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. |
circle4(self, lat) Get the equatorial or a parallel circle of latitude. |
degrees2m(self, deg, lat=0) Convert an angle to the distance along the equator or along a parallel of (geodetic) 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 a (geodetic) latitude, see `L`. |
Lmeridian(self, lat) Return the meridional length, the distance along a meridian between the equator and a (geodetic) latitude, see `L`. |
m2degrees(self, distance, lat=0) Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. |
m2radians(self, distance, lat=0) Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. |
radians2m(self, rad, lat=0) Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. |
Rgeocentric(self, lat) Compute the geocentric earth radius of (geodetic) latitude. |
Rlat(self, lat) Approximate the earth radius of (geodetic) latitude. |
roc1_(self, sa, ca=None) Compute the prime-vertical, normal radius of curvature of (geodetic) latitude, unscaled. |
roc2(self, lat, scaled=False) Compute the meridional and prime-vertical, normal radii of curvature of (geodetic) latitude. |
roc2_(self, phi, scaled=False) Compute the meridional and prime-vertical, normal radii of curvature of (geodetic) latitude. |
rocBearing(self, lat, bearing) Compute the directional radius of curvature of (geodetic) latitude and compass direction. |
rocGauss(self, lat) Compute the Gaussian radius of curvature of (geodetic) latitude. |
rocMean(self, lat) Compute the mean radius of curvature of (geodetic) latitude. |
rocMeridional(self, lat) Compute the meridional radius of curvature of (geodetic) latitude. |
rocPrimeVertical(self, lat) Compute the prime-vertical, normal radius of curvature of (geodetic) latitude, aka the transverse radius of curvature. |
rocTransverse(self, lat) Compute the prime-vertical, normal radius of curvature of (geodetic) 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`, `rename`, `toStr2` Inherited from `object`: `__delattr__`, `__format__`, `__getattribute__`, `__hash__`, `__new__`, `__reduce__`, `__reduce_ex__`, `__setattr__`, `__sizeof__`, `__subclasshook__`

Class Variables
| _f_ = `0` hash(x)

Properties
| a Get the equatorial radius, semi-axis (`meter`).
| equatoradius Get the equatorial radius, semi-axis (`meter`).
| a2 Get the equatorial radius squared (`meter**2`), a**2.
| a2_ Get the inverse of the equatorial radius squared (`meter**2`), 1 / a**2.
| a_b Get the ratio equatorial over polar radius (`float`), a / b == 1 / (1 - f).
| a2_b Get the polar meridional radius of curvature (`meter`), a**2 / b.
| a2_b2 Get the ratio equatorial over polar radius squared (`float`), (a / b)**2 == 1 / (1 - e**2) == 1 / (1 - e2) == 1 / e12.
| a_f Get the equatorial radius and flattening (a_f2Tuple).
| A Get the UTM meridional (or rectifying) radius (`meter`).
| AlphaKs Get the Krüger Alpha series coefficients (KsOrder`-tuple`).
| area Get the ellipsoid's surface area (`meter**2`), 4 * PI * c2.
| areax Get the ellipsoid's surface area (`meter**2`), 4 * PI * c2x, more accurate for very oblate ellipsoids.
| b Get the polar radius, semi-axis (`meter`).
| polaradius Get the polar radius, semi-axis (`meter`).
| b_a Get the ratio polar over equatorial radius (`float`), b / a == 1 - f.
| b2 Get the polar radius squared (`float`), b**2.
| b2_a Get the equatorial meridional radius of curvature (`meter`), b**2 / a, see rocMeridional`(0)`.
| b2_a2 Get the ratio polar over equatorial radius squared (`float`), (b / a)**2 == (1 - f)**2 == 1 - e**2 == e12.
| BetaKs Get the Krüger Beta series coefficients (KsOrder`-tuple`).
| c DEPRECATED, use property R2 or Rauthalic.
| c2 Get the authalic earth radius squared (`meter**2`).
| c2x Get the authalic earth radius squared (`meter**2`), more accurate for very oblate ellipsoids.
| 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.
| e2abs Get `abs` value of the (1st) eccentricity squared (`float`).
| e12 Get 1 less (1st) eccentricity squared (`float`), 1 - e**2 == (1 - f)**2 == b**2 / a**2, see b2_a2.
| e22 Get the 2nd eccentricity squared (`float`), e2 / (1 - e2) == e2 / (1 - f)**2 == (a / b)**2 - 1, see a_b2e22.
| eccentricity2nd2 Get the 2nd eccentricity squared (`float`), e2 / (1 - e2) == e2 / (1 - f)**2 == (a / b)**2 - 1, see a_b2e22.
| e22abs Get `abs` value of the (2nd) eccentricity squared (`float`).
| 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.
| e32abs Get `abs` value of the (3rd) eccentricity squared (`float`).
| 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, negative for prolate.
| flattening Get the flattening (`float`), (a - b) / a, `0` for spherical, negative for prolate.
| 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.
| geodesicx Get this ellipsoid's GeodesicExact.
| geodsolve Get this ellipsoid's wrapped utility GeodSolve, provided the path to the GeodSolve executable is specified with env variable `PYGEODESY_GEODSOLVE`.
| 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`), see properties `AlphaKs`, `BetaKs`.
| 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.
| Mabcd Get the OSGR meridional coefficients (`4-Tuple`), `Airy130` only.
| majoradius DEPRECATED, use property a or Requatorial.
| minoradius DEPRECATED, use property 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.
| quarteradius DEPRECATED, use property L or Llat.
| 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(c2).
| Rauthalic Get the authalic earth radius (`meter`), sqrt(c2).
| R2x Get the authalic earth radius (`meter`), sqrt(c2x).
| Rauthalicx Get the authalic earth radius (`meter`), sqrt(c2x).
| 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 radius, semi-axis (`meter`).
| Rgeometric Get the geometric mean earth radius (`meter`), sqrt(a * b).
| Rpolar Get the polar radius, semi-axis (`meter`).
| Rquadratic Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2).
| Rr DEPRECATED, use property Rrectifying.
| Rrectifying Get the rectifying earