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# -*- coding: utf-8 -*- 

 

u'''Triaxal ellipsoid classes L{JacobiConformal}, Jacobi's conformal projection, trancoded 

from I{Charles Karney}'s C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/C++/ 

doc/classGeographicLib_1_1JacobiConformal.html#details>} to pure Python, I{ordered} L{Triaxial} 

and I{unordered} L{Triaxial_} and miscellaneous classes L{BetaOmega2Tuple}, L{BetaOmega3Tuple}, 

L{Jacobi2Tuple} and L{TriaxialError}. 

 

@see: U{Geodesics on a triaxial ellipsoid<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid# 

Geodesics_on_a_triaxial_ellipsoid>} and U{Triaxial coordinate systems and their geometrical 

interpretation<https://www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

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

from __future__ import division as _; del _ # PYCHOK semicolon 

 

# from pygeodesy.basics import isscalar, map1, _zip # from .fsums, .namedTuples, .streprs 

from pygeodesy.constants import EPS, EPS0, EPS02, EPS4, _EPS2e4, INT0, PI2, PI_3, PI4, \ 

_0_0, _0_5, _1_0, _N_1_0, isfinite, isnear1, \ 

_4_0 # PYCHOK used! 

from pygeodesy.datums import Datum, Ellipsoid, _spherical_datum, _WGS84 

# from pygeodesy.ellipsoids import Ellipsoid # from .datums 

# from pygeodesy.elliptic import Elliptic # ._MODS 

# from pygeodesy.errors import _ValueError # from .streprs 

from pygeodesy.fmath import Fdot, fdot, fmean_, hypot, hypot_, hypot2, hypot2_, norm2 

from pygeodesy.fsums import Fsum, fsum, fsum_, isscalar, Property_RO 

from pygeodesy.interns import NN, _a_, _b_, _c_, _distant_, _height_, _inside_, \ 

_near_, _not_, _NOTEQUAL_, _null_, _opposite_, _outside_, \ 

_SPACE_, _spherical_, _too_, _x_, _y_ 

# from pygeodesy.lazily import _ALL_LAZY, _ALL_MODS as _MODS # from .vector3d 

from pygeodesy.named import _NamedBase, _NamedTuple, _Pass 

from pygeodesy.namedTuples import LatLon3Tuple, map1, Vector3Tuple, Vector4Tuple 

# from pygeodesy.props import Property_RO # from .fsums 

from pygeodesy.streprs import Fmt, _ValueError, _zip 

from pygeodesy.units import Degrees, Float, Height_, Meter, Meter2, Meter3, Radians, Radius 

from pygeodesy.utily import asin1, atan2d, km2m, sincos2, sincos2d, sincos2d_ 

from pygeodesy.vector3d import _ALL_LAZY, _MODS, _otherV3d, Vector3d 

 

from math import atan2, fabs, sqrt 

 

__all__ = _ALL_LAZY.triaxials 

__version__ = '22.11.03' 

 

_not_ordered_ = _not_('ordered') 

_TRIPS = 256 # max 55, Eberly 1074? 

 

 

class _ToNamedBase(_NamedBase): 

'''(INTERNAL) C{-.toDegrees}, C{-.toRadians} base. 

''' 

def _toDegrees(self, a, b, *c, **toDMS_kwds): 

if toDMS_kwds: 

toDMS = _MODS.dms.toDMS 

a = toDMS(a.toDegrees(), **toDMS_kwds) 

b = toDMS(b.toDegrees(), **toDMS_kwds) 

elif isinstance(a, Degrees) and \ 

isinstance(b, Degrees): 

return self 

else: 

a, b = a.toDegrees(), b.toDegrees() 

return self.classof(a, b, *c, name=self.name) 

 

def _toRadians(self, a, b, *c): 

return self if isinstance(a, Radians) and \ 

isinstance(b, Radians) else \ 

self.classof(a.toRadians(), b.toRadians(), 

*c, name=self.name) 

 

 

class BetaOmega2Tuple(_NamedTuple, _ToNamedBase): 

'''2-Tuple C{(beta, omega)} with I{ellipsoidal} lat- and 

longitude C{beta} and C{omega} both in C{Radians} (or 

C{Degrees}). 

''' 

_Names_ = ('beta', 'omega') 

_Units_ = (_Pass, _Pass) 

 

def toDegrees(self, **toDMS_kwds): 

'''Convert this L{BetaOmega2Tuple} to C{Degrees} or C{toDMS}. 

 

@return: L{BetaOmega2Tuple}C{(beta, omega)} with 

C{beta} and C{omega} both in C{Degrees} 

or as an L{toDMS} string provided some 

B{C{toDMS_kwds}} are supplied. 

''' 

return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) 

 

def toRadians(self): 

'''Convert this L{BetaOmega2Tuple} to C{Radians}. 

 

@return: L{BetaOmega2Tuple}C{(beta, omega)} with 

C{beta} and C{omega} both in C{Radians}. 

''' 

return _ToNamedBase._toRadians(self, *self) 

 

 

class BetaOmega3Tuple(_NamedTuple, _ToNamedBase): 

'''3-Tuple C{(beta, omega, height)} with I{ellipsoidal} lat- and 

longitude C{beta} and C{omega} both in C{Radians} (or C{Degrees}) 

and the C{height}, rather the (signed) I{distance} to the triaxial's 

surface (measured along the radial line to the triaxial's center) 

in C{meter}, conventionally. 

''' 

_Names_ = BetaOmega2Tuple._Names_ + (_height_,) 

_Units_ = BetaOmega2Tuple._Units_ + ( Meter,) 

 

def toDegrees(self, **toDMS_kwds): 

'''Convert this L{BetaOmega3Tuple} to C{Degrees} or C{toDMS}. 

 

@return: L{BetaOmega3Tuple}C{(beta, omega, height)} with 

C{beta} and C{omega} both in C{Degrees} or as an 

L{toDMS} string provided some B{C{toDMS_kwds}} 

are supplied. 

''' 

return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) 

 

def toRadians(self): 

'''Convert this L{BetaOmega3Tuple} to C{Radians}. 

 

@return: L{BetaOmega3Tuple}C{(beta, omega, height)} with 

C{beta} and C{omega} both in C{Radians}. 

''' 

return _ToNamedBase._toRadians(self, *self) 

 

def to2Tuple(self): 

'''Reduce this L{BetaOmega3Tuple} to a L{BetaOmega2Tuple}. 

''' 

return BetaOmega2Tuple(*self[:2]) 

 

 

class Jacobi2Tuple(_NamedTuple, _ToNamedBase): 

'''2-Tuple C{(x, y)} with a Jacobi Conformal C{x} and C{y} 

projection, both in C{Radians} (or C{Degrees}). 

''' 

_Names_ = (_x_, _y_) 

_Units_ = (_Pass, _Pass) 

 

def toDegrees(self, **toDMS_kwds): 

'''Convert this L{Jacobi2Tuple} to C{Degrees} or C{toDMS}. 

 

@return: L{Jacobi2Tuple}C{(x, y)} with C{x} and C{y} 

both in C{Degrees} or as an L{toDMS} string 

provided some B{C{toDMS_kwds}} are supplied. 

''' 

return _ToNamedBase._toDegrees(self, *self, **toDMS_kwds) 

 

def toRadians(self): 

'''Convert this L{Jacobi2Tuple} to C{Radians}. 

