Package pygeodesy :: Module ellipsoids :: Class Ellipsoid |
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# Class Ellipsoid

      object --+
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    named._Named --+
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    named._NamedBase --+
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    named._NamedEnumItem --+
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                          Ellipsoid

Known Subclasses:
  - Ellipsoid2

Ellipsoid with equatorial and polar radius, flattening, inverse flattening and other, often used, cached attributes, supporting spherical and oblate and prolate ellipsoidal models.

Instance Methods
__init__(self, a, b=None, f_=None, name=`'````'`) New Ellipsoid from equatorial and polar radius or equatorial radius and inverse flattening. |
__eq__(self, other) Compare this and an other ellipsoid. |
auxAuthalic(self, lat, inverse=False) Compute the authalic auxiliary latitude or inverse thereof. |
auxConformal(self, lat, inverse=False) Compute the conformal auxiliary latitude or inverse thereof. |
auxGeocentric(self, lat, inverse=False) Compute the geocentric auxiliary latitude or inverse thereof. |
auxIsometric(self, lat, inverse=False) Compute the isometric auxiliary latitude or inverse thereof. |
auxParametric(self, lat, inverse=False) Compute the parametric auxiliary latitude or inverse thereof. |
auxReduced(self, lat, inverse=False) Compute the parametric auxiliary latitude or inverse thereof. |
auxRectifying(self, lat, inverse=False) Compute the rectifying auxiliary latitude or inverse thereof. |
degrees2m(self, deg, lat=0) Convert an angle to the distance along the equator or along a parallel at the given latitude. |
distance2(self, lat0, lon0, lat1, lon1) Approximate the distance and (initial) bearing between two points based on the local, flat earth approximation aka Hubeny formula. |
ecef(self, Ecef=None) Return ECEF converter. |
e2s(self, s) Compute norm sqrt(1 - e2 * s**2). |
e2s2(self, s) Compute 1 - e2 * s**2. |
es_atanh(self, x) Compute es * atanh(es * x) where es is the signed (1st) eccentricity. |
es_tauf(self, taup) Compute Karney's equations (19), (20) and (21). |
es_taupf(self, tau) Compute Karney's equations (7), (8) and (9). |
_hubeny2_(self, phi2, phi1, lam21) (INTERNAL) like function flatLocal_/hubeny_ but returning the angular distance in `radians squared`. |
Llat(self, lat) Return the meridional length, the distance along a meridian between the equator and the given latitude (`Meter`). |
Lmeridian(self, lat) Return the meridional length, the distance along a meridian between the equator and the given latitude (`Meter`). |
m2degrees(self, meter, lat=0) Convert distance to angle along equator or along a parallel at an other latitude. |
Rgeocentric(self, lat) Compute the geocentric earth radius at the given latitude. |
Rlat(self, lat) Approximate the earth radius at the given latitude. |
roc2(self, lat, scaled=False) Compute the meridional and prime-vertical, normal radii of curvature at the given latitude. |
roc2_(self, phi, scaled=False) Compute the meridional and prime-vertical, normal radii of curvature at the given latitude. |
rocBearing(self, lat, bearing) Compute the directional radius of curvature at a given latitude and compass direction. |
rocGauss(self, lat) Compute the Gaussian radius of curvature at the given latitude. |
rocMean(self, lat) Compute the mean radius of curvature at the given latitude. |
rocMeridional(self, lat) Compute the meridional radius of curvature at the given latitude. |
rocPrimeVertical(self, lat) Compute the prime-vertical, normal radius of curvature at the given latitude, aka the transverse radius of curvature. |
rocTransverse(self, lat) Compute the prime-vertical, normal radius of curvature at the given latitude, aka the transverse radius of curvature. |
toStr(self, prec=8) Return this ellipsoid as a text string. |
Inherited from `named._NamedEnumItem`: `__ne__`, `unregister` Inherited from `named._NamedBase`: `__repr__`, `__str__`, `others`, `toRepr` Inherited from `named._Named`: `_dot_`, `attrs`, `classof`, `copy`, `toStr2` Inherited from `object`: `__delattr__`, `__format__`, `__getattribute__`, `__hash__`, `__new__`, `__reduce__`, `__reduce_ex__`, `__setattr__`, `__sizeof__`, `__subclasshook__`

