# Coverage for pygeodesy/triaxials.py : 92%

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''' # make sure int/int division yields float quotient, see .basics from __future__ import division as _; del _ # PYCHOK semicolon # from pygeodesy.basics import isscalar # from .fsums from pygeodesy.constants import EPS, EPS0, EPS02, PI2, PI_3, PI4, _0_0, _0_5, \ _1_0, _3_0, _4_0, isfinite, isnear1 from pygeodesy.elliptic import Elliptic, Property_RO # from pygeodesy.errors import _ValueError # from .fsums from pygeodesy.fmath import fdot, Fmt, hypot, hypot_, hypot2, hypot2_, norm2 from pygeodesy.fsums import Fsum, fsum_, isscalar, _ValueError from pygeodesy.interns import NN, _distant_, _height_, _inside_, _near_, _not_, _null_, \ _opposite_, _outside_, _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, Vector3Tuple, Vector4Tuple # from pygeodesy.streprs import Fmt # from .fmath from pygeodesy.units import Degrees, Height_, Meter, Meter2, Meter3, Radians, Radius, Scalar 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__ = '22.10.24' _not_ordered_ = _not_('ordered') _TRIPS = 1074 # Eberly root finders class BetaOmega2Tuple(_NamedTuple): '''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 _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 _toRadians(self, *self) class BetaOmega3Tuple(_NamedTuple): '''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 _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 _toRadians(self, *self) def to2Tuple(self): '''Reduce this L{BetaOmega3Tuple} to a L{BetaOmega2Tuple}. ''' return BetaOmega2Tuple(*self[:2]) class JacobiError(_ValueError): '''Raised for L{JacobiConformal} issues. ''' pass # ... class Jacobi2Tuple(_NamedTuple): '''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 _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 _toRadians(self, *self) class TriaxialError(_ValueError): '''Raised for L{Triaxial} issues. ''' pass # ... class Triaxial(_NamedBase): # _NamedEnumItem '''Triaxial ellipsoid with 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, 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: Geodetic lat- and longitudes are in C{degrees}, I{ellipsoidal} lat- and longitude C{beta} and C{omega} are in C{Radians} by default (or C{Degrees}. ''' _Error = TriaxialError def __init__(self, a, b, c, name=NN): '''New L{Triaxial}. @arg a: The largest semi-axis (C{meter}, conventionally). @arg b: The middle semi-axis (C{meter}, same units as B{C{a}}). @arg c: The smallest semi-axis (C{meter}, same units as B{C{a}}). @kwarg name: Optional name (C{str}). @note: The semi-axes must be ordered as C{B{a} >= B{b} >= B{c} > 0} and be ellipsoidal, C{B{a} > B{c}}. @raise TriaxialError: Semi-axes not ordered, spherical or invalid. ''' if name: self.name = name E = self._Error self._a = a = Radius(a=a, Error=E) self._b = b = Radius(b=b, Error=E) self._c = c = Radius(c=c, Error=E) if not (isfinite(a) and a >= b >= c > 0): raise E(a=a, b=b, c=c, txt=_not_ordered_) self._a2b2 = (a - b) * (a + b) # == a**2 - b**2 == E_sub_e**2 self._b2c2 = (b - c) * (b + c) # == b**2 - c**2 == E_sub_y**2 self._a2c2 = (a - c) * (a + c) # == a**2 - c**2 == E_sub_x**2 if not (a > c and self._a2c2 > 0 and self.e2ac > 0): raise E(a=a, c=c, e2ac=self.e2ac, txt=_spherical_) self._a2_b2 = _1_0 if a == b else (a / b)**2 self._c2_b2 = _1_0 if c == b else (c / b)**2 def __str__(self): return self.toStr() @Property_RO def a(self): '''Get the largest semi-axis (C{meter}, conventionally). ''' return self._a @Property_RO def area(self): '''Get the surface area (C{meter} I{squared}). @see: U{Surface area<https://WikiPedia.org/wiki/Ellipsoid#Surface_area>}. ''' kp2, k2 = self._k2_kp2 # swapped! aE = 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) a = self.c**2 + (aE.fE(r) * s2 + aE.fF(r) * c2) / s * self.a * self.b return Meter2(area=PI2 * a) def area_p(self, p=1.6075): '''I{Approximate} the surface area (C{meter} I{squared}). @kwarg p: Exponent (C{scalar}), 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, _p = self.a, self.b, self.c, pow a = _p(fsum_(_p(a * b, p), _p(a * c, p), _p(b * c, p)) / _3_0, _1_0 / p) return Meter2(area_p=PI4 * a) @Property_RO def b(self): '''Get the middle semi-axis (C{meter}, same units as B{C{a}}). ''' return self._b @Property_RO def c(self): '''Get the