 

@return: L{Jacobi2Tuple}C{(x, y)} with C{x} 

and C{y} both in C{Radians}. 

''' 

return _ToNamedBase._toRadians(self, *self) 

 

 

class Triaxial_(_NamedBase): # _NamedEnumItem 

'''I{Unordered} triaxial ellipsoid and base class. 

 

Triaxial ellipsoids with right-handed semi-axes C{a}, C{b} and C{c}, oriented 

such that the large principal ellipse C{ab} is the equator I{Z}=0, I{beta}=0, 

while the small principal ellipse C{ac} is the prime meridian, plane I{Y}=0, 

I{omega}=0. 

 

The four umbilic points, C{abs}(I{omega}) = C{abs}(I{beta}) = C{PI/2}, lie on 

the middle principal ellipse C{bc} in plane I{X}=0, I{omega}=C{PI/2}. 

 

@note: I{Geodetic} C{lat}- and C{lon}gitudes are in C{degrees}, I{geodetic} 

C{phi} and C{lam}bda are in C{radians}, but I{ellipsoidal} lat- and 

longitude C{beta} and C{omega} are in C{Radians} by default (or in 

C{Degrees} if converted). 

''' 

_unordered = True 

 

def __init__(self, a_triax, b=None, c=None, name=NN): 

'''New I{unordered} L{Triaxial_}. 

 

@arg a_triax: C{X} semi-axis (C{scalar}, conventionally in C{meter}) 

or an other L{Triaxial} or L{Triaxial_} instance. 

@kwarg b: C{Y} semi-axis (C{meter}, same units as B{C{a}}), required 

if C{B{a_triax} is scalar}, ignored otherwise. 

@kwarg c: C{Z} semi-axis (C{meter}, same units as B{C{a}}), required 

if C{B{a_triax} is scalar}, ignored otherwise. 

@kwarg name: Optional name (C{str}). 

 

@raise TriaxialError: Invalid semi-axis or -axes. 

''' 

try: 

a = a_triax 

t = a._abc3 if isinstance(a, Triaxial_) else map1(float, a, b, c) 

except (TypeError, ValueError) as x: 

raise TriaxialError(a=a, b=b, c=c, cause=x) 

if name: 

self.name = name 

 

a, b, c = self._abc3 = t 

if self._unordered: # == not isinstance(self, Triaxial) 

s, _, t = sorted(t) 

if not (isfinite(t) and s > 0): 

raise TriaxialError(a=a, b=b, c=c) # txt=_invalid_ 

elif not (isfinite(a) and a >= b >= c > 0): 

raise TriaxialError(a=a, b=b, c=c, txt=_not_ordered_) 

elif not (a > c and self._a2c2 > 0 and self.e2ac > 0): 

raise TriaxialError(a=a, c=c, e2ac=self.e2ac, txt=_spherical_) 

 

def __str__(self): 

return self.toStr() 

 

@Property_RO 

def a(self): 

'''Get the (largest) C{x} semi-axis (C{meter}, conventionally). 

''' 

a, _, _ = self._abc3 

return Radius(a=a) 

 

@Property_RO 

def _a2b2(self): 

'''(INTERNAL) Get C{a**2 - b**2} == E_sub_e**2. 

''' 

a, b, _ = self._abc3 

return ((a - b) * (a + b)) if a != b else _0_0 

 

@Property_RO 

def _a2_b2(self): 

'''(INTERNAL) Get C{(a/b)**2}. 

''' 

a, b, _ = self._abc3 

return (a / b)**2 if a != b else _1_0 

 

@Property_RO 

def _a2c2(self): 

'''(INTERNAL) Get C{a**2 - c**2} == E_sub_x**2. 

''' 

a, _, c = self._abc3 

return ((a - c) * (a + c)) if a != c else _0_0 

 

@Property_RO 

def area(self): 

'''Get the surface area (C{meter} I{squared}). 

''' 

c, b, a = sorted(self._abc3) 

if a > c: 

a = Triaxial(a, b, c).area if a > b else \ 

Ellipsoid(a, b=c).areax # a == b 

else: # a == c == b 

a = Meter2(area=a**2 * PI4) 

return a 

 

def area_p(self, p=1.6075): 

'''I{Approximate} the surface area (C{meter} I{squared}). 

 

@kwarg p: Exponent (C{scalar} > 0), 1.6 for near-spherical or 1.5849625007 

for "near-flat" triaxials. 

 

@see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Approximate_formula>}. 

''' 

a, b, c = self._abc3 

if a == b == c: 

a *= a 

else: 

_p = pow 

a = _p(fmean_(_p(a * b, p), _p(a * c, p), _p(b * c, p)), _1_0 / p) 

return Meter2(area_p=a * PI4) 

 

@Property_RO 

def b(self): 

'''Get the (middle) C{y} semi-axis (C{meter}, same units as B{C{a}}). 

''' 

_, b, _ = self._abc3 

return Radius(b=b) 

 

@Property_RO 

def _b2c2(self): 

'''(INTERNAL) Get C{b**2 - c**2} == E_sub_y**2. 

''' 

_, b, c = self._abc3 

return ((b - c) * (b + c)) if b != c else _0_0 

 

@Property_RO 

def c(self): 

'''Get the (smallest) C{z} semi-axis (C{meter}, same units as B{C{a}}). 

''' 

_, _, c = self._abc3 

return Radius(c=c) 

 

@Property_RO 

def _c2_b2(self): 

'''(INTERNAL) Get C{(c/b)**2}. 

''' 

_, b, c = self._abc3 

return (c / b)**2 if b != c else _1_0 

 

@Property_RO 

def e2ab(self): 

'''Get the C{ab} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (b/a)**2}. 

''' 

return Float(e2ab=(_1_0 - self._1e2ab) or _0_0) 

 

@Property_RO 

def _1e2ab(self): 

'''(INTERNAL) Get C{1 - e2ab} == C{(b/a)**2}. 

''' 

a, b, _ = self._abc3 

return (b / a)**2 if a != b else _1_0 

 

@Property_RO 

def e2bc(self): 

'''Get the C{bc} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/b)**2}. 

''' 

return Float(e2bc=(_1_0 - self._1e2bc) or _0_0) 

 

_1e2bc = _c2_b2 # C{1 - e2bc} == C{(c/b)**2} 

 

@Property_RO 

def e2ac(self): 

'''Get the C{ac} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c/a)**2}. 

''' 

return Float(e2ac=(_1_0 - self._1e2ac) or _0_0) 

 

@Property_RO 

def _1e2ac(self): 

'''(INTERNAL) Get C{1 - e2ac} == C{(c/a)**2}. 

''' 

a, _, c = self._abc3 

return (c / a)**2 if a != c else _1_0 

 

@Property_RO 

def _Elliptic(self): 

'''(INTERNAL) Get class L{Elliptic} once. 

''' 

return _MODS.elliptic.Elliptic 

 

def hartzell4(self, pov, los=None, name=NN): 

'''Compute the intersection of this triaxial's surface with a Line-Of-Sight 

from a Point-Of-View in space. 

 

@see: Function L{pygeodesy.hartzell4} for further details. 

''' 

return hartzell4(pov, los=los, tri_biax=self, name=name) 

 

def height4(self, x_xyz, y=None, z=None, normal=True, eps=EPS): 

'''Compute the projection on and the height of a cartesian above or below 

this triaxial's surface. 