Class Variables
| _f_ = `0` hash(x)
| _a2_ = `None` hash(x)

Properties
| a Get the equatorial (major) radius, semi-axis (`meter`).
| equatoradius Get the equatorial (major) radius, semi-axis (`meter`).
| a2 Get the equatorial (major) radius squared (`float`), a**2.
| a2_ Get the inverse of the equatorial (major) radius squared (`float`), 1 / a**2.
| a_b Get ratio a / b (`float`).
| a2_b Get the polar meridional radius of curvature (`meter`), a**2 / b, see `rocPolar`.
| a_f Get equatorial radius and flattening (a_f2Tuple).
| A Get the UTM meridional radius (`meter`).
| area Get the ellipsoid's surface area (`meter**2`), 4 * PI * R2**2 or 4 * PI * a**2.
| AlphaKs Get the Krüger Alpha series coefficients (`KsOrder`-tuple).
| b Get the polar (minor) radius, semi-axis (`meter`).
| polaradius Get the polar (minor) radius, semi-axis (`meter`).
| b_a Get ratio b / a (`float`), 1 - f.
| b2 Get the polar (minor) radius squared (`float`), b**2.
| b2_a Get the equatorial meridional radius of curvature (`meter`), b**2 / a, see `rocMeridional(0)`.
| b2_a2 Get ratio (b / a)**2 (`float`), (1 - f)**2, see `e12`.
| BetaKs Get the Krüger Beta series coefficients (`KsOrder`-tuple).
| c Get the authalic earth radius (`meter`), see `R2`.
| e Get the (1st) eccentricity (`float`), sqrt(1 - (b / a)**2)), see `a_b2e`.
| eccentricity Get the (1st) eccentricity (`float`), sqrt(1 - (b / a)**2)), see `a_b2e`.
| e2 Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`.
| eccentricity2 Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`.
| e22 Get the 2nd eccentricity squared (`float`), e2 / (1 - e2) == (a / b)**2 - 1, see `a_b2e22`.
| eccentricity2nd2 Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`.
| e32 Get the 3rd eccentricity squared (`float`), e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2), see `a_b2e32`.
| eccentricity3rd2 Get the 3rd eccentricity squared (`float`), e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2), see `a_b2e32`.
| e12 Get 1 - e**2 (`float`), (1 - f)**2, b**2 / a**2, see `b2_a2`.
| e4 Get the (1st) eccentricity to 4th power (`float`), e**4 == e2**2.
| es Get the (1st) eccentricity signed (`float`).
| es_c Get (1 - f) * exp(es_atanh(1)) (`float`), b_a * exp(es_atanh(1)).
| f Get the flattening (`float`), (a - b) / a, `0` for spherical.
| flattening Get the flattening (`float`), (a - b) / a, `0` for spherical.
| f_ Get the inverse flattening (`float`), 1 / f == a / (a - b), `0` for spherical, see `a_b2f_`.
| f2 Get the 2nd flattening (`float`), (a - b) / b == f / (1 - f), `0` for spherical, see `a_b2f2`.
| flattening2nd Get the 2nd flattening (`float`), (a - b) / b == f / (1 - f), `0` for spherical, see `a_b2f2`.
| geodesic Get this ellipsoid's wrapped Karney Geodesic, provided the geographiclib package is installed.
| isEllipsoidal Is this model ellipsoidal (`bool`)?
| isOblate Is this ellipsoid oblate (`bool`)? Prolate or spherical otherwise.
| isProlate Is this ellipsoid prolate (`bool`)? Oblate or spherical otherwise.
| isSpherical Is this model spherical (`bool`)?
| KsOrder Property to get and set the Krüger series' order (`int`).
| L Get the quarter meridian `L` (aka polar distance) the distance along a meridian between the equator and a pole (`meter`), b * Elliptic(-e2 / (1 - e2)).E or a * PI / 2.
| quarteradius DEPRECATED, use `L`.
| Mabcd Get the OSGR meridional coefficients (`4-Tuple`), `Airy130` only.
| majoradius DEPRECATED, use `a` or `Requatorial`.
| minoradius DEPRECATED, use `b` or `Rpolar`.
| n Get the 3rd flattening (`float`), f / (2 - f) == (a - b) / (a + b), see `a_b2n`.
| flattening3rd Get the 3rd flattening (`float`), f / (2 - f) == (a - b) / (a + b), see `a_b2n`.
| R1 Get the mean earth radius per IUGG (`meter`), (2 * a + b) / 3.
| Rmean Get the mean earth radius per IUGG (`meter`), (2 * a + b) / 3.
| R2 Get the authalic earth radius (`meter`), sqrt((a**2 + b**2 * atanh(e) / e) / 2).
| Rauthalic Get the authalic earth radius (`meter`), sqrt((a**2 + b**2 * atanh(e) / e) / 2).
| R3 Get the volumetric earth radius (`meter`), (a * a * b)**(1/3).
| Rvolumetric Get the volumetric earth radius (`meter`), (a * a * b)**(1/3).
| Rbiaxial Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2).
| Requatorial Get the equatorial (major) radius, semi-axis (`meter`).
| Rgeometric Get the geometric mean earth radius (`meter`), sqrt(a * b).
| Rs DEPRECATED, use `Rgeometric`.
| Rpolar Get the polar (minor) radius, semi-axis (`meter`).
| Rquadratic Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2).
| Rrectifying Get the rectifying earth radius (`meter`), ((a**(3/2) + b**(3/2)) / 2)**(2/3).
| Rr DEPRECATED, use `Rrectifying`.
| rocPolar Get the polar radius of curvature (`meter`), see `a2_b`.
| volume Get the ellipsoid's volume (`meter**3`), 4 / 3 * PI * a**2 * b.
Inherited from `named._NamedEnumItem`: `name` Inherited from `named._Named`: `classname`, `classnaming`, `named`, `named2`, `named3`, `named4` Inherited from `object`: `__class__`