smallest semi-axis (C{meter}, same units as B{C{a}}). ''' return self._c @Property_RO def e2ab(self): '''Get the C{ab} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (b / a)**2}. ''' return Scalar(e2ab=(_1_0 - self._1e2ab) if self.b != self.a else _0_0) @Property_RO def _1e2ab(self): '''(INTERNAL) Get C{1 - e2ab}. ''' return (_1_0 / self._a2_b2) if self.b != self.a 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 Scalar(e2bc=(_1_0 - self._1e2bc) if self.c != self.b else _0_0) @Property_RO def _1e2bc(self): '''(INTERNAL) Get C{1 - e2bc}. ''' return self._c2_b2 @Property_RO def e2ac(self): '''Get the C{ac} ellipse' I{(1st) eccentricity squared} (C{scalar}), M{1 - (c / a)**2}. ''' return Scalar(e2ac=(_1_0 - self._1e2ac) if self.c != self.a else _0_0) @Property_RO def _1e2ac(self): '''(INTERNAL) Get C{1 - e2ac}. ''' return (self.c / self.a)**2 if self.c != self.a else _1_0 @Property_RO def _Exycr4(self): '''(INTERNAL) Helper for C{.forwardBetOmg}. ''' return self._Exyur4(self.c) def _Exyur4(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, (self._a2b2 / self._a2c2) 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: U{Expressions (23-25)<https://www.Topo.Auth.GR/wp-content/ uploads/sites/111/2021/12/09_Panou.pdf>}. ''' sa, ca = _sincos2(beta) sb, cb = _sincos2(omega) if height: h = Height_(height=height, low=-self.c, Error=self._Error) x, y, z, r = self._Exyur4(h + self.c) else: x, y, z, r = self._Exycr4 if z: # and x and y 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 ellipsoid'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)}. @see: 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>}. ''' 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) x = self.a * cb * self._sqrt(x) y = self.b * ca * sb z = Fsum(c2_a2, sb**2, b2_a2 * cb**2).fover(a2c2_a2) z = self.c * sa * self._sqrt(z) return Vector3Tuple(x, y, z, name=name) 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: 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: 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_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_sb**2) N = self.a / self._sqrt(t) # prime vertical x = (h + N) * ca * cb y = (h + N * self._1e2ab) * ca_sb z = (h + N * self._1e2ac) * sa return Vector3Tuple(x, y, z, name=name) def hartzell(self, pov, los=None, name=NN): '''Compute the intersection of a Line-Of-Sight from a Point-Of-View in space with the surface of this triaxial. @arg pov: Point-Of-View outside this triaxial (C{Cartesian}, L{Ecef9Tuple} or L{Vector3d}). @kwarg los: Line-Of-Sight, I{direction} to this triaxial (L{Vector3d}) or C{None} to point to this triaxial's center. @kwarg name: Optional name (C{str}). @return: L{Vector4Tuple}C{(x, y, z, h)} on this triaxial's surface. @raise TriaxialError: Null B{C{pov}} or B{C{los}} vector, B{C{pov}} is inside this triaxial or B{C{los}} points outside this triaxial or points in an opposite direction. @raise TypeError: Invalid B{C{pov}} or B{C{los}}. @see: Function L{pygeodesy.tyr3d} for B{C{los}} and Hartzell, S. U{I{Satellite Line-of-Sight Intersection with Earth}<https://StephenHartzell.Medium.com/ satellite-line-of-sight-intersection-with-earth-d786b4a6a9b6>}. ''' def _Error(txt): return TriaxialError(pov=pov, los=los, triaxial=self, txt=txt) v, h = _hartzell3d2(pov, los, self.a, self.b, self.c, _Error) return Vector4Tuple(v.x, v.y, v.z, h, name=name or self.hartzell.__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}), needed if B{C{x_xyz}} if C{scalar}. @kwarg z: Z component (C{scalar}), needed if B{C{x_xyz}} if C{scalar}. @kwarg normal: If C{True} the projection is the nearest point on this triaxial's surface, otherwise the intersection of the radial line to the center and this triaxial's surface. @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}} or no convergence in root finding. @see: U{Eberly<https://www.GeometricTools.com/Documentation/DistancePointEllipseEllipsoid.pdf>}. ''' v = Vector3d(x_xyz, y, z) if isscalar(x_xyz) else \ _otherV3d(x_xyz=x_xyz) if normal: # perpendicular to triaxial x, y, z, h, i = _normalTo5(v.x, v.y, v.z, self, eps=eps) else: # radial to triaxial's center x, y, z = self.forwardBetaOmega_(v.z, hypot(v.x, v.y), v.y, v.x) h, i = v.minus_(x, y, z).length, None if h and self._sideOf(v.x, v.y, v.z, eps=0) < 0: # XXX eps=fabs(eps)? h = -h # below the surface return Vector4Tuple(x, y, z, h, iteration=i, name=self.height4.