 

@arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

@kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

@kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

@kwarg normal: If C{True} the projection is perpendicular to (the nearest 

point on) this triaxial's surface, otherwise the C{radial} 

line to this triaxial's center (C{bool}). 

@kwarg eps: Tolerance for root finding and validation (C{scalar}), use a 

negative value to skip validation. 

 

@return: L{Vector4Tuple}C{(x, y, z, h)} with the cartesian coordinates 

C{x}, C{y} and C{z} of the projection on or the intersection 

with and with the height C{h} above or below the triaxial's 

surface in C{meter}, conventionally. 

 

@raise TriaxialError: Non-cartesian B{C{xyz}}, invalid B{C{eps}}, no 

convergence in root finding or validation failed. 

 

@see: Method L{Ellipsoid.height4} and I{Eberly}'s U{Distance from a Point 

to ... an Ellipsoid ...<https://www.GeometricTools.com/Documentation/ 

DistancePointEllipseEllipsoid.pdf>}. 

''' 

v = Vector3d(x_xyz, y, z) if isscalar(x_xyz) else _otherV3d(x_xyz=x_xyz) 

 

i, h = None, v.length 

if h < EPS0: # EPS 

x = y = z = _0_0 

h -= min(self._abc3) 

elif self.isSpherical: 

r, _, _ = self._abc3 

x, y, z = v.times(r / h).xyz 

h -= r 

else: 

x, y, z = v.xyz 

if normal: # perpendicular to triaxial 

try: 

x, y, z, h, i = _normalTo5(x, y, z, self, eps=eps) 

except Exception as e: 

raise TriaxialError(x=x, y=y, z=z, cause=e) 

else: # radially to triaxial's center 

x, y, z = self._radialTo3(z, hypot(x, y), y, x) 

h = v.minus_(x, y, z).length 

if h > 0 and self.sideOf(v, eps=EPS0) < 0: 

h = -h # below the surface 

return Vector4Tuple(x, y, z, h, iteration=i, name=self.height4.__name__) 

 

@Property_RO 

def isOrdered(self): 

'''Is this triaxial I{ordered} and not I{spherical} (C{bool})? 

''' 

a, b, c = self._abc3 

return bool(a >= b > c) # b > c! 

 

@Property_RO 

def isSpherical(self): 

'''Is this triaxial I{spherical} (C{bool})? 

''' 

a, b, c = self._abc3 

return bool(a == b == c) 

 

def normal3d(self, x, y, z, length=_1_0): 

'''Get a 3-D vector at a cartesian on, perpendicular to this triaxial's surface. 

 

@arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). 

@arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). 

@arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). 

@kwarg length: Optional length and in-/outward direction (C{scalar}). 

 

@return: A C{Vector3d(x_, y_, z_)} normalized to B{C{length}}, pointing 

in- or outward for neg- respectively positive B{C{length}}. 

 

@note: Cartesian location C{(B{x}, B{y}, B{z})} must be on this triaxial's 

surface, use method L{Triaxial.sideOf} to validate. 

''' 

# n = 2 * (x / a2, y / b2, z / c2) 

# == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # iff ordered 

# == 2 * (x, y / _1e2ab, z / _1e2ac) / a2 

# == unit(x, y / _1e2ab, z / _1e2ac).times(length) 

n = self._normal3d.times_(x, y, z) 

if n.length < EPS0: 

raise TriaxialError(x=x, y=y, z=z, txt=_null_) 

return n.times(length / n.length) 

 

@Property_RO 

def _normal3d(self): 

'''(INTERNAL) Get M{Vector3d((d/a)**2, (d/b)**2, (d/c)**2)}, M{d = max(a, b, c)}. 

''' 

d = max(self._abc3) 

t = tuple(((d / x)**2 if x != d else _1_0) for x in self._abc3) 

return Vector3d(*t) 

 

def _norm2(self, s, c, *a): 

'''(INTERNAL) Normalize C{s} and C{c} iff not already. 

''' 

if fabs(s) > _1_0 or fabs(c) > _1_0: # or \ 

\ 

# fabs(fsum(s**2, c**2, _N_1_0)) > EPS: 

s, c = norm2(s, c) 

if a: 

s, c = norm2(s * self.b, c * a[0]) 

return (s or _0_0), (c or _0_0) 

 

def _radialTo3(self, sbeta, cbeta, somega, comega): 

'''(INTERNAL) I{Unordered} helper for C{.height4}. 

''' 

def _rphi(a, b, sphi, cphi): 

# <https://WikiPedia.org/wiki/Ellipse#Polar_form_relative_to_focus> 

# polar form: radius(phi) = a * b / hypot(a * sphi, b * cphi) 

return (b / hypot(sphi, b / a * cphi)) if a > b else ( 

(a / hypot(cphi, a / b * sphi)) if a < b else a) 

 

sa, ca = self._norm2(sbeta, cbeta) 

sb, cb = self._norm2(somega, comega) 

 

a, b, c = self._abc3 

if a != b: 

a = _rphi(a, b, sb, cb) 

if a != c: 

c = _rphi(a, c, sa, ca) 

z, r = c * sa, c * ca 

x, y = r * cb, r * sb 

return x, y, z 

 

def sideOf(self, x_xyz, y=None, z=None, eps=EPS4): 

'''Is a cartesian above, below or on the surface of this triaxial? 

 

@arg x_xyz: X component (C{scalar}) or a cartesian (C{Cartesian}, 

L{Ecef9Tuple}, L{Vector3d}, L{Vector3Tuple} or L{Vector4Tuple}). 

@kwarg y: Y component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

@kwarg z: Z component (C{scalar}), required if B{C{x_xyz}} if C{scalar}. 

@kwarg eps: Near surface tolerance(C{scalar}). 

 

@return: C{INT0} if C{(B{x}, B{y}, B{z})} is near this triaxial's surface 

within tolerance B{C{eps}}, otherwise a neg- or positive C{float} 

if in- respectively outside this triaxial. 

 

@see: Methods L{Triaxial.height4} and L{Triaxial.normal3d}. 

''' 

def _x2_a2_1(xyz, abc): 

for x, a in _zip(xyz, abc): # strict=True 

yield (x / a)**2 

yield _N_1_0 

 

t = (x_xyz, y, z) if isscalar(x_xyz) else _otherV3d(x_xyz=x_xyz).xyz 

s = fsum(_x2_a2_1(t, self._abc3), floats=True) 

return s if s > eps or s < -eps else INT0 

 

def _sqrt(self, x): 

'''(INTERNAL) Helper. 

''' 

if x < 0: 

raise TriaxialError(Fmt.PAREN(sqrt=x)) 

return _0_0 if x < EPS02 else sqrt(x) 

 

def toEllipsoid(self, name=NN): 

'''Convert this triaxial to an L{Ellipsoid}, provided C{a == b} or C{b == c}. 

 

@return: An L{Ellipsoid} with north along this C{Z} axis if C{a == b}, 

this C{Y} axis if C{a == c} or this C{X} axis if C{b == c}. 

 

@raise TriaxialError: This C{a} != C{b}, C{b} != C{c} and C{c} != C{a}. 

 

@see: Method L{Ellipsoid.toTriaxial}. 

