y, z)}, case C{C}, optional L{EcefMatrix} 

C{M} and C{datum} if available. 

 

@raise EcefError: If B{C{xyz}} not L{Ecef9Tuple} or C{scalar} C{x} 

or B{C{y}} and/or B{C{z}} not C{scalar} for C{scalar} 

B{C{xyz}}. 

 

@see: Note at method L{EcefKarney.reverse}. 

''' 

xyz_n = _xyzn4(xyz, y, z, Error=EcefError) 

x, y, z = self.M.unrotate(xyz_n[:3], *self._t0[:3]) # .x, .y, .z 

t = self.ecef.reverse(x, y, z, M=M) 

 

m = self.M.multiply(t.M) if M else None 

r = Ecef9Tuple(x, y, z, t.lat, t.lon, t.height, t.C, m, self.ecef.datum) 

return self._xnamed(r, name=xyz_n[3]) 

 

def toStr(self, prec=9): # PYCHOK signature 

'''Return this L{EcefCartesian} as a string. 

 

@kwarg prec: Optional precision, number of decimal digits (0..9). 

 

@return: This L{EcefCartesian} representation (C{str}). 

''' 

return self.attrs(_lat0_, _lon0_, 'height0', _M_, 'ecef', _name_, prec=prec) 

 

 

class EcefMatrix(_NamedTuple): 

'''A rotation matrix. 

''' 

_Names_ = ('_0_0_', '_0_1_', '_0_2_', # row-order 

'_1_0_', '_1_1_', '_1_2_', 

'_2_0_', '_2_1_', '_2_2_') 

_Units_ = (Scalar,) * len(_Names_) 

 

def _validate(self, **_OK): # PYCHOK unused 

'''(INTERNAL) Allow C{_Names_} with leading underscore. 

''' 

_NamedTuple._validate(self, _OK=True) 

 

def __new__(cls, sa, ca, sb, cb, *_m): 

'''New L{EcefMatrix} matrix. 

 

@arg sa: C{sin(phi)} (C{float}). 

@arg ca: C{cos(phi)} (C{float}). 

@arg sb: C{sin(lambda)} (C{float}). 

@arg cb: C{cos(lambda)} (C{float}). 

@arg _m: (INTERNAL) from C{.multiply}. 

 

@raise EcefError: If B{C{sa}}, B{C{ca}}, B{C{sb}} or 

B{C{cb}} outside M{[-1.0, +1.0]}. 

''' 

t = sa, ca, sb, cb 

if _m: # all 9 matrix elements ... 

t += _m # ... from .multiply 

 

elif max(map(abs, t)) > 1: 

raise EcefError(EcefMatrix.__name__, t) 

 

else: # build matrix from the following quaternion operations 

# qrot(lam, [0,0,1]) * qrot(phi, [0,-1,0]) * [1,1,1,1]/2 

# or 

# qrot(pi/2 + lam, [0,0,1]) * qrot(-pi/2 + phi, [-1,0,0]) 

# where 

# qrot(t,v) = [cos(t/2), sin(t/2)*v[1], sin(t/2)*v[2], sin(t/2)*v[3]] 

 

# Local X axis (east) in geocentric coords 

# M[0] = -slam; M[3] = clam; M[6] = 0; 

# Local Y axis (north) in geocentric coords 

# M[1] = -clam * sphi; M[4] = -slam * sphi; M[7] = cphi; 

# Local Z axis (up) in geocentric coords 

# M[2] = clam * cphi; M[5] = slam * cphi; M[8] = sphi; 

t = (-sb, -cb * sa, cb * ca, 

cb, -sb * sa, sb * ca, 

_0_0, ca, sa) 

 

return _NamedTuple.__new__(cls, *t) 

 

def copy(self, **unused): # PYCHOK signature 

'''Make a shallow or deep copy of this instance. 

 

@return: The copy (C{This class} or subclass thereof). 