Method Details

### __init__ (self, a, b=None, f_=None, name=`'````'`) (Constructor) |
New Ellipsoid from equatorial and polar radius or equatorial radius and inverse flattening. Arguments: `a` - Equatorial (major) radius, semi-axis (`meter`). `b` - Optional, polar (minor) radius, semi-axis (`meter`). `f_` - Inverse flattening: a / (a - b) (`float` >>> 1.0). `name` - Optional, unique name (`str`). Raises: `NameError` - Ellipsoid with that `name` already exists. `ValueError` - Invalid `a`, `b` or `f_`. Overrides: object.__init__ Note: abs(f_) > 1 / EPS or abs(1 / f_) < EPS is forced to 1 / f_ = 0, spherical.

### __eq__ (self, other) (Equality operator) |
Compare this and an other ellipsoid. Arguments: `other` - The other ellipsoid (Ellipsoid or Ellipsoid2). Returns: `True` if equal, `False` otherwise.

### auxAuthalic (self, lat, inverse=False) |
Compute the authalic auxiliary latitude or inverse thereof. Arguments: `lat` - The geodetic (or authalic) latitude (`degrees90`). `inverse` - If `True`, `lat` is the authalic and return the geodetic latitude (`bool`). Returns: The authalic (or geodetic) latitude in `degrees90`. See Also: Inverse-/AuthalicLatitude, Authalic latitude, and Snyder, p 16.

### auxConformal (self, lat, inverse=False) |
Compute the conformal auxiliary latitude or inverse thereof. Arguments: `lat` - The geodetic (or conformal) latitude (`degrees90`). `inverse` - If `True`, `lat` is the conformal and return the geodetic latitude (`bool`). Returns: The conformal (or geodetic) latitude in `degrees90`. See Also: Inverse-/ConformalLatitude, Conformal latitude, and Snyder, pp 15-16.

### auxGeocentric (self, lat, inverse=False) |
Compute the geocentric auxiliary latitude or inverse thereof. Arguments: `lat` - The geodetic (or geocentric) latitude (`degrees90`). `inverse` - If `True`, `lat` is the geocentric and return the geocentric latitude (`bool`). Returns: The geocentric (or geodetic) latitude in `degrees90`. See Also: Inverse-/GeocentricLatitude, Geocentric latitude, and Snyder<, pp 17-18.