''' 

a, b, c = self._abc3 

if a == b: # N = Z 

b = c 

elif b == c: # N = X 

a, b = b, a 

elif a != c: 

t = _SPACE_(_a_, _NOTEQUAL_, _b_, _NOTEQUAL_, _c_) 

raise TriaxialError(a=a, b=b, c=c, txt=t) 

return Ellipsoid(a, b=b, name=name or self.name) 

 

def toStr(self, prec=9, name=NN, **unused): # PYCHOK signature 

'''Return this C{Triaxial} as a string. 

 

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

@kwarg name: Override name (C{str}) or C{None} to exclude 

this triaxial's name. 

 

@return: This C{Triaxial}'s attributes (C{str}). 

''' 

T = Triaxial_ 

t = T.a, T.b, T.c, T.e2ab, T.e2bc, T.e2ac 

if isinstance(self, JacobiConformal): 

t += JacobiConformal.xyQ2, 

t += T.volume, T.area 

return self._instr(name, prec, props=t, area_p=self.area_p()) 

 

@Property_RO 

def volume(self): 

'''Get the volume (C{meter**3}), M{4 / 3 * PI * a * b * c}. 

''' 

a, b, c = self._abc3 

return Meter3(volume=a * b * c * PI_3 * _4_0) 

 

 

class Triaxial(Triaxial_): 

'''I{Ordered} triaxial ellipsoid. 

 

@see: L{Triaxial_} for more information. 

''' 

_unordered = False 

 

def __init__(self, a_triax, b=None, c=None, name=NN): 

'''New I{ordered} L{Triaxial}. 

 

@arg a_triax: Largest semi-axis (C{scalar}, conventionally in C{meter}) 

or an other L{Triaxial} or L{Triaxial_} instance. 

@kwarg b: Middle semi-axis (C{meter}, same units as B{C{a}}), required 

if C{B{a_triax} is scalar}, ignored otherwise. 

@kwarg c: Smallest semi-axis (C{meter}, same units as B{C{a}}), required 

if C{B{a_triax} is scalar}, ignored otherwise. 

@kwarg name: Optional name (C{str}). 

 

@note: The semi-axes must be ordered as C{B{a} >= B{b} >= B{c} > 0} and 

must be ellipsoidal, C{B{a} > B{c}}. 

 

@raise TriaxialError: Semi-axes not ordered, spherical or invalid. 

''' 

Triaxial_.__init__(self, a_triax, b=b, c=c, name=name) 

 

@Property_RO 

def _a2b2_a2c2(self): 

''' @see: Method C{forwardBetaOmega}. 

''' 

return self._a2b2 / self._a2c2 

 

@Property_RO 

def area(self): 

'''Get the surface area (C{meter} I{squared}). 

 

@see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Surface_area>}. 

''' 

a, b, c = self._abc3 

if a != b: 

kp2, k2 = self._k2_kp2 # swapped! 

aE = self._Elliptic(k2, _0_0, kp2, _1_0) 

c2 = self._1e2ac # cos(phi)**2 == (c/a)**2 

s2 = self. e2ac # sin(phi)**2 == 1 - c2 

s = sqrt(s2) 

r = asin1(s) # phi == atan2(sqrt(c2), s) 

b *= fsum_(aE.fE(r) * s, aE.fF(r) * c2 / s, c / a * c / b, floats=True) 

a = Meter2(area=a * b * PI2) 

else: # a == b > c 

a = Ellipsoid(a, b=c).areax 

return a 

 

def _exyz3(self, u): 

'''(INTERNAL) Helper for C{.forwardBetOmg}. 

''' 

if u > 0: 

u2 = u**2 

x = self._sqrt(_1_0 + self._a2c2 / u2) * u 

y = self._sqrt(_1_0 + self._b2c2 / u2) * u 

else: 

x = y = u = _0_0 

return x, y, u 

 

def forwardBetaOmega(self, beta, omega, height=0, name=NN): 

'''Convert I{ellipsoidal} lat- and longitude C{beta}, C{omega} 

and height to cartesian. 

 

@arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

@arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

@arg height: Height above or below the ellipsoid's surface (C{meter}, same 

units as this triaxial's C{a}, C{b} and C{c} semi-axes). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: Method L{Triaxial.reverseBetaOmega} and U{Expressions (23-25)<https:// 

www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

if height: 

h = Height_(height=height, low=-self.c, Error=TriaxialError) 

x, y, z = self._exyz3(h + self.c) 

else: 

x, y, z = self._abc3 # == self._exyz3(self.c) 

if z: # and x and y: 

sa, ca = _SinCos2(beta) 

sb, cb = _SinCos2(omega) 

 

r = self._a2b2_a2c2 

x *= cb * self._sqrt(ca**2 + r * sa**2) 

y *= ca * sb 

z *= sa * self._sqrt(_1_0 - r * cb**2) 

return Vector3Tuple(x, y, z, name=name) 

 

def forwardBetaOmega_(self, sbeta, cbeta, somega, comega, name=NN): 

'''Convert I{ellipsoidal} lat- and longitude C{beta} and C{omega} 

to cartesian coordinates I{on the triaxial's surface}. 

 

@arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

@arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

@arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

@arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)} on the surface. 

 

@raise TriaxialError: This triaxial is near-spherical. 

 

@see: Method L{Triaxial.reverseBetaOmega}, U{Triaxial ellipsoid coordinate 

system<https://WikiPedia.org/wiki/Geodesics_on_an_ellipsoid# 

Triaxial_ellipsoid_coordinate_system>} and U{expressions (23-25)<https:// 

www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

t = self._radialTo3(sbeta, cbeta, somega, comega) 

return Vector3Tuple(*t, name=name) 

 

def forwardCartesian(self, x, y, z, name=NN, **normal_eps): 

'''Project a cartesian on this triaxial. 

 

@arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). 

@arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). 

@arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). 

@kwarg name: Optional name (C{str}). 

@kwarg normal_eps: Optional keyword arguments C{B{normal}=True} and 

C{B{eps}=EPS}, see method L{Triaxial.height4}. 

 

@see: Method L{Triaxial.height4} for further information and method 

L{Triaxial.reverseCartesian} to reverse the projection. 

''' 

t = self.height4(x, y, z, **normal_eps) 

_ = t.rename(name) 

return t 

 

def forwardLatLon(self, lat, lon, height=0, name=NN): 

'''Convert I{geodetic} lat-, longitude and heigth to cartesian. 

 

@arg lat: Geodetic latitude (C{degrees}). 

@arg lon: Geodetic longitude (C{degrees}). 

@arg height: Height above the ellipsoid (C{meter}, same units 

as this triaxial's C{a}, C{b} and C{c} axes). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https:// 

www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

return self._forward3(height, name, *sincos2d_(lat, lon)) 

 

def forwardLatLon_(self, slat, clat, slon, clon, height=0, name=NN): 

'''Convert I{geodetic} lat-, longitude and heigth to cartesian. 

 

@arg slat: Geodetic latitude C{sin(lat)} (C{scalar}). 

@arg clat: Geodetic latitude C{cos(lat)} (C{scalar}). 

@arg slon: Geodetic longitude C{sin(lon)} (C{scalar}). 

@arg clon: Geodetic longitude C{cos(lon)} (C{scalar}). 

@arg height: Height above the ellipsoid (C{meter}, same units 

as this triaxial's axes C{a}, C{b} and C{c}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: Method L{Triaxial.reverseLatLon} and U{Expressions (9-11)<https:// 

www.Topo.Auth.GR/wp-content/uploads/sites/111/2021/12/09_Panou.pdf>}. 