''' 

return self.classof(*self) 

 

__copy__ = __deepcopy__ = copy 

 

def multiply(self, other): 

'''Matrix multiply M{M0' ⋅ M} this matrix transposed with 

an other matrix. 

 

@arg other: The other matrix (L{EcefMatrix}). 

 

@return: The matrix product (L{EcefMatrix}). 

 

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

''' 

_xinstanceof(EcefMatrix, other=other) 

 

# like LocalCartesian.MatrixMultiply, transposed(self) x other 

# <https://GeographicLib.SourceForge.io/html/LocalCartesian_8cpp_source.html> 

M = (fdot(self[r::3], *other[c::3]) for r in range(3) for c in range(3)) 

return EcefMatrix(*M) 

 

def rotate(self, xyz, *xyz0): 

'''Forward rotation M{M0' ⋅ ([x, y, z] - [x0, y0, z0])'}. 

 

@arg xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). 

@arg xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). 

 

@return: Rotated C{(x, y, z)} location (C{3-tuple}). 

 

@raise LenError: Unequal C{len(B{xyz})} and C{len(B{xyz0})}. 

''' 

if xyz0: 

if len(xyz0) != len(xyz): 

raise LenError(self.rotate, xyz0=len(xyz0), xyz=len(xyz)) 

 

xyz = tuple(c - c0 for c, c0 in zip(xyz, xyz0)) 

 

# x' = M[0] * x + M[3] * y + M[6] * z 

# y' = M[1] * x + M[4] * y + M[7] * z 

# z' = M[2] * x + M[5] * y + M[8] * z 

return (fdot(xyz, *self[0::3]), 

fdot(xyz, *self[1::3]), 

fdot(xyz, *self[2::3])) 

 

def unrotate(self, xyz, *xyz0): 

'''Inverse rotation M{[x0, y0, z0] + M0 ⋅ [x,y,z]'}. 

 

@arg xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). 

@arg xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). 

 

@return: Unrotated C{(x, y, z)} location (C{3-tuple}). 

 

@raise LenError: Unequal C{len(B{xyz})} and C{len(B{xyz0})}. 

''' 

if xyz0: 

if len(xyz0) != len(xyz): 

raise LenError(self.unrotate, xyz0=len(xyz0), xyz=len(xyz)) 

 

_xyz = (_1_0,) + xyz 

# x' = x0 + M[0] * x + M[1] * y + M[2] * z 

# y' = y0 + M[3] * x + M[4] * y + M[5] * z 

# z' = z0 + M[6] * x + M[7] * y + M[8] * z 

xyz_ = (fdot(_xyz, xyz0[0], *self[0:3]), 

fdot(_xyz, xyz0[1], *self[3:6]), 

fdot(_xyz, xyz0[2], *self[6:])) 

else: 

# x' = M[0] * x + M[1] * y + M[2] * z 

# y' = M[3] * x + M[4] * y + M[5] * z 

# z' = M[6] * x + M[7] * y + M[8] * z 

xyz_ = (fdot(xyz, *self[0:3]), 

fdot(xyz, *self[3:6]), 

fdot(xyz, *self[6:])) 

return xyz_ 

 

 

class Ecef9Tuple(_NamedTuple): # .ecef.py 

'''9-Tuple C{(x, y, z, lat, lon, height, C, M, datum)} with geocentric 

coordinates C{x}, C{y} and C{z}, geodetic coordinates C{lat}, C{lon} 

and C{height}, case C{C} (see the C{Ecef*.reverse} methods) and 

optionally, the L{EcefMatrix} C{M} and C{datum}, with C{lat} and 

C{lon} in C{degrees} and C{x}, C{y}, C{z} and C{height} in C{meter}, 

conventionally. 

''' 

_Names_ = (_x_, _y_, _z_, _lat_, _lon_, _height_, _C_, _M_, _datum_) 

_Units_ = ( Meter, Meter, Meter, Lat, Lon, Height, Int, _Pass, _Pass) 

 

def convertDatum(self, datum2): 

'''Convert this C{Ecef9Tuple} to an other datum. 