__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 _norm2(self, s, c, *a): '''(INTERNAL) Normalize C{s} and C{c} iff needed. ''' if fabs(s) > _1_0 or fabs(c) > _1_0: 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 _normal3d(self, x, y, z): '''(INTERNAL) Normal at C{(x, y, z)} on this triaxial's surface. ''' # n = 2 * (x / a2, y / b2, z / c2) # == 2 * (x, y * a2 / b2, z * a2 / c2) / a2 # == 2 * (x, y / _1e2ab, z / _1e2ac) / a2 # == unit(x, y / _1e2ab, z / _1e2ac) return Vector3d(x, y / self._1e2ab, z / self._1e2ac).unit() 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 the axes). @arg z: Z coordinate along C{c}-axis (C{meter}, same units as the axes). @kwarg name: Optional name (C{str}). @return: A L{BetaOmega3Tuple}C{(beta, omega, height)} with C{beta} and C{omega} in C{Radians} and C{height} in C{meter}, same units as this triaxial's axes. @see: 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 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 the axes). @arg z: Z coordinate along C{c}-axis (C{meter}, same units as the axes). @kwarg name: Optional name (C{str}). @return: A L{LatLon3Tuple}C{(lat, lon, height)} with C{lat} and C{lon} in C{degrees} and C{height} in C{meter}, same units as this triaxial's axes. @see: 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 def _sideOf(self, x, y, z, eps=EPS): '''(INTERNAL) Return C{e} = M{(x / a)**2 + (y / b)**2 + (z / c)**2 - 1} iff C{abs(e) > B{eps}} else C{0}. ''' e = hypot2_(x / self.a, y / self.b, z / self.c) - _1_0 return e if fabs(e) > eps else 0 def _sqrt(self, x): '''(INTERNAL) Helper. ''' if x < 0: raise self._Error(Fmt.PAREN(sqrt=x)) return sqrt(x) if x > EPS02 else _0_0 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 ellipsoid's name. @return: This C{Triaxial}'s attributes (C{str}). ''' T = self.__class__ try: t = (T.xyQ2.name,) except AttributeError: t = () return self._instr(name, prec, T.a.name, T.b.name, T.c.name, T.e2ab.name, T.e2bc.name, T.e2ac.name, T.area.name, T.volume.name, *t) @Property_RO def volume(self): '''Get the volume (C{meter**3}), M{4 / 3 * PI * a * b * c}. ''' return Meter3(volume=self.a * self.b * self.c * PI_3 * _4_0) class JacobiConformal(Triaxial): '''This is a conformal projection of the ellipsoid to a plane in which the grid lines are straight, see Jacobi, C. G. J. U{I{Vorlesungen über Dynamik} <https://Books.Google.com/books?id=ryEOAAAAQAAJ&pg=PA212>}, page 212ff. 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>} and U{Jacobi's conformal projection<https://GeographicLib.SourceForge.io/C++/doc/jacobi.html>}. ''' _Error = JacobiError # @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 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) 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 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) @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 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) def _hartzell3d2(pov, los, a, b, c, Error): '''(INTERNAL) Hartzell's "Satellite Lin-of-Sight Intersection ..." ''' a2 = a**2 if a == b: b2 = a2 p2 = _1_0 else: b2 = b**2 p2 = b2 / a2 c2 = c**2 q2 = c2 / a2 V3 = _MODS.vector3d._otherV3d p3 = V3(pov=pov) u3 = V3(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 Error(_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 Error(_opposite_ if max(ux, vy, wz) > 0 else _outside_) n = fdot(t, ux, vy, wz) # a2 factored out d = (n + r) / m # (n - r) / m for antipode if d > 0: # POV inside or LOS missing the triaxial raise Error(_inside_ if min(x2 - a2, y2 - b2, z2 - c2) < EPS else _outside_) elif fsum_(x2, y2, z2) < d**2: # d past triaxial's center raise Error(_too_(_distant_)) v = p3.minus(u3.times(d)) # Vector3d h = p3.minus(v).length return v, h def _normalTo4(x, y, a, b, eps=EPS): # MCCABE 13 '''(INTERNAL) Nearest point on and distance to a 2-D ellipse. @see: Function C{.ellipsoids._normalTo3} and U{Eberly<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 e = _a(t0 - t1) if t in (t0, t1) or e < 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 t = _root2d.