''' 

sa, ca = self._norm2(slat, clat) 

sb, cb = self._norm2(slon, clon) 

return self._forward3(height, name, sa, ca, sb, cb) 

 

def _forward3(self, h, name, sa, ca, sb, cb): 

'''(INTERNAL) Helper for C{.forwardLatLon} and C{.forwardLatLon_}. 

''' 

ca_x_sb = ca * sb 

# 1 - (1 - (c/a)**2) * sa**2 - (1 - (b/a)**2) * ca**2 * sb**2 

t = fsum_(_1_0, -self.e2ac * sa**2, -self.e2ab * ca_x_sb**2, floats=True) 

N = self.a / self._sqrt(t) # prime vertical 

x = (h + N) * ca * cb 

y = (h + N * self._1e2ab) * ca_x_sb 

z = (h + N * self._1e2ac) * sa 

return Vector3Tuple(x, y, z, name=name) 

 

@Property_RO 

def _k2_kp2(self): 

'''(INTERNAL) Get C{k2} and C{kp2} for C{._xE}, C{._yE} and C{.area}. 

''' 

# k2 = a2b2 / a2c2 * c2_b2 

# kp2 = b2c2 / a2c2 * a2_b2 

# b2 = b**2 

# xE = Elliptic(k2, -a2b2 / b2, kp2, a2_b2) 

# yE = Elliptic(kp2, +b2c2 / b2, k2, c2_b2) 

# aE = Elliptic(kp2, 0, k2, 1) 

return (self._a2b2 / self._a2c2 * self._c2_b2, 

self._b2c2 / self._a2c2 * self._a2_b2) 

 

def _radialTo3(self, sbeta, cbeta, somega, comega): 

'''(INTERNAL) Convert I{ellipsoidal} lat- and longitude C{beta} and 

C{omega} to cartesian coordinates I{on the triaxial's surface}, 

also I{ordered} helper for C{.height4}. 

''' 

sa, ca = self._norm2(sbeta, cbeta) 

sb, cb = self._norm2(somega, comega) 

 

b2_a2 = self._1e2ab # == (b/a)**2 

c2_a2 = -self._1e2ac # == -(c/a)**2 

a2c2_a2 = self. e2ac # (a**2 - c**2) / a**2 == 1 - (c/a)**2 

 

x = Fsum(_1_0, -b2_a2 * sa**2, c2_a2 * ca**2).fover(a2c2_a2) 

z = Fsum(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2) 

 

x = self.a * cb * self._sqrt(x) 

y = self.b * ca * sb 

z = self.c * sa * self._sqrt(z) 

return x, y, z 

 

def reverseBetaOmega(self, x, y, z, name=NN): 

'''Convert cartesian to I{ellipsoidal} lat- and longitude, C{beta}, C{omega} 

and height. 

 

@arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). 

@arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). 

@arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and 

C{omega} in C{Radians} and (radial) C{height} in C{meter}, same 

units as this triaxial's axes. 

 

@see: Methods L{Triaxial.forwardBetaOmega} and L{Triaxial.forwardBetaOmega_} 

and U{Expressions (21-22)<https://www.Topo.Auth.GR/wp-content/uploads/ 

sites/111/2021/12/09_Panou.pdf>}. 

''' 

v = Vector3d(x, y, z) 

a, b, h = self._reverse3(x, y, z, atan2, v, self.forwardBetaOmega_) 

return BetaOmega3Tuple(Radians(beta=a), Radians(omega=b), h, name=name) 

 

def reverseCartesian(self, x, y, z, h, normal=True, eps=_EPS2e4, name=NN): 

'''"Unproject" a cartesian on to a cartesion off this triaxial's surface. 

 

@arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). 

@arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). 

@arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). 

@arg h: Height above or below this triaxial's surface (C{meter}, same units 

as B{C{x}}). 

@kwarg normal: If C{True} the height is C{normal} to the surface, otherwise 

C{radially} to the center of this triaxial (C{bool}). 

@kwarg eps: Tolerance for surface test (C{scalar}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Vector3Tuple}C{(x, y, z)}. 

 

@see: Methods L{Triaxial.forwardCartesian} and L{Triaxial.height4}. 

''' 

v = Vector3d(x, y, z, name=name) 

s = self.sideOf(v.x, v.y, v.z, eps=eps) 

if s: # PYCHOK no cover 

t = _SPACE_((_inside_ if s < 0 else _outside_), self) 

raise TriaxialError(eps=eps, sideOf=s, x=x, y=y, z=z, txt=t) 

if h: 

if normal: 

v = v.plus(self.normal3d(v.x, v.y, v.z, length=h)) 

elif v.length > EPS0: 

v = v.times(_1_0 + (h / v.length)) 

return v.xyz # Vector3Tuple 

 

def reverseLatLon(self, x, y, z, name=NN): 

'''Convert cartesian to I{geodetic} lat-, longitude and height. 

 

@arg x: X coordinate along C{a}-axis (C{meter}, same units as the axes). 

@arg y: Y coordinate along C{b}-axis (C{meter}, same units as B{C{x}}). 

@arg z: Z coordinate along C{c}-axis (C{meter}, same units as B{C{x}}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{LatLon3Tuple}C{(lat, lon, height)} with C{lat} and C{lon} 

in C{degrees} and (radial) C{height} in C{meter}, same units 

as this triaxial's axes. 

 

@see: Methods L{Triaxial.forwardLatLon} and L{Triaxial.forwardLatLon_} 

and U{Expressions (4-5)<https://www.Topo.Auth.GR/wp-content/uploads/ 

sites/111/2021/12/09_Panou.pdf>}. 

''' 

v = Vector3d(x, y, z) 

x *= self._1e2ac # == 1 - e_sub_x**2 

y *= self._1e2bc # == 1 - e_sub_y**2 

t = self._reverse3(x, y, z, atan2d, v, self.forwardLatLon_) 

return LatLon3Tuple(*t, name=name) 

 

def _reverse3(self, x, y, z, atan2_, v, forward_): 

'''(INTERNAL) Helper for C{.reverseBetOmg} and C{.reverseLatLon}. 

''' 

d = hypot( x, y) 

a = atan2_(z, d) 

b = atan2_(y, x) 

h = v.minus_(*forward_(z, d, y, x)).length 

return a, b, h 

 

 

class JacobiConformal(Triaxial): 

'''This is a conformal projection of a triaxial ellipsoid to a plane in which the 

C{X} and C{Y} grid lines are straight. 

 

Ellipsoidal coordinates I{beta} and I{omega} are converted to Jacobi Conformal 

I{y} respectively I{x} separately. Jacobi's coordinates have been multiplied 

by C{sqrt(B{a}**2 - B{c}**2) / (2 * B{b})} so that the customary results are 

returned in the case of an ellipsoid of revolution (or a sphere, I{currently 

not supported}). 

 

Copyright (C) U{Charles Karney<mailto:Charles@Karney.com>} (2014-2020) and 

licensed under the MIT/X11 License. 

 

@note: This constructor can not be used to specify a sphere. 