 

@arg datum2: Datum to convert I{to} (L{Datum}). 

 

@return: The converted 9-Tuple (C{Ecef9Tuple}). 

 

@raise TypeError: The B{C{datum2}} is not a L{Datum}. 

''' 

if self.datum in (None, datum2): # PYCHOK _Names_ 

r = self.copy() 

else: 

from pygeodesy.cartesianBase import CartesianBase 

c = CartesianBase(self, datum=self.datum) # PYCHOK _Names_ 

# c.toLatLon converts datum, x, y, z, lat, lon, etc. 

# and returns another Ecef9Tuple iff LatLon is None 

r = c.toLatLon(datum=datum2, LatLon=None) 

return self._xnamed(r) 

 

@property_RO 

def lam(self): 

'''Get the longitude in C{radians} (C{float}). 

''' 

return self.philam.lam 

 

@property_RO 

def lamVermeille(self): 

'''Get the longitude in radians M{[-PI*3/2..+PI*3/2]} after U{Vermeille 

<https://Search.ProQuest.com/docview/639493848>} (2004), p 95. 

 

@see: U{Karney<https://GeographicLib.SourceForge.io/html/geocentric.html>}, 

U{Vermeille<https://Search.ProQuest.com/docview/847292978>} 2011, pp 112-113, 116 

and U{Featherstone, et.al.<https://Search.ProQuest.com/docview/872827242>}, p 7. 

''' 

x, y = self.x, self.y 

if y > 0: 

r = -_2_0 * atan2(x, hypot(y, x) + y) + PI_2 

elif y < 0: 

r = _2_0 * atan2(x, hypot(y, x) - y) - PI_2 

else: # y == 0 

r = PI if x < 0 else _0_0 

return r 

 

@property_RO 

def latlon(self): 

'''Get the lat-, longitude in C{degrees} (L{LatLon2Tuple}C{(lat, lon)}). 

''' 

return self._xnamed(LatLon2Tuple(self.lat, self.lon)) 

 

@property_RO 

def latlonheight(self): 

'''Get the lat-, longitude in C{degrees} and height (L{LatLon3Tuple}C{(lat, lon, height)}). 

''' 

return self._xnamed(self.latlon.to3Tuple(self.height)) 

 

@property_RO 

def latlonheightdatum(self): 

'''Get the lat-, longitude in C{degrees} with height and datum (L{LatLon4Tuple}C{(lat, lon, height, datum)}). 

''' 

return self._xnamed(self.latlon.to3Tuple(self.height).to4Tuple(self.datum)) 

 

@property_RO 

def lonVermeille(self): 

'''Get the longitude in degrees M{[-225..+225]} after U{Vermeille 

<https://Search.ProQuest.com/docview/639493848>} (2004), p 95. 

 

@see: Property C{lamVermeille}. 

''' 

return degrees(self.lamVermeille) 

 

@property_RO 

def phi(self): 

'''Get the latitude in C{radians} (C{float}). 

''' 

return self.philam.phi 

 

@property_RO 

def philam(self): 

'''Get the lat-, longitude in C{radians} (L{PhiLam2Tuple}C{(phi, lam)}). 

''' 

return self._xnamed(PhiLam2Tuple(radians(self.lat), radians(self.lon))) 

 

@property_RO 

def philamheight(self): 

'''Get the lat-, longitude in C{radians} and height (L{PhiLam3Tuple}C{(phi, lam, height)}). 

''' 

return self._xnamed(self.philam.to3Tuple(self.height)) 

 

@property_RO 

def philamheightdatum(self): 

'''Get the lat-, longitude in C{radians} with height and datum (L{PhiLam4Tuple}C{(phi, lam, height, datum)}). 

''' 

return self._xnamed(self.philam.to3Tuple(self.height).to4Tuple(self.datum)) 

 

def toCartesian(self, Cartesian, **Cartesian_kwds): 

'''Return the geocentric C{(x, y, z)} coordinates as an ellipsoidal or spherical 

C{Cartesian}. 