### auxIsometric (self, lat, inverse=False) |
Compute the isometric auxiliary latitude or inverse thereof. Arguments: `lat` - The geodetic (or isometric) latitude (`degrees`). `inverse` - If `True`, `lat` is the isometric and return the geodetic latitude (`bool`). Returns: The isometric (or geodetic) latitude in `degrees`. Note: The isometric latitude for geodetic `+/-90` is far outside the `[-90..+90]` range but the inverse thereof is the original geodetic latitude. See Also: Inverse-/IsometricLatitude, Isometric latitude, and Snyder, pp 15-16.

### auxParametric (self, lat, inverse=False) |
Compute the parametric auxiliary latitude or inverse thereof. Arguments: `lat` - The geodetic (or parametric) latitude (`degrees90`). `inverse` - If `True`, `lat` is the parametric and return the geodetic latitude (`bool`). Returns: The parametric (or geodetic) latitude in `degrees90`. See Also: Inverse-/ParametricLatitude, Parametric latitude, and Snyder, p 18.

### auxReduced (self, lat, inverse=False) |
Compute the parametric auxiliary latitude or inverse thereof. Arguments: `lat` - The geodetic (or parametric) latitude (`degrees90`). `inverse` - If `True`, `lat` is the parametric and return the geodetic latitude (`bool`). Returns: The parametric (or geodetic) latitude in `degrees90`. See Also: Inverse-/ParametricLatitude, Parametric latitude, and Snyder, p 18.

### auxRectifying (self, lat, inverse=False) |
Compute the rectifying auxiliary latitude or inverse thereof. Arguments: `lat` - The geodetic (or rectifying) latitude (`degrees90`). `inverse` - If `True`, `lat` is the rectifying and return the geodetic latitude (`bool`). Returns: The rectifying (or geodetic) latitude in `degrees90`. See Also: Inverse-/RectifyingLatitude, Rectifying latitude, and Snyder, pp 16-17.

### degrees2m (self, deg, lat=0) |
Convert an angle to the distance along the equator or along a parallel at the given latitude. Arguments: `deg` - Angle (`degrees`). `lat` - Parallel latitude (`degrees90`, `str`). Returns: Angle (`degrees`). Raises: `RangeError` - Latitude `lat` outside valid range and rangerrors set to `True`. `ValueError` - Invalid `deg` or `lat`.

### distance2 (self, lat0, lon0, lat1, lon1) |
Approximate the distance and (initial) bearing between two points based on the local, flat earth approximation aka Hubeny formula. Suitable only for distances of several hundred Km or Miles and only between points not near-polar. Arguments: `lat0` - From latitude (`degrees`). `lon0` - From longitude (`degrees`). `lat1` - To latitude (`degrees`). `lon1` - To longitude (`degrees`). Returns: A Distance2Tuple`(distance, initial)` with `distance` in same units as this ellipsoid's axes. Note: The meridional and prime_vertical radii of curvature are taken and scaled at the initial latitude, see `roc2`. See Also: Function flatLocal/hubeny.

### ecef (self, Ecef=None) |
Return ECEF converter. Arguments: `Ecef` - ECEF class to use (EcefKarney, EcefVeness or EcefYou). Returns: An ECEF converter for this `ellipsoid` (EcefKarney, EcefVeness or EcefYou). Raises: `TypeError` - Invalid `Ecef`.

### e2s (self, s) |
Compute norm sqrt(1 - e2 * s**2). Arguments: `s` - S value (`scalar`). Returns: Norm (`float`). Raises: `ValueError` - Invalid `s`.

### e2s2 (self, s) |
Compute 1 - e2 * s**2. Arguments: `s` - S value (`scalar`). Returns: Result (`float`). Raises: `ValueError` - Invalid `s`.

### es_atanh (self, x) |
Compute es * atanh(es * x) where es is the signed (1st) eccentricity. Raises: `ValueError` - Invalid `x`. See Also: Function Math::eatanhe.

### es_tauf (self, taup) |
Compute Karney's equations (19), (20) and (21). See Also: Math::tauf.

### es_taupf (self, tau) |
Compute Karney's equations (7), (8) and (9). See Also: Math::taupf.

### Llat (self, lat) |
Return the meridional length, the distance along a meridian between the equator and the given latitude (`Meter`). Arguments: `lat` - Geodetic latitude (`degrees90`). Returns: The meridional length at `lat`, negative on southern hemisphere (`meter`).