 

@see: L{Triaxial}, C++ class U{JacobiConformal<https://GeographicLib.SourceForge.io/ 

C++/doc/classGeographicLib_1_1JacobiConformal.html#details>}, U{Jacobi's conformal 

projection<https://GeographicLib.SourceForge.io/C++/doc/jacobi.html>} and Jacobi, 

C. G. J. I{U{Vorlesungen über Dynamik<https://Books.Google.com/books? 

id=ryEOAAAAQAAJ&pg=PA212>}}, page 212ff, 

''' 

# @Property_RO 

# def ab(self): 

# '''Get relative magnitude C{ab} (C{None} or C{meter}, same units as B{C{a}}). 

# ''' 

# return self._ab 

 

# @Property_RO 

# def bc(self): 

# '''Get relative magnitude C{bc} (C{None} or C{meter}, same units as B{C{a}}). 

# ''' 

# return self._bc 

 

@Property_RO 

def _xE(self): 

'''(INTERNAL) Get the x-elliptic function. 

''' 

k2, kp2 = self._k2_kp2 

# -a2b2 / b2 == (b2 - a2) / b2 == 1 - a2 / b2 == 1 - a2_b2 

return self._Elliptic(k2, _1_0 - self._a2_b2, kp2, self._a2_b2) 

 

def xR(self, omega): 

'''Compute a Jacobi Conformal C{x} projection. 

 

@arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

 

@return: The C{x} projection (C{Radians}). 

''' 

return self.xR_(*_SinCos2(omega)) 

 

def xR_(self, somega, comega): 

'''Compute a Jacobi Conformal C{x} projection. 

 

@arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

@arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

 

@return: The C{x} projection (C{Radians}). 

''' 

s, c = self._norm2(somega, comega, self.a) 

return Radians(x=self._xE.fPi(s, c) * self._a2_b2) 

 

@Property_RO 

def xyQ2(self): 

'''Get the Jacobi Conformal quadrant size (L{Jacobi2Tuple}C{(x, y)}). 

''' 

return Jacobi2Tuple(Radians(x=self._a2_b2 * self._xE.cPi), 

Radians(y=self._c2_b2 * self._yE.cPi), 

name=JacobiConformal.xyQ2.name) 

 

def xyR2(self, beta, omega, name=NN): 

'''Compute a Jacobi Conformal C{x} and C{y} projection. 

 

@arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

@arg omega: Ellipsoidal longitude (C{radians} or L{Degrees}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Jacobi2Tuple}C{(x, y)}. 

''' 

return self.xyR2_(*(_SinCos2(beta) + _SinCos2(omega)), 

name=name or self.xyR2.__name__) 

 

def xyR2_(self, sbeta, cbeta, somega, comega, name=NN): 

'''Compute a Jacobi Conformal C{x} and C{y} projection. 

 

@arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

@arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

@arg somega: Ellipsoidal longitude C{sin(omega)} (C{scalar}). 

@arg comega: Ellipsoidal longitude C{cos(omega)} (C{scalar}). 

@kwarg name: Optional name (C{str}). 

 

@return: A L{Jacobi2Tuple}C{(x, y)}. 

''' 

return Jacobi2Tuple(self.xR_(somega, comega), 

self.yR_(sbeta, cbeta), 

name=name or self.xyR2_.__name__) 

 

@Property_RO 

def _yE(self): 

'''(INTERNAL) Get the x-elliptic function. 

''' 

kp2, k2 = self._k2_kp2 # swapped! 

# b2c2 / b2 == (b2 - c2) / b2 == 1 - c2 / b2 == e2bc 

return self._Elliptic(k2, self.e2bc, kp2, self._c2_b2) 

 

def yR(self, beta): 

'''Compute a Jacobi Conformal C{y} projection. 

 

@arg beta: Ellipsoidal latitude (C{radians} or L{Degrees}). 

 

@return: The C{y} projection (C{Radians}). 

''' 

return self.yR_(*_SinCos2(beta)) 

 

def yR_(self, sbeta, cbeta): 

'''Compute a Jacobi Conformal C{y} projection. 

 

@arg sbeta: Ellipsoidal latitude C{sin(beta)} (C{scalar}). 

@arg cbeta: Ellipsoidal latitude C{cos(beta)} (C{scalar}). 

 

@return: The C{y} projection (C{Radians}). 

''' 

s, c = self._norm2(sbeta, cbeta, self.c) 

return Radians(y=self._yE.fPi(s, c) * self._c2_b2) 

 

 

class TriaxialError(_ValueError): 

'''Raised for L{Triaxial} issues. 

''' 

pass # ... 

 

 

def _hartzell3d2(pov, los, abc3): # MCCABE 13 in .formy.hartzell 

'''(INTERNAL) Hartzell's "Satellite Lin-of-Sight Intersection ...", I{unordered}. 

''' 

def _order3d(ijk, v, **reverse): # reverse=False 

# @return: Vector3d(x, y, z) un-/ordered 

return Vector3d(*_order3(ijk, *v.xyz, **reverse)) if ijk else v 

 

a, b, c, ijk = _order4(*abc3) 

 

a2 = a**2 # largest, factored out 

b2, p2 = (b**2, (b / a)**2) if b != a else (a2, _1_0) 

c2, q2 = c**2, (c / a)**2 

 

p3 = _order3d(ijk, _otherV3d(pov=pov)) 

u3 = _order3d(ijk, _otherV3d(los=los)) if los else p3.negate() 

u3 = u3.unit() # unit vector, opposing signs 

 

x2, y2, z2 = p3.x2y2z2 # p3.times_(p3).xyz 

ux, vy, wz = u3.times_(p3).xyz 

u2, v2, w2 = u3.x2y2z2 # u3.times_(u3).xyz 

 

t = (p2 * c2), c2, b2 

m = fdot(t, u2, v2, w2) # a2 factored out 

if m < EPS0: # zero or near-null LOS vector 

raise _ValueError(_near_(_null_)) 

 

r = fsum_(b2 * w2, c2 * v2, -v2 * z2, vy * wz * 2, 

-w2 * y2, b2 * u2 * q2, -u2 * z2 * p2, ux * wz * 2 * p2, 

-w2 * x2 * p2, -u2 * y2 * q2, -v2 * x2 * q2, ux * vy * 2 * q2, floats=True) 

if r > 0: # a2 factored out 

r = sqrt(r) * b * c # == a * a * b * c / a2 

elif r < 0: # LOS pointing away from or missing the triaxial 

raise _ValueError(_opposite_ if max(ux, vy, wz) > 0 else _outside_) 

 

d = Fdot(t, ux, vy, wz).fadd_(r).fover(m) # -r for antipode, a2 factored out 

if d > 0: # POV inside or LOS missing, outside the triaxial 

raise _ValueError(_outside_ if max(x2 - a2, y2 - b2, z2 - c2) > EPS else _inside_) 

elif fsum_(x2, y2, z2, floats=True) < d**2: # d past triaxial's center 

raise _ValueError(_too_(_distant_)) 

 

v = p3.minus(u3.times(d)) # Vector3d 

h = p3.minus(v).length # distance to triaxial 

return _order3d(ijk, v, reverse=True), h 

 

 

E = _WGS84.ellipsoid 

_WGS84 = Triaxial(E.a + 35, E.a - 35, E.b, name=E.name + '+/-35') 

del E 

 

 

def hartzell4(pov, los=None, tri_biax=_WGS84, name=NN): 

'''Compute the intersection of a tri-/biaxial ellipsoid and a Line-Of-Sight 

from a Point-Of-View in space. 