 

@arg Cartesian: L{ellipsoidalKarney.Cartesian}, L{ellipsoidalNvector.Cartesian}, 

L{ellipsoidalVincenty.Cartesian}, L{sphericalNvector.Cartesian} or 

L{sphericalTrigonometry.Cartesian} class to return the C{(x, y, z)} 

coordinates. 

@kwarg Cartesian_kwds: Optional B{C{Cartesian}} keyword arguments. 

 

@return: A B{C{Cartesian}}C{(x, y, z)} instance. 

 

@raise TypeError: Invalid B{C{Cartesian}} or B{C{Cartesian_kwds}}. 

''' 

from pygeodesy.cartesianBase import CartesianBase 

_xsubclassof(CartesianBase, Cartesian=Cartesian) 

r = Cartesian(self, **Cartesian_kwds) 

return self._xnamed(r) 

 

def toLatLon(self, LatLon=None, **LatLon_height_datum_kwds): 

'''Return the geodetic C{(lat, lon, height[, datum])} coordinates. 

 

@kwarg LatLon: Optional class to return C{(lat, lon, height[, 

datum])} or C{None}. 

@kwarg LatLon_height_datum_kwds: Optional B{C{height}}, B{C{datum}} 

and other B{C{LatLon}} keyword 

arguments. 

 

@return: An instance of C{LatLon}C{(lat, lon, **_height_datum_kwds)} 

or if B{C{LatLon}} is C{None}, a L{LatLon3Tuple}C{(lat, lon, 

height)} respectively L{LatLon4Tuple}C{(lat, lon, height, 

datum)} depending on whether C{datum} is un-/available. 

 

@raise TypeError: Invalid B{C{LatLon}} or B{C{LatLon_height_datum_kwds}}. 

''' 

kwds = _xkwds(LatLon_height_datum_kwds, height=self.height, datum=self.datum) # PYCHOK Ecef9Tuple 

d = kwds[_datum_] 

if LatLon is None: 

r = LatLon3Tuple(self.lat, self.lon, kwds[_height_]) # PYCHOK Ecef9Tuple 

if d: 

r = r.to4Tuple(d) # checks type(d) 

else: 

if d is None: # remove datum 

_ = kwds.pop[_datum_] 

r = LatLon(self.lat, self.lon, **kwds) # PYCHOK Ecef9Tuple 

_datum_datum(getattr(r, _datum_, self.datum), self.datum) # PYCHOK Ecef9Tuple 

return self._xnamed(r) 

 

def toVector(self, Vector=None, **Vector_kwds): 

'''Return the geocentric C{(x, y, z)} coordinates as vector. 

 

@kwarg Vector: Optional vector class to return C{(x, y, z)} or 

C{None}. 

@kwarg Vector_kwds: Optional, additional B{C{Vector}} keyword 

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

 

@return: A C{Vector}C{(x, y, z, **Vector_kwds)} instance or a 

L{Vector3Tuple}C{(x, y, z)} if B{C{Vector}} is C{None}. 

 

@see: Propertes C{xyz} and C{xyzh} 

''' 

return self.xyz if Vector is None else \ 

self._xnamed(Vector(self.x, self.y, self.z, **Vector_kwds)) # PYCHOK Ecef9Tuple 

 

@property_RO 

def xyz(self): 

'''Get the geocentric C{(x, y, z)} coordinates (L{Vector3Tuple}C{(x, y, z)}). 

''' 

return self._xnamed(Vector3Tuple(self.x, self.y, self.z)) 

 

@property_RO 

def xyzh(self): 

'''Get the geocentric C{(x, y, z)} coordinates and height (L{Vector4Tuple}C{(x, y, z, h)}) 

''' 

return self._xnamed(self.xyz.to4Tuple(self.height)) 

 

 

class EcefVeness(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates transcribed from I{Chris Veness}' 

JavaScript classes U{LatLonEllipsoidal, Cartesian 

<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}. 