### Lmeridian (self, lat) |
Return the meridional length, the distance along a meridian between the equator and the given latitude (`Meter`). Arguments: `lat` - Geodetic latitude (`degrees90`). Returns: The meridional length at `lat`, negative on southern hemisphere (`meter`).

### m2degrees (self, meter, lat=0) |
Convert distance to angle along equator or along a parallel at an other latitude. Arguments: `meter` - Distance (`meter`). `lat` - Parallel latitude (`degrees90`, `str`). Returns: Angle (`degrees`). Raises: `RangeError` - Latitude `lat` outside valid range and rangerrors set to `True`. `ValueError` - Invalid `meter` or `lat`.

### Rgeocentric (self, lat) |
Compute the geocentric earth radius at the given latitude. Arguments: `lat` - Latitude (`degrees90`). Returns: Geocentric earth radius (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Geocentric Radius

### Rlat (self, lat) |
Approximate the earth radius at the given latitude. Arguments: `lat` - Latitude (`degrees90`). Returns: Approximate earth radius (`meter`). Raises: `ValueError` - Invalid `lat`.

### roc2 (self, lat, scaled=False) |
Compute the meridional and prime-vertical, normal radii of curvature at the given latitude. Arguments: `lat` - Latitude (`degrees90`). `scaled` - Scale prime_vertical by `cos(radians(lat))` (`bool`). Returns: An Curvature2Tuple`(meridional, prime_vertical)` with the radii of curvature. Raises: `ValueError` - Invalid `lat`. See Also: Method `roc2_`, Local, flat earth approximation and Radii of Curvature.

### roc2_ (self, phi, scaled=False) |
Compute the meridional and prime-vertical, normal radii of curvature at the given latitude. Arguments: `phi` - Latitude (`radians`). `scaled` - Scale prime_vertical by `cos(phi)` (`bool`). Returns: An Curvature2Tuple`(meridional, prime_vertical)` with the radii of curvature. Raises: `ValueError` - Invalid `phi`. See Also: Method `roc2`, Local, flat earth approximation and meridional and prime vertical Radii of Curvature.

### rocBearing (self, lat, bearing) |
Compute the directional radius of curvature at a given latitude and compass direction. Arguments: `lat` - Latitude (`degrees90`). `bearing` - Direction (compass `degrees360`). Returns: Directional radius of curvature (`meter`). Raises: `RangeError` - Latitude `lat` outside valid range and rangerrors set to `True`. `ValueError` - Invalid `lat` or `bearing`. See Also: Radii of Curvature

### rocGauss (self, lat) |
Compute the Gaussian radius of curvature at the given latitude. Arguments: `lat` - Latitude (`degrees90`). Returns: Gaussian radius of curvature (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Radii of Curvature

### rocMean (self, lat) |
Compute the mean radius of curvature at the given latitude. Arguments: `lat` - Latitude (`degrees90`). Returns: Mean radius of curvature (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Radii of Curvature

### rocMeridional (self, lat) |
Compute the meridional radius of curvature at the given latitude. Arguments: `lat` - Latitude (`degrees90`). Returns: Meridional radius of curvature (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Local, flat earth approximation and Radii of Curvature.

### rocPrimeVertical (self, lat) |
Compute the prime-vertical, normal radius of curvature at the given latitude, aka the transverse radius of curvature. Arguments: `lat` - Latitude (`degrees90`). Returns: Prime-vertical radius of curvature (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Local, flat earth approximation and Radii of Curvature.

### rocTransverse (self, lat) |
Compute the prime-vertical, normal radius of curvature at the given latitude, aka the transverse radius of curvature. Arguments: `lat` - Latitude (`degrees90`). Returns: Prime-vertical radius of curvature (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Local, flat earth approximation and Radii of Curvature.