 

@arg pov: Point-Of-View outside the tri-/biaxial (C{Cartesian}, L{Ecef9Tuple} 

or L{Vector3d}). 

@kwarg los: Line-Of-Sight, I{direction} to the tri-/biaxial (L{Vector3d}) or 

C{None} to point to the tri-/biaxial's center. 

@kwarg tri_biax: A triaxial (L{Triaxial}, L{Triaxial_}, L{JacobiConformal}) 

or biaxial ellipsoid (L{Datum}, L{Ellipsoid}, L{Ellipsoid2}, 

L{a_f2Tuple} or C{scalar} radius in C{meter}). 

@kwarg name: Optional name (C{str}). 

 

@return: L{Vector4Tuple}C{(x, y, z, h)} on the tri-/biaxial's surface, with C{h} 

the distance from B{C{pov}} to C{(x, y, z)} along B{C{los}}. 

 

@raise TriaxialError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}} is inside 

the tri-/biaxial or B{C{los}} points outside the 

tri-/biaxial or points in an opposite direction. 

 

@raise TypeError: Invalid B{C{pov}} or B{C{los}}. 

 

@see: Function L{pygeodesy.hartzell}, L{pygeodesy.tyr3d} for B{C{los}} and 

U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell. 

Medium.com/satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. 

''' 

if isinstance(tri_biax, Triaxial_): 

T = tri_biax 

else: 

D = tri_biax if isinstance(tri_biax, Datum) else \ 

_spherical_datum(tri_biax, name=hartzell4.__name__) 

E = D.ellipsoid 

T = Triaxial_(E.a, E.a, E.b, name=E.name) 

 

try: 

v, h = _hartzell3d2(pov, los, T._abc3) 

except Exception as x: 

raise TriaxialError(pov=pov, los=los, tri_biax=tri_biax, cause=x) 

return Vector4Tuple(v.x, v.y, v.z, h, name=name or hartzell4.__name__) 

 

 

def _normalTo4(x, y, a, b, eps=EPS): # MCCABE 14 

'''(INTERNAL) Nearest point on and distance to a 2-D ellipse, I{unordered}. 

 

@see: Function C{pygeodesy.ellipsoids._normalTo3} and I{Eberly}'s U{Distance 

from a Point to ... an Ellipsoid ...<https://www.GeometricTools.com/ 

Documentation/DistancePointEllipseEllipsoid.pdf>}. 

''' 

def _root2d(r, u, v, g, eps): 

# robust root finder 

_1, __2 = _1_0, _0_5 

_a, _h2 = fabs, hypot2 

u *= r 

t0 = v - _1 

t1 = _0_0 if g < 0 else (hypot(u, v) - _1) 

for i in range(1, _TRIPS): 

t = (t0 + t1) * __2 

if t in (t0, t1) or _a(t0 - t1) < eps: 

break 

g = _h2(u / (t + r), v / (t + _1)) - _1 

if g > 0: 

t0 = t 

elif g < 0: 

t1 = t 

else: 

break 

else: # PYCHOK no cover 

e = _a(t0 - t1) 

t = _root2d.__name__ 

raise _ValueError(Fmt.no_convergence(e, eps), txt=t) 

return t, i 

 

if a < b: 

b, a, d, i = _normalTo4(y, x, b, a, eps=eps) 

return a, b, d, i 

 

if not (isfinite(a) and b > 0): 

raise _ValueError(a=a, b=b) 

 

i = None 

if y: 

if x: 

u = fabs(x / a) 

v = fabs(y / b) 

g = hypot2(u, v) - _1_0 

if g: 

r = (a / b)**2 

t, i = _root2d(r, u, v, g, eps) 

a = x / (t / r + _1_0) 

b = y / (t + _1_0) 

d = hypot(x - a, y - b) 

else: # on the ellipse 

a, b, d = x, y, _0_0 

else: # x == 0 

if y < 0: 

b = -b 

a, d = x, fabs(y - b) 

 

else: # y == 0 

n = a * x 

d = (a + b) * (a - b) 

if d > fabs(n): # PYCHOK no cover 

r = n / d 

a *= r 

b *= sqrt(_1_0 - r**2) 

d = hypot(x - a, b) 

else: 

if x < 0: 

a = -a 

b, d = y, fabs(x - a) 

return a, b, d, i 

 

 

def _normalTo5(x, y, z, T, eps=EPS): # MCCABE 24 

'''(INTERNAL) Nearest point on and distance to an I{unordered} triaxial. 

 

@see: I{Eberly}'s U{Distance from a Point to ... an Ellipsoid ...<https:// 

www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. 

''' 

def _root3d(r, s, u, v, w, g, eps): 

# robust root finder 

_1, __2 = _1_0, _0_5 

_a, _h2 = fabs, hypot2_ 

u *= r 

v *= s 

t0 = w - _1 

t1 = _0_0 if g < 0 else (hypot_(u, v, w) - _1) 

for i in range(1, _TRIPS): 

t = (t0 + t1) * __2 

if t in (t0, t1) or _a(t0 - t1) < eps: 

break 

g = _h2(u / (t + r), v / (t + s), w / (t + _1)) - _1 

if g > 0: 

t0 = t 

elif g < 0: 

t1 = t 

else: 

break 

else: # PYCHOK no cover 

e = _a(t0 - t1) 

t = _root3d.__name__ 

raise _ValueError(Fmt.no_convergence(e, eps), txt=t) 

return t, i 

 

a, b, c, ijk = _order4(*T._abc3) 

if ijk: 

t = _order3(ijk, x, y, z) + (Triaxial_(a, b, c),) 

a, b, c, d, i = _normalTo5(*t, eps=eps) 

return _order3(ijk, a, b, c, reverse=True) + (d, i) 

 

if not (isfinite(a) and c > 0): 

raise _ValueError(a=a, b=b, c=c) 

 

if eps > 0: 

val = max(eps * 1e8, EPS) 

else: # no validation 

val, eps = 0, -eps 

 

i = None 

if z: 

if y: 

if x: 

u = fabs(x / a) 

v = fabs(y / b) 

w = fabs(z / c) 

g = hypot2_(u, v, w) - _1_0 

if g: 

r = T._1e2ac # (c / a)**2 

s = T._1e2bc # (c / b)**2 

t, i = _root3d(_1_0 / r, _1_0 / s, u, v, w, g, eps) 

a = x / (t * r + _1_0) 

b = y / (t * s + _1_0) 

c = z / (t + _1_0) 

d = hypot_(x - a, y - b, z - c) 

else: # on the ellipsoid 

a, b, c, d = x, y, z, _0_0 

else: # x == 0 

a = x # 0 

b, c, d, i = _normalTo4(y, z, b, c, eps=eps) 

elif x: # y == 0 

b = y # 0 

a, c, d, i = _normalTo4(x, z, a, c, eps=eps) 

else: # x == y == 0 

if z < 0: 

c = -c 

a, b, d = x, y, fabs(z - c) 

 

else: # z == 0 

t = False 

n = a * x 

d = T._a2c2 # (a + c) * (a - c) 

if d > fabs(n): 

u = n / d 

n = b * y 

d = T._b2c2 # (b + c) * (b - c) 

if d > fabs(n): 

v = n / d 

d = _1_0 - hypot2(u, v) 

if d > 0: 

a *= u 

b *= v 

c *= sqrt(d) 

d = hypot_(x - a, y - b, c) 

t = True 

if not t: 

c = z # 0 

a, b, d, i = _normalTo4(x, y, a, b, eps=eps) 

 

if val > 0: # validate 

e = T.sideOf(a, b, c, eps=val) 

if e: # not near the ellipsoid's surface 

raise _ValueError(a=a, b=b, c=c, d=d, 

sideOf=e, eps=val) 

if d: # angle of delta and normal vector 

m = Vector3d(x, y, z).minus_(a, b, c) 

if m.euclid > val: 

m = m.unit() 

n = T.normal3d(a, b, c) 

e = n.dot(m) # n.negate().dot(m) 

if not isnear1(fabs(e), eps1=val): 

raise _ValueError(n=n, m=m, 

dot=e, eps=val) 

return a, b, c, d, i 

 

 

def _order3(ijk, *abc, **reverse): # reverse=False 

'''(INTERNAL) Order or un-order C{a}, C{b} and C{c} by C{ijk}. 