 

@see: U{A Guide to Coordinate Systems in Great Britain 

<https://www.OrdnanceSurvey.co.UK/documents/resources/guide-coordinate-systems-great-britain.pdf>}, 

section I{B) Converting between 3D Cartesian and ellipsoidal 

latitude, longitude and height coordinates}. 

''' 

 

def __init__(self, a_ellipsoid, f=None, name=NN): 

'''New L{EcefVeness}/L{EcefSudano} converter. 

 

@arg a_ellipsoid: An ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum} 

or L{a_f2Tuple}) or C{scalar} for the equatorial 

(major) radius of the ellipsoid (C{meter}). 

@kwarg f: C{None} or the ellipsoid flattening (C{scalar}), required 

for C{scalar} B{C{a_ellipsoid}}, B{C{f=0}} represents a 

sphere, negative B{C{f}} a prolate ellipsoid. 

@kwarg name: Optional name (C{str}). 

 

@raise EcefError: If B{C{a_ellipsoid}} not L{Ellipsoid}, L{Ellipsoid2}, 

L{Datum} or L{a_f2Tuple} or C{scalar} or B{C{f}} not 

C{scalar} or if C{scalar} B{C{a_ellipsoid}} not positive 

or B{C{f}} not less than 1.0. 

''' 

_EcefBase.__init__(self, a_ellipsoid, f, name) 

 

def forward(self, latlonh, lon=None, height=0, M=False): 

'''Convert from geodetic C{(lat, lon, height)} to geocentric C{(x, y, z)}. 

 

@arg latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} 

latitude in C{degrees}. 

@kwarg lon: Optional C{scalar} longitude in C{degrees} for C{scalar} 

B{C{latlonh}}. 

@kwarg height: Optional height in C{meter}, vertically above (or 

below) the surface of the ellipsoid. 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geocentric C{(x, y, z)} coordinates for the given 

geodetic ones C{(lat, lon, height)}, case C{C} 0, 

L{EcefMatrix} C{M} and C{datum} if available. 

 

@raise EcefError: If B{C{latlonh}} not C{LatLon}, L{Ecef9Tuple} or 

C{scalar} or B{C{lon}} not C{scalar} for C{scalar} 

B{C{latlonh}} or C{abs(lat)} exceeds 90°. 

''' 

lat, lon, h, name = _llhn4(latlonh, lon, height) 

sa, ca, sb, cb = sincos2d(lat, lon) 

 

E = self.ellipsoid 

 

# radius of curvature in prime vertical 

t = E.e2s2(sa) # r, _ = E.roc2_(sa, 1) 

r = _0_0 if t < EPS else (E.a if t > EPS1 else (E.a / sqrt(t))) 

x = (h + r) * ca 

z = (h + r * E.e12) * sa 

 

m = self._Matrix(sa, ca, sb, cb) if M else None 

r = Ecef9Tuple(x * cb, x * sb, z, lat, lon, h, 0, m, self.datum) 

return self._xnamed(r, name=name) 

 

def reverse(self, xyz, y=None, z=None, **no_M): # PYCHOK unused M 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} 

transcribed from I{Chris Veness}' U{JavaScript 

<https://www.Movable-Type.co.UK/scripts/geodesy/docs/latlon-ellipsoidal.js.html>}. 

 

Uses B. R. Bowring’s formulation for μm precision in concise 

form: U{'The accuracy of geodetic latitude and height equations' 

<https://www.ResearchGate.net/publication/ 

233668213_The_Accuracy_of_Geodetic_Latitude_and_Height_Equations>}, 

Survey Review, Vol 28, 218, Oct 1985. 

 

@arg xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} 

ECEF C{x} coordinate in C{meter}. 

@kwarg y: ECEF C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{z}}. 

@kwarg z: ECEF C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{y}}. 

