# Coverage for pygeodesy/ecef.py : 93%

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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 origin of local cartesian coordinate system is at C{lat0}, C{lon0} and 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.datum import Datum, Datums, Ellipsoid from pygeodesy.fmath import cbrt, EPS, EPS1, EPS_2, fdot, fStr, fsum_, hypot1, \ _IsNotError, isscalar, map1 from pygeodesy.lazily import _ALL_DOCS, _ALL_LAZY from pygeodesy.named import LatLon3Tuple, LatLon4Tuple, _NamedBase, \ _NamedTuple, Vector3Tuple from pygeodesy.utily import degrees90, degrees180, property_RO, \ sincos2, sincos2d, _TypeError, unStr from pygeodesy.vector3d import _xyzn4 from math import atan2, copysign, cos, degrees, hypot, sqrt # all public contants, classes and functions __all__ = _ALL_LAZY.ecef + _ALL_DOCS('_EcefBase', 'Ecef9Tuple') __version__ = '19.12.22' class EcefError(ValueError): '''An ECEF issue. ''' pass def _afStr(this, prec, *attrs): def _fStr(a): return '%s=%s' % (a, fStr(getattr(this, a), prec=prec)) return tuple(map(_fStr, attrs)) def _aRepr(this, *attrs): def _repr(a): return '%s=%r' % (a, getattr(this, a)) return tuple(map(_repr, attrs)) def _llhn4(latlonh, lon, height, suffix): '''(INTERNAL) Get an C{(lat, lon, h, name)} 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_ or height_'.replace('_', suffix) raise EcefError('%s invalid: %r, %s' % (t, llh, x)) if abs(lat) > 90: raise EcefError('%s%s out of range: %.6g' % ('lat', suffix, lat)) return lat, lon, h, getattr(latlonh, 'name', '') 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, 1 return s, c, h 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: if isinstance(E, Datum): self._datum = E E = E.ellipsoid elif not isinstance(E, Ellipsoid): raise TypeError self.name = E.name elif isscalar(E) and isscalar(f): a = float(E) f_ = (1.0 / f) if f else 0 # sphere b = None if f_ else a E = Ellipsoid(a, b, f_, name='_' + name) else: raise ValueError if not (E.a > 0 and E.f < 1): raise ValueError except (TypeError, ValueError): t = unStr(self.classname, a=E, f=f) raise EcefError('%s invalid: %s' % ('ellipsoid', t)) 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}. ''' return self._datum @property_RO def ellipsoid(self): '''Get C{E}, the ellipsoid (L{Ellipsoid}). ''' 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 _Matrix(self, sa, ca, sb, cb): '''Creation a rotation matrix. @return: A L{EcefMatrix}. ''' # This matrix is given by 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; return self._xnamed(EcefMatrix(sa, ca, sb, cb)) def toStr(self, prec=9): # PYCHOK signature '''Return this ECEF as a string. @keyword prec: Optional precision, number of decimal digits (0..9). @return: This ECEF as C{str}. ''' t = _afStr(self, prec, 'a', 'f') + \ _aRepr(self, 'datum','ellipsoid','name') return ', '.join(t) 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 def __init__(self, a_ellipsoid, f=None, name=''): '''New L{EcefKarney} converter. @param a_ellipsoid: An ellipsoid (L{Ellipsoid}), a datum (L{Datum}) or C{scalar} for the major, equatorial radius of the ellipsoid (C{meter}). @keyword f: C{None} or the ellipsoid flattening (C{scalar}), required for C{scalar} B{C{a_datum_ellipsoid}}, B{C{f}}=0 represents a sphere, negative B{C{f}} a prolate ellipsoid. @keyword name: Optional name (C{str}). @raise EcefError: If B{C{a_ellipsoid}} not L{Ellipsoid}, L{Datum} 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)}. @param latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} latitude in C{degrees}. @keyword lon: Optional C{scalar} longitude in C{degrees} for C{scalar} B{C{latlonh}}. @keyword height: Optional height in C{meter}, vertically above (or below) the surface of the ellipsoid. @keyword 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(B{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 / 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) def reverse(self, xyz, y=None, z=None, M=False): '''Convert from geocentric C{(x, y, z)} to geodetic C{(lat, lon, height)}. @param xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} ECEF C{x} coordinate in C{meter}. @keyword y: ECEF C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{z}}. @keyword z: ECEF C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{y}}. @keyword 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: # We really far away (> 12 million 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 / 2, x / 2) sa, ca, _ = _sch3(z / 2, 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 # s = r^3 * s u = r = r / 6 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 * r * cos(atan2(sqrt(-disc), -t3) / 3) 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 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, d) h -= self.a C = 4 r = Ecef9Tuple(x, y, z, degrees(atan2(sa, ca)), degrees(atan2(sb, cb)), h, C, self._Matrix(sa, ca, sb, cb) if