### toStr (self, prec=8) |
Return this ellipsoid as a text string. Arguments: `prec` - Optional number of decimals, unstripped (`int`). Returns: Ellipsoid attributes (`str`). Raises: `AssertionError` - Always, see function notOverloaded. Overrides: named._Named.toStr

Property Details

### a Get the equatorial (major) radius, semi-axis (`meter`). Get method: a(self) - Get the equatorial (major) radius, semi-axis (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### equatoradius Get the equatorial (major) radius, semi-axis (`meter`). Get method: a(self) - Get the equatorial (major) radius, semi-axis (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### a2 Get the equatorial (major) radius squared (`float`), a**2. Get method: a2(self) - Get the equatorial (major) radius squared (`float`), a**2. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### a2_ Get the inverse of the equatorial (major) radius squared (`float`), 1 / a**2. Get method: a2_(self) - Get the inverse of the equatorial (major) radius squared (`float`), 1 / a**2. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### a_b Get ratio a / b (`float`). Get method: a_b(self) - Get ratio a / b (`float`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### a2_b Get the polar meridional radius of curvature (`meter`), a**2 / b, see `rocPolar`. Get method: a2_b(self) - Get the polar meridional radius of curvature (`meter`), a**2 / b, see `rocPolar`. Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Radii of Curvature and Moritz, H. (1980), Geodetic Reference System 1980. Note: Symbol `c` is used by IUGG and IERS for the polar radius of curvature.

### a_f Get equatorial radius and flattening (a_f2Tuple). Get method: a_f(self) - Get equatorial radius and flattening (a_f2Tuple). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### A Get the UTM meridional radius (`meter`). Get method: A(self) - Get the UTM meridional radius (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### area Get the ellipsoid's surface area (`meter**2`), 4 * PI * R2**2 or 4 * PI * a**2. Get method: area(self) - Get the ellipsoid's surface area (`meter**2`), 4 * PI * R2**2 or 4 * PI * a**2. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### AlphaKs Get the Krüger Alpha series coefficients (`KsOrder`-tuple). Get method: AlphaKs(self) - Get the Krüger Alpha series coefficients (`KsOrder`-tuple). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### b Get the polar (minor) radius, semi-axis (`meter`). Get method: b(self) - Get the polar (minor) radius, semi-axis (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### polaradius Get the polar (minor) radius, semi-axis (`meter`). Get method: b(self) - Get the polar (minor) radius, semi-axis (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### b_a Get ratio b / a (`float`), 1 - f. Get method: b_a(self) - Get ratio b / a (`float`), 1 - f. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### b2 Get the polar (minor) radius squared (`float`), b**2. Get method: b2(self) - Get the polar (minor) radius squared (`float`), b**2. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### b2_a Get the equatorial meridional radius of curvature (`meter`), b**2 / a, see `rocMeridional(0)`. Get method: b2_a(self) - Get the equatorial meridional radius of curvature (`meter`), b**2 / a, see `rocMeridional(0)`. Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Radii of Curvature.

### b2_a2 Get ratio (b / a)**2 (`float`), (1 - f)**2, see `e12`. Get method: b2_a2(self) - Get ratio (b / a)**2 (`float`), (1 - f)**2, see `e12`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### BetaKs Get the Krüger Beta series coefficients (`KsOrder`-tuple). Get method: BetaKs(self) - Get the Krüger Beta series coefficients (`KsOrder`-tuple). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### c Get the authalic earth radius (`meter`), see `R2`. Get method: c(self) - Get the authalic earth radius (`meter`), see `R2`. Set method: immutable(inst, value) - Throws an `AttributeError`, always. Notes: Symbol `c` in equation 60. Symbol `c` is used by IUGG and IERS for the polar radius of curvature, see `rocPolar`.