 

@return: 3-Tuple C{(a, b, c)} ordered by C{ijk} or 

un-ordered by (reverse-ordered) C{ijk} if 

C{B{reverse}=True}. 

''' 

def _kwds(reverse=False): 

return reverse 

 

if ijk: 

if reverse and _kwds(**reverse): 

_, _, _, ijk = _order4(*ijk, **reverse) 

# abc = tuple(abc[i] for i in ijk) 

abc = tuple(map(abc.__getitem__, ijk)) 

return abc 

 

 

def _order4(a, b, c, reverse=False): 

'''(INTERNAL) Order or un-order C{a}, C{b} and C{c}. 

 

@return: 4-Tuple C{(a, b, c, ijk)} with C{a} >= C{b} >= C{c} 

and C{ijk} a 3-tuple with the original indices. 

''' 

i, j, k = 0, 1, 2 

if a < b: 

a, b, i, j = b, a, j, i 

if a < c: 

a, c, i, k = c, a, k, i 

if b < c: 

b, c, j, k = c, b, k, j 

return a, b, c, ((k, j, i) if reverse else (None if i < j < k else (i, j, k))) 

 

 

def _SinCos2(x): 

'''Get C{sin} and C{cos} of C{x} in C{Degrees}, C{Radians} or {radians}. 

''' 

return sincos2d(x) if isinstance(x, Degrees) else ( 

sincos2(x) if isinstance(x, Radians) else 

sincos2(float(x))) # assume C{radians} 

 

 

if __name__ == '__main__': 

 

from pygeodesy import printf 

 

# <https://ArxIV.org/pdf/1909.06452.pdf> Table 1 Semi-axes in km # Planet 

# <https://www.JPS.NASA.gov/education/images/pdf/ss-moons.pdf> 

# <https://link.Springer.com/article/10.1007/s00190-022-01650-9> 

for n, a, b, c in (('Amalthea', 125.0, 73.0, 64), # Jupiter 

('Ariel', 581.1, 577.9, 577.7), # Uranus 

('Earth', 6378.173435, 6378.1039, 6356.7544), 

('Enceladus', 256.6, 251.4, 248.3), # Saturn 

('Europa', 1564.13, 1561.23, 1560.93), # Jupiter 

('Io', 1829.4, 1819.3, 1815.7), # Jupiter 

('Mars', 3394.6, 3393.3, 3376.3), 

('Mimas', 207.4, 196.8, 190.6), # Saturn 

('Miranda', 240.4, 234.2, 232.9), # Uranus 

('Moon', 1735.55, 1735.324, 1734.898), # Earth 

('Tethys', 535.6, 528.2, 525.8)): # Saturn 

t = Triaxial(km2m(a), km2m(b), km2m(c), name=n) 

printf('# %r', t) 

if n == 'Earth': 

printf('# %r', JacobiConformal(t.a, t.b, t.c, name=n)) 

printf('# %r', _WGS84) 

 

# **) MIT License 

# 

# Copyright (C) 2022-2023 -- 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. 

 

# % python3 -m pygeodesy.triaxials 

 

# Triaxial(name='Amalthea', a=125000, b=73000, c=64000, e2ab=0.658944, e2bc=0.231375493, e2ac=0.737856, volume=2446253479595252, area=93239507787.490371704, area_p=93212299402.670425415) 

# Triaxial(name='Ariel', a=581100, b=577900, c=577700, e2ab=0.01098327, e2bc=0.000692042, e2ac=0.011667711, volume=812633172614203904, area=4211301462766.580078125, area_p=4211301574065.829589844) 

# Triaxial(name='Earth', a=6378173.435, b=6378103.9, c=6356754.399999999, e2ab=0.000021804, e2bc=0.006683418, e2ac=0.006705077, volume=1083208241574987694080, area=510065911057441.0625, area_p=510065915922713.6875) 

# JacobiConformal(name='Earth', a=6378173.435, b=6378103.9, c=6356754.399999999, e2ab=0.000021804, e2bc=0.006683418, e2ac=0.006705077, xyQ2=xyQ2(x=1.572084, y=4.249876), volume=1083208241574987694080, area=510065911057441.0625, area_p=510065915922713.6875) 

# Triaxial(name='Enceladus', a=256600, b=251400, c=248300, e2ab=0.040119337, e2bc=0.024509841, e2ac=0.06364586, volume=67094551514082248, area=798618496278.596679688, area_p=798619018175.109863281) 

# Triaxial(name='Europa', a=1564130, b=1561230, c=1560930, e2ab=0.003704694, e2bc=0.000384275, e2ac=0.004087546, volume=15966575194402123776, area=30663773697323.51953125, area_p=30663773794562.45703125) 

# Triaxial(name='Io', a=1829400, b=1819300, c=1815700, e2ab=0.011011391, e2bc=0.003953651, e2ac=0.014921506, volume=25313121117889765376, area=41691875849096.7421875, area_p=41691877397441.2109375) 

# Triaxial(name='Mars', a=3394600, b=3393300, c=3376300, e2ab=0.000765776, e2bc=0.009994646, e2ac=0.010752768, volume=162907283585817247744, area=144249140795107.4375, area_p=144249144150662.15625) 

# Triaxial(name='Mimas', a=207400, b=196800, c=190600, e2ab=0.09960581, e2bc=0.062015624, e2ac=0.155444317, volume=32587072869017956, area=493855762247.691894531, area_p=493857714107.9375) 

# Triaxial(name='Miranda', a=240400, b=234200, c=232900, e2ab=0.050915557, e2bc=0.011070811, e2ac=0.061422691, volume=54926187094835456, area=698880863325.756958008, area_p=698881306767.950317383) 

# Triaxial(name='Moon', a=1735550, b=1735324, c=1734898, e2ab=0.000260419, e2bc=0.000490914, e2ac=0.000751206, volume=21886698675223740416, area=37838824729886.09375, area_p=37838824733332.2265625) 

# Triaxial(name='Tethys', a=535600, b=528200, c=525800, e2ab=0.027441672, e2bc=0.009066821, e2ac=0.036259685, volume=623086233855821440, area=3528073490771.394042969, area_p=3528074261832.738769531) 

# Triaxial(name='WGS84+/-35', a=6378172, b=6378102, c=6356752.314245179, e2ab=0.00002195, e2bc=0.006683478, e2ac=0.006705281, volume=1083207319768789942272, area=510065621722018.125, area_p=510065626587483.3125)