M else None, self.datum) return self._xnamed(r, 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)} transcibed from on 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 = None _t0 = None def __init__(self, latlonh0=0, lon0=0, height0=0, ecef=EcefKarney(Datums.WGS84), name=''): '''New L{EcefCartesian} converter. @keyword latlonh0: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} latitude in C{degrees} of the cartesian origin. @keyword lon0: Optional C{scalar} longitude of the cartesian origin in C{degrees} for C{scalar} B{C{latlonh0}}. @keyword height0: Optional height of the cartesian origin in C{meter}, vertically above (or below) the surface of the ellipsoid. @keyword ecef: An EXEC converter (L{EcefKarney}). @keyword 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(B{lat0})} exceeds 90°. @raise TypeError: Invalid B{C{ecef}}, not L{EcefKarney}. ''' _TypeError(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)}. @param latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} latitude in C{degrees}. @keyword lon: Optional C{scalar} longitude in C{degrees} for C{scalar} B{C{latlonh}}. @keyword height: Optional height in C{meter}, vertically above (or below) the surface of the ellipsoid. @keyword 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(B{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) @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=''): '''Reset the local cartesian origin. @keyword latlonh0: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} latitude in C{degrees} of the cartesian origin. @keyword lon0: Optional, C{scalar} longitude of the cartesian origin in C{degrees} for C{scalar} B{C{latlonh0}}. @keyword height0: Optional, height of the cartesian origin in C{meter}, vertically above (or below) the surface of the ellipsoid. @keyword 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(B{lat})} exceeds 90°. ''' lat0, lon0, height0, n = _llhn4(latlonh0, lon0, height0, '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)}. @param xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} local cartesian C{x} coordinate in C{meter}. @keyword y: Local cartesian C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{z}}. @keyword z: Local cartesian C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{y}}. @keyword 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, xyz_n[3]) def toStr(self, prec=9): # PYCHOK signature '''Return this L{EcefCartesian} as a string. @keyword prec: Optional precision, number of decimal digits (0..9). @return: This L{EcefCartesian} as C{str}. ''' t = _afStr(self, prec, 'lat0', 'lon0', 'height0') + \ _aRepr(self, 'M', 'ecef', 'name') return ', '.join(t) 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') def __new__(cls, sa, ca, sb, cb, *_m): '''New L{EcefMatrix} matrix. @param sa: C{sin(phi)} (C{float}). @param ca: C{cos(phi)} (C{float}). @param sb: C{sin(lambda)} (C{float}). @param cb: C{cos(lambda)} (C{float}). @param _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('%s invalid: %r' % ('sa, ca, sb or cb', t)) else: # build matrix t = (-sb, -cb * sa, cb * ca, cb, -sb * sa, sb * ca, 0, ca, sa) return _NamedTuple.__new__(cls, *t) def multiply(self, other): '''Matrix multiply M{M0' ⋅ M} this matrix transposed with an other matrix. @param other: The other matrix (L{EcefMatrix}). @return: The matrix product (L{EcefMatrix}). @raise TypeError: If B{C{other}} is not L{EcefMatrix}. ''' _TypeError(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])'}. @param xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). @param xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). @return: Rotated C{(x, y, z)} location (C{3-tuple}). ''' if xyz0: # and len(xyz) == len(xyz0) 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(self[0::3], *xyz), fdot(self[1::3], *xyz), fdot(self[2::3], *xyz)) def unrotate(self, xyz, *xyz0): '''Inverse rotation M{[x0, y0, z0] + M0 ⋅ [x,y,z]'}. @param xyz: Local C{(x, y, z)} coordinates (C{3-tuple}). @param xyz0: Optional, local C{(x0, y0, z0)} origin (C{3-tuple}). @return: Unrotated C{(x, y, z)} location (C{3-tuple}). ''' # 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(self[0:3], *xyz), fdot(self[3:6], *xyz), fdot(self[6:], *xyz)) if xyz0: # and len(xyz) == len(xyz0) xyz = tuple(c0 + c for c0, c in zip(xyz0, xyz)) 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} 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}, usually. ''' _Names_ = ('x', 'y', 'z', 'lat', 'lon', 'height', 'C', 'M', 'datum') def toCartesian(self, Cartesian): '''Return the geocentric C{(x, y, z)} coordinates as an ellipsoidal or spherical C{Cartesian}. @param Cartesian: L{ellipsoidalKarney.Cartesian}, L{ellipsoidalNvector.Cartesian}, L{ellipsoidalVincenty.Cartesian}, L{sphericalNvector.Cartesian} or L{sphericalTrigonometry.Cartesian} (sub-)class to return C{(x, y, z)}. @return: A B{C{Cartesian}}C{(x, y, z)} instance. @raise TypeError: Invalid B{C{Cartesian}}. ''' from pygeodesy.cartesianBase import CartesianBase as _CB if not issubclass(Cartesian, _CB): raise _IsNotError(_CB.