### e Get the (1st) eccentricity (`float`), sqrt(1 - (b / a)**2)), see `a_b2e`. Get method: e(self) - Get the (1st) eccentricity (`float`), sqrt(1 - (b / a)**2)), see `a_b2e`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### eccentricity Get the (1st) eccentricity (`float`), sqrt(1 - (b / a)**2)), see `a_b2e`. Get method: e(self) - Get the (1st) eccentricity (`float`), sqrt(1 - (b / a)**2)), see `a_b2e`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### e2 Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`. Get method: e2(self) - Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### eccentricity2 Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`. Get method: e2(self) - Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### e22 Get the 2nd eccentricity squared (`float`), e2 / (1 - e2) == (a / b)**2 - 1, see `a_b2e22`. Get method: e22(self) - Get the 2nd eccentricity squared (`float`), e2 / (1 - e2) == (a / b)**2 - 1, see `a_b2e22`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### eccentricity2nd2 Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`. Get method: e2(self) - Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see `a_b2e2`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### e32 Get the 3rd eccentricity squared (`float`), e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2), see `a_b2e32`. Get method: e32(self) - Get the 3rd eccentricity squared (`float`), e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2), see `a_b2e32`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### eccentricity3rd2 Get the 3rd eccentricity squared (`float`), e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2), see `a_b2e32`. Get method: e32(self) - Get the 3rd eccentricity squared (`float`), e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2), see `a_b2e32`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### e12 Get 1 - e**2 (`float`), (1 - f)**2, b**2 / a**2, see `b2_a2`. Get method: e12(self) - Get 1 - e**2 (`float`), (1 - f)**2, b**2 / a**2, see `b2_a2`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### e4 Get the (1st) eccentricity to 4th power (`float`), e**4 == e2**2. Get method: e4(self) - Get the (1st) eccentricity to 4th power (`float`), e**4 == e2**2. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### es Get the (1st) eccentricity signed (`float`). Get method: es(self) - Get the (1st) eccentricity signed (`float`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### es_c Get (1 - f) * exp(es_atanh(1)) (`float`), b_a * exp(es_atanh(1)). Get method: es_c(self) - Get (1 - f) * exp(es_atanh(1)) (`float`), b_a * exp(es_atanh(1)). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### f Get the flattening (`float`), (a - b) / a, `0` for spherical. Get method: f(self) - Get the flattening (`float`), (a - b) / a, `0` for spherical. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### flattening Get the flattening (`float`), (a - b) / a, `0` for spherical. Get method: f(self) - Get the flattening (`float`), (a - b) / a, `0` for spherical. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### f_ Get the inverse flattening (`float`), 1 / f == a / (a - b), `0` for spherical, see `a_b2f_`. Get method: f_(self) - Get the inverse flattening (`float`), 1 / f == a / (a - b), `0` for spherical, see `a_b2f_`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### f2 Get the 2nd flattening (`float`), (a - b) / b == f / (1 - f), `0` for spherical, see `a_b2f2`. Get method: f2(self) - Get the 2nd flattening (`float`), (a - b) / b == f / (1 - f), `0` for spherical, see `a_b2f2`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### flattening2nd Get the 2nd flattening (`float`), (a - b) / b == f / (1 - f), `0` for spherical, see `a_b2f2`. Get method: f2(self) - Get the 2nd flattening (`float`), (a - b) / b == f / (1 - f), `0` for spherical, see `a_b2f2`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### geodesic Get this ellipsoid's wrapped Karney Geodesic, provided the geographiclib package is installed. Get method: geodesic(self) - Get this ellipsoid's wrapped Karney Geodesic, provided the geographiclib package is installed. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### isEllipsoidal Is this model ellipsoidal (`bool`)? Get method: isEllipsoidal(self) - Is this model ellipsoidal (`bool`)? Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### isOblate Is this ellipsoid oblate (`bool`)? Prolate or spherical otherwise. Get method: isOblate(self) - Is this ellipsoid oblate (`bool`)? Prolate or spherical otherwise. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### isProlate Is this ellipsoid prolate (`bool`)? Oblate or spherical otherwise. Get method: isProlate(self) - Is this ellipsoid prolate (`bool`)? Oblate or spherical otherwise. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### isSpherical Is this model spherical (`bool`)? Get method: isSpherical(self) - Is this model spherical (`bool`)? Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### KsOrder Property to get and set the Krüger series' order (`int`). Get method: KsOrder(self) - Get the Krüger series order (`int` 4, 6 or 8). Set method: KsOrder(self, order) - Set the Krüger series' order.