@kwarg no_M: Rotation matrix C{M} not available. 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geodetic coordinates C{(lat, lon, height)} for the given 

geocentric ones C{(x, y, z)}, case C{C}, L{EcefMatrix} C{M} 

always C{None} and C{datum} if available. 

 

@raise EcefError: If B{C{xyz}} not L{Ecef9Tuple} or C{scalar} C{x} 

or B{C{y}} and/or B{C{z}} not C{scalar} for C{scalar} 

B{C{xyz}}. 

 

@see: Ralph M. Toms U{'An Efficient Algorithm for Geocentric to Geodetic 

Coordinate Conversion'<https://www.OSTI.gov/scitech/biblio/110235>}, 

Sept 1995 and U{'An Improved Algorithm for Geocentric to Geodetic 

Coordinate Conversion'<https://www.OSTI.gov/scitech/servlets/purl/231228>}, 

Apr 1996, both from Lawrence Livermore National Laboratory (LLNL) and 

John J. Sudano U{An exact conversion from an Earth-centered coordinate 

system to latitude longitude and altitude<https://www.ResearchGate.net/ 

publication/3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, Error=EcefError) 

 

E = self.ellipsoid 

 

p = hypot(x, y) # distance from minor axis 

r = hypot(p, z) # polar radius 

if min(p, r) > EPS: 

# parametric latitude (Bowring eqn 17, replaced) 

t = (E.b * z) / (E.a * p) * (_1_0 + E.e22 * E.b / r) 

c = _1_0 / hypot1(t) 

s = t * c 

 

# geodetic latitude (Bowring eqn 18) 

a = atan2(z + E.e22 * E.b * s**3, 

p - E.e2 * E.a * c**3) 

 

# height above ellipsoid (Bowring eqn 7) 

sa, ca = sincos2(a) 

# r = E.a / E.e2s(sa) # length of normal terminated by minor axis 

# h = p * ca + z * sa - (E.a * E.a / r) 

h = fsum_(p * ca, z * sa, -E.a * E.e2s(sa)) 

 

C, lat, lon = 1, degrees90(a), atan2d(y, x) 

 

# see <https://GIS.StackExchange.com/questions/28446> 

elif p > EPS: # lat arbitrarily zero 

C, lat, lon, h = 2, _0_0, atan2d(y, x), p - E.a 

 

else: # polar lat, lon arbitrarily zero 

C, lat, lon, h = 3, copysign(_90_0, z), _0_0, abs(z) - E.b 

 

r = Ecef9Tuple(x, y, z, lat, lon, h, C, None, self.datum) 

return self._xnamed(r, name=name) 

 

 

class EcefSudano(EcefVeness): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates based on I{John J. Sudano}'s U{paper 

<https://www.ResearchGate.net/publication/ 

3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}. 

''' 

 

if _FOR_DOCS: 

__init__ = EcefVeness.__init__ 

forward = EcefVeness.forward 

 

def reverse(self, xyz, y=None, z=None, **no_M): # PYCHOK unused M 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} using 

I{Sudano}'s U{iterative method<https://www.ResearchGate.net/publication/ 

3709199_An_exact_conversion_from_an_Earth-centered_coordinate_system_to_latitude_longitude_and_altitude>}. 

 

@arg xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} 

ECEF C{x} coordinate in C{meter}. 

@kwarg y: ECEF C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{z}}. 

@kwarg z: ECEF C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{y}}. 

@kwarg no_M: Rotation matrix C{M} not available. 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geodetic coordinates C{(lat, lon, height)} for the given 

geocentric ones C{(x, y, z)}, iteration C{C}, L{EcefMatrix} C{M} 

always C{None} and C{datum} if available. 