__name__ raise TriaxialError(Fmt.no_convergence(e, eps), txt=t) return t, i if not (a >= b > 0): raise TriaxialError(a=a, b=b, eps=eps) 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): 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 23 '''(INTERNAL) Nearest point on and distance to a 3-D triaxial. @see: U{Eberly<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 e = _a(t0 - t1) if t in (t0, t1) or e < 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 t = _root3d.__name__ raise TriaxialError(Fmt.no_convergence(e, eps), txt=t) return t, i a, b, c = T.a, T.b, T.c if not (a >= b >= c > 0): raise TriaxialError(a=a, b=b, c=c, eps=eps) 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 = (a / c)**2 s = (b / c)**2 t, i = _root3d(r, 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 = (a + c) * (a - c) if d > fabs(n): u = n / d n = b * y d = (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 TriaxialError(a=a, b=b, c=c, d=d, x=x, y=y, z=z, eps=val, sideOf=e, txt=T.toRepr()) if d: # compare non-null 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 TriaxialError(n=n, m=m, eps=val, dot=e, txt=T.toRepr()) return a, b, c, d, i def _sincos2(x): '''Get C{sin} and C{cos} of C{x} in C{radians} or {degrees}. ''' return sincos2d(x) if isinstance(x, Degrees) else ( sincos2(x) if isinstance(x, Radians) else sincos2(float(x))) # assume C{radians} def _toDegrees(inst, a, b, *c, **toDMS_kwds): '''Helper for L{BetaOmega3Tuple} and L{Jacobi2Tuple} ''' 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 inst else: a, b = a.toDegrees(), b.toDegrees() return inst.classof(a, b, *c, name=inst.name) def _toRadians(inst, a, b, *c): '''Helper for L{BetaOmega3Tuple} and L{Jacobi2Tuple} ''' return inst if isinstance(a, Radians) and \ isinstance(b, Radians) else \ inst.classof(a.toRadians(), b.toRadians(), *c, name=inst.name) if __name__ == '__main__': from pygeodesy import printf, signBit # <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, area_p=%g', t, t.area_p()) t = Triaxial(3, 2, 1) # Eberly n = t.height4.__name__ s = signBit.__name__ for x in (-2, 0, 2): for y in (-4, 0, 4): for z in (-3, 0, 3): p = t.height4(x, y, z) printf('%s%s: %s, iteration %s', n, (x, y, z), p, p.iteration, nl=1) e = NN if signBit(p.x) == signBit(x) and \ signBit(p.y) == signBit(y) and \ signBit(p.z) == signBit(z) else '!= !!!' printf('%s%s: %s %s', s, (x, y, z), p, e) # **) 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, area=93239507787.490386963, volume=2446253479595252), area_p=9.32123e+10 # Triaxial(name='Ariel', a=581100, b=577900, c=577700, e2ab=0.01098327, e2bc=0.000692042, e2ac=0.011667711, area=4211301462766.580078125, volume=812633172614203904), area_p=4.2113e+12 # Triaxial(name='Earth', a=6378173.435, b=6378103.9, c=6356754.399999999, e2ab=0.000021804, e2bc=0.006683418, e2ac=0.006705077, area=510065911057441.0625, volume=1083208241574987694080), area_p=5.10066e+14 # Triaxial(name='Enceladus', a=256600, b=251400, c=248300, e2ab=0.040119337, e2bc=0.024509841, e2ac=0.06364586, area=798618496278.596557617, volume=67094551514082248), area_p=7.98619e+11 # Triaxial(name='Europa', a=1564130, b=1561230, c=1560930, e2ab=0.003704694, e2bc=0.000384275, e2ac=0.004087546, area=30663773697323.52734375, volume=15966575194402123776), area_p=3.06638e+13 # Triaxial(name='Io', a=1829400, b=1819300, c=1815700, e2ab=0.011011391, e2bc=0.003953651, e2ac=0.014921506, area=41691875849096.7421875, volume=25313121117889765376), area_p=4.16919e+13 # Triaxial(name='Mars', a=3394600, b=3393300, c=3376300, e2ab=0.000765776, e2bc=0.009994646, e2ac=0.010752768, area=144249140795107.4375, volume=162907283585817247744), area_p=1.44249e+14 # Triaxial(name='Mimas', a=207400, b=196800, c=190600, e2ab=0.09960581, e2bc=0.062015624, e2ac=0.155444317, area=493855762247.692016602, volume=32587072869017956), area_p=4.93858e+11 # Triaxial(name='Miranda', a=240400, b=234200, c=232900, e2ab=0.050915557, e2bc=0.011070811, e2ac=0.061422691, area=698880863325.756835938, volume=54926187094835456), area_p=6.98881e+11 # Triaxial(name='Moon', a=1735550, b=1735324, c=1734898, e2ab=0.000260419, e2bc=0.000490914, e2ac=0.000751206, area=37838824729886.09375, volume=21886698675223740416), area_p=3.78388e+13 # Triaxial(name='Tethys', a=535600, b=528200, c=525800, e2ab=0.027441672, e2bc=0.009066821, e2ac=0.036259685, area=3528073490771.394042969, volume=623086233855821440), area_p=3.52807e+12