### L Get the quarter meridian `L` (aka polar distance) the distance along a meridian between the equator and a pole (`meter`), b * Elliptic(-e2 / (1 - e2)).E or a * PI / 2. Get method: L(self) - Get the quarter meridian `L` (aka polar distance) the distance along a meridian between the equator and a pole (`meter`), b * Elliptic(-e2 / (1 - e2)).E or a * PI / 2. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### quarteradius DEPRECATED, use `L`. Get method: L(self) - Get the quarter meridian `L` (aka polar distance) the distance along a meridian between the equator and a pole (`meter`), b * Elliptic(-e2 / (1 - e2)).E or a * PI / 2. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### Mabcd Get the OSGR meridional coefficients (`4-Tuple`), `Airy130` only. Get method: Mabcd(self) - Get the OSGR meridional coefficients (`4-Tuple`), `Airy130` only. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### majoradius DEPRECATED, use `a` or `Requatorial`. Get method: a(self) - Get the equatorial (major) radius, semi-axis (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### minoradius DEPRECATED, use `b` or `Rpolar`. Get method: b(self) - Get the polar (minor) radius, semi-axis (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### n Get the 3rd flattening (`float`), f / (2 - f) == (a - b) / (a + b), see `a_b2n`. Get method: n(self) - Get the 3rd flattening (`float`), f / (2 - f) == (a - b) / (a + b), see `a_b2n`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### flattening3rd Get the 3rd flattening (`float`), f / (2 - f) == (a - b) / (a + b), see `a_b2n`. Get method: n(self) - Get the 3rd flattening (`float`), f / (2 - f) == (a - b) / (a + b), see `a_b2n`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### R1 Get the mean earth radius per IUGG (`meter`), (2 * a + b) / 3. Get method: R1(self) - Get the mean earth radius per IUGG (`meter`), (2 * a + b) / 3. Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Earth radius and method `Rgeometric`.

### Rmean Get the mean earth radius per IUGG (`meter`), (2 * a + b) / 3. Get method: R1(self) - Get the mean earth radius per IUGG (`meter`), (2 * a + b) / 3. Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Earth radius and method `Rgeometric`.

### R2 Get the authalic earth radius (`meter`), sqrt((a**2 + b**2 * atanh(e) / e) / 2). Get method: R2(self) - Get the authalic earth radius (`meter`), sqrt((a**2 + b**2 * atanh(e) / e) / 2). Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Earth radius, `area` and c.

### Rauthalic Get the authalic earth radius (`meter`), sqrt((a**2 + b**2 * atanh(e) / e) / 2). Get method: R2(self) - Get the authalic earth radius (`meter`), sqrt((a**2 + b**2 * atanh(e) / e) / 2). Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Earth radius, `area` and c.

### R3 Get the volumetric earth radius (`meter`), (a * a * b)**(1/3). Get method: R3(self) - Get the volumetric earth radius (`meter`), (a * a * b)**(1/3). Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Earth radius.

### Rvolumetric Get the volumetric earth radius (`meter`), (a * a * b)**(1/3). Get method: R3(self) - Get the volumetric earth radius (`meter`), (a * a * b)**(1/3). Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Earth radius.

### Rbiaxial Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2). Get method: Rbiaxial(self) - Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### Requatorial Get the equatorial (major) radius, semi-axis (`meter`). Get method: a(self) - Get the equatorial (major) radius, semi-axis (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### Rgeometric Get the geometric mean earth radius (`meter`), sqrt(a * b). Get method: Rgeometric(self) - Get the geometric mean earth radius (`meter`), sqrt(a * b). Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Method `R1`.

### Rs DEPRECATED, use `Rgeometric`. Get method: Rgeometric(self) - Get the geometric mean earth radius (`meter`), sqrt(a * b). Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Method `R1`.

### Rpolar Get the polar (minor) radius, semi-axis (`meter`). Get method: b(self) - Get the polar (minor) radius, semi-axis (`meter`). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### Rquadratic Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2). Get method: Rbiaxial(self) - Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2). Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### Rrectifying Get the rectifying earth radius (`meter`), ((a**(3/2) + b**(3/2)) / 2)**(2/3). Get method: Rrectifying(self) - Get the rectifying earth radius (`meter`), ((a**(3/2) + b**(3/2)) / 2)**(2/3). Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Earth radius.

### Rr DEPRECATED, use `Rrectifying`. Get method: Rrectifying(self) - Get the rectifying earth radius (`meter`), ((a**(3/2) + b**(3/2)) / 2)**(2/3). Set method: immutable(inst, value) - Throws an `AttributeError`, always. See Also: Earth radius.

### rocPolar Get the polar radius of curvature (`meter`), see `a2_b`. Get method: rocPolar(self) - Get the polar radius of curvature (`meter`), see `a2_b`. Set method: immutable(inst, value) - Throws an `AttributeError`, always.

### volume Get the ellipsoid's volume (`meter**3`), 4 / 3 * PI * a**2 * b. Get method: volume(self) - Get the ellipsoid's volume (`meter**3`), 4 / 3 * PI * a**2 * b. Set method: immutable(inst, value) - Throws an `AttributeError`, always.