 

@raise EcefError: If B{C{xyz}} not L{Ecef9Tuple} or C{scalar} C{x} 

or B{C{y}} and/or B{C{z}} not C{scalar} for C{scalar} 

B{C{xyz}} or no convergence. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, Error=EcefError) 

 

E = self.ellipsoid 

e = E.e2 * E.a 

h = hypot(x, y) # Rh 

d = e - h 

 

a = atan2(z, h * E.e12) 

sa, ca = sincos2(abs(a)) 

# Sudano's Eq (A-6) and (A-7) refactored/reduced, 

# replacing Rn from Eq (A-4) with n = E.a / ca: 

# N = ca**2 * ((z + E.e2 * n * sa) * ca - h * sa) 

# = ca**2 * (z * ca + E.e2 * E.a * sa - h * sa) 

# = ca**2 * (z * ca + (E.e2 * E.a - h) * sa) 

# D = ca**3 * (E.e2 * n / E.e2s2(sa)) - h 

# = ca**2 * (E.e2 * E.a / E.e2s2(sa) - h / ca**2) 

# N / D = (z * ca + (E.e2 * E.a - h) * sa) / 

# (E.e2 * E.a / E.e2s2(sa) - h / ca**2) 

S = Fsum(sa) 

for C in range(1, _TRIPS): 

ca2 = _1_0 - sa**2 

if ca2 < EPS_2: # PYCHOK no cover 

ca = _0_0 

break 

ca = sqrt(ca2) 

t = e / E.e2s2(sa) - h / ca2 

if abs(t) < EPS_2: 

break 

sa, t = S.fsum2_(-(z * ca + d * sa) / t) 

if abs(t) < EPS: 

break 

else: 

raise EcefError(_no_convergence_, txt=unstr(self.reverse.__name__, x=x, y=y, z=z)) 

 

if C: 

a = copysign(asin(sa), z) 

h = fsum_(h * ca, abs(z * sa), -E.a * E.e2s(sa)) # use Veness', 

# Sudano's Eq (7) doesn't seem to provide the correct height 

# h = (abs(z) + h - E.a * cos(a + E.e12) * sa / ca) / (ca + sa) 

 

r = Ecef9Tuple(x, y, z, degrees90(a), atan2d(y, x), h, C, None, self.datum) 

r._iteration = C 

return self._xnamed(r, name=name) 

 

 

class EcefYou(_EcefBase): 

'''Conversion between geodetic and geocentric, aka I{Earth-Centered, 

Earth-Fixed} (ECEF) coordinates using I{Rey-Jer You}'s U{transformations 

<https://www.ResearchGate.net/publication/240359424>}. 

 

@see: W.E. Featherstone, S.J. (Sten) Claessens U{Closed-form transformation 

between geodetic and ellipsoidal coordinates 

<https://espace.Curtin.edu.AU/bitstream/handle/20.500.11937/11589/115114_9021_geod2ellip_final.pdf>} 

Studia Geophysica et Geodaetica, 2008, 52, 1-18 and U{PyMap3D 

<https://PyPI.org/project/pymap3d>}. 

''' 

 

def __init__(self, a_ellipsoid, f=None, name=NN): 

'''New L{EcefYou} converter. 

 

@arg a_ellipsoid: An ellipsoid (L{Ellipsoid}, L{Ellipsoid2}, L{Datum} 

or L{a_f2Tuple}) or C{scalar} for the equatorial 

(major) radius of the ellipsoid (C{meter}). 

@kwarg f: C{None} or the ellipsoid flattening (C{scalar}), required 

for C{scalar} B{C{a_ellipsoid}}, B{C{f=0}} represents a 

sphere, negative B{C{f}} a prolate ellipsoid. 

@kwarg name: Optional name (C{str}). 

 

@raise EcefError: If B{C{a_ellipsoid}} not L{Ellipsoid}, L{Ellipsoid2}, 

L{Datum} or L{a_f2Tuple} or C{scalar} or B{C{f}} not 

C{scalar} or if C{scalar} B{C{a_ellipsoid}} not positive 

or B{C{f}} not less than 1.0. 

''' 

_EcefBase.__init__(self, a_ellipsoid, f, name) 

 

def forward(self, latlonh, lon=None, height=0, M=False): 

'''Convert from geodetic C{(lat, lon, height)} to geocentric C{(x, y, z)}. 