__name__, Cartesian=Cartesian) r = Cartesian(self) return self._xnamed(r) def toLatLon(self, LatLon=None): '''Return the geodetic C{(lat, lon, height[, datum])} coordinates. @keyword LatLon: Optional (sub-)class to return C{(lat, lon, height[, datum])} or C{None}. @return: An instance of C{LatLon}C{(lat, lon, height[, datum])} if B{C{LatLon}} is not C{None} or a L{LatLon3Tuple}C{(lat, lon, height)} or a L{LatLon4Tuple}C{(lat, lon, height, datum)} if C{datum} is unavailable respectively available. ''' lat, lon = self.lat, self.lon # PYCHOK expected h, d = self.height, self.datum # PYCHOK expected if d is None: r = LatLon3Tuple(lat, lon, h) if LatLon is None else \ LatLon(lat, lon, h) elif isinstance(d, Datum): r = LatLon4Tuple(lat, lon, h, d) if LatLon is None else \ LatLon(lat, lon, height=h, datum=d) else: raise AssertionError('%r.%s: %r' % (self, 'datum', d)) return self._xnamed(r) def toVector(self, Vector=None): '''Return the geocentric C{(x, y, z)} coordinates as vector. @keyword Vector: Optional vector (sub-)class to return C{(x, y, z)} or C{None}. @return: A C{Vector}C{(x, y, z)} instance or if B{C{Vector}} is C{None} a L{Vector3Tuple}C{(x, y, z)}. ''' r = Vector3Tuple(self.x, self.y, self.z) if Vector is None else \ Vector(self.x, self.y, self.z) # PYCHOK x, y, z return self._xnamed(r) 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=''): '''New L{EcefVeness} converter. @param a_ellipsoid: An ellipsoid (L{Ellipsoid}), a datum (L{Datum}) or C{scalar} for the major, equatorial radius of the ellipsoid (C{meter}). @keyword f: C{None} or the ellipsoid flattening (C{scalar}), required for C{scalar} B{C{a_datum_ellipsoid}}, B{C{f}}=0 represents a sphere, negative B{C{f}} a prolate ellipsoid. @keyword name: Optional name (C{str}). @raise EcefError: If B{C{a_ellipsoid}} not L{Ellipsoid}, L{Datum} 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)}. @param latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} latitude in C{degrees}. @keyword lon: Optional C{scalar} longitude in C{degrees} for C{scalar} B{C{latlonh}}. @keyword height: Optional height in C{meter}, vertically above (or below) the surface of the ellipsoid. @keyword 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(B{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 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) 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. @param xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} ECEF C{x} coordinate in C{meter}. @keyword y: ECEF C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{z}}. @keyword z: ECEF C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{y}}. @keyword 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). ''' 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 + E.e22 * E.b / r) c = 1 / 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) b = atan2(y, x) # ... and longitude # 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), degrees180(b) # see <https://GIS.StackExchange.com/questions/28446> elif p > EPS: # lat arbitrarily zero C, lat, lon, h = 2, 0.0, degrees180(atan2(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) 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=''): '''New L{EcefYou} converter. @param a_ellipsoid: An ellipsoid (L{Ellipsoid}), a datum (L{Datum}) or C{scalar} for the major, equatorial radius of the ellipsoid (C{meter}). @keyword f: C{None} or the ellipsoid flattening (C{scalar}), required for C{scalar} B{C{a_datum_ellipsoid}}, B{C{f}}=0 represents a sphere, negative B{C{f}} a prolate ellipsoid. @keyword name: Optional name (C{str}). @raise EcefError: If B{C{a_ellipsoid}} not L{Ellipsoid}, L{Datum} 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)}. @param latlonh: Either a C{LatLon}, an L{Ecef9Tuple} or C{scalar} latitude in C{degrees}. @keyword lon: Optional C{scalar} longitude in C{degrees} for C{scalar} B{C{latlonh}}. @keyword height: Optional height in C{meter}, vertically above (or below) the surface of the ellipsoid. @keyword 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(B{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) 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. @param xyz: Either an L{Ecef9Tuple}, an C{(x, y, z)} 3-tuple or C{scalar} ECEF C{x} coordinate in C{meter}. @keyword y: ECEF C{y} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{z}}. @keyword z: ECEF C{z} coordinate in C{meter} for C{scalar} B{C{xyz}} and B{C{y}}. @keyword 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, degrees(atan2(E.a * sB, E.b * cB)), # atan(E.a_b * tan(B)) degrees(atan2(y, x)), h, 1, None, self.datum) return self._xnamed(r, name) # **) 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 # 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.