 

@arg latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} 

latitude in C{degrees}. 

@kwarg lon: Optional C{scalar} longitude in C{degrees} for C{scalar} 

B{C{latlonh}}. 

@kwarg height: Optional height in C{meter}, vertically above (or 

below) the surface of the ellipsoid. 

@kwarg M: Optionally, return the rotation L{EcefMatrix} (C{bool}). 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geocentric C{(x, y, z)} coordinates for the given 

geodetic ones C{(lat, lon, height)}, case C{C} 0, 

L{EcefMatrix} C{M} and C{datum} if available. 

 

@raise EcefError: If B{C{latlonh}} not C{LatLon}, L{Ecef9Tuple} or 

C{scalar} or B{C{lon}} not C{scalar} for C{scalar} 

B{C{latlonh}} or C{abs(lat)} exceeds 90°. 

''' 

lat, lon, h, name = _llhn4(latlonh, lon, height) 

sa, ca, sb, cb = sincos2d(lat, lon) 

 

E = self.ellipsoid 

n = _1_0 / hypot(E.a * ca, E.b * sa) 

z = (n * E.b2 + h) * sa 

x = (n * E.a2 + h) * ca 

 

m = self._Matrix(sa, ca, sb, cb) if M else None 

r = Ecef9Tuple(x * cb, x * sb, z, lat, lon, h, 0, m, self.datum) 

return self._xnamed(r, name=name) 

 

def reverse(self, xyz, y=None, z=None, **no_M): # PYCHOK unused M 

'''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)} 

using I{Rey-Jer You}'s transformation. 

 

@arg xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} 

ECEF C{x} coordinate in C{meter}. 

@kwarg y: ECEF C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} 

and B{C{z}}. 

@kwarg z: ECEF C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} 

and B{C{y}}. 

@kwarg no_M: Rotation matrix C{M} not available. 

 

@return: An L{Ecef9Tuple}C{(x, y, z, lat, lon, height, C, M, datum)} 

with geodetic coordinates C{(lat, lon, height)} for the given 

geocentric ones C{(x, y, z)}, case C{C} 1, L{EcefMatrix} C{M} 

always C{None} and C{datum} if available. 

 

@raise EcefError: If B{C{xyz}} not L{Ecef9Tuple} or C{scalar} C{x} 

or B{C{y}} and/or B{C{z}} not C{scalar} for C{scalar} 

B{C{xyz}}. 

''' 

x, y, z, name = _xyzn4(xyz, y, z, Error=EcefError) 

 

x2, y2, z2 = x**2, y**2, z**2 

r2 = fsum_(x2, y2, z2) # = hypot3(x2, y2, z2)**2 

 

E = self.ellipsoid 

e = sqrt(E.a2 - E.b2) 

e2 = e**2 

 

u = sqrt(fsum_(r2, -e2, hypot(r2 - e2, 2 * e * z)) / 2) 

p = hypot(u, e) 

q = hypot(x, y) 

B = atan2(p * z, u * q) # beta0 = atan(p / u * z / q) 

sB, cB = sincos2(B) 

p *= E.a 

B += fsum_(u * E.b, -p, e2) * sB / (p / cB - e2 * cB) 

sB, cB = sincos2(B) 

 

h = hypot(z - E.b * sB, q - E.a * cB) 

if fsum_(x2, y2, z2 * E.a_b**2) < E.a2: 

h = -h # inside ellipsoid 

 

r = Ecef9Tuple(x, y, z, atan2d(E.a * sB, E.b * cB), # atan(E.a_b * tan(B)) 

atan2d(y, x), h, 1, # C=1 

None, # M=None 

self.datum) 

return self._xnamed(r, name=name) 

 

 

__all__ += _ALL_DOCS(_EcefBase, Ecef9Tuple) 

 

# **) MIT License 

# 

# Copyright (C) 2016-2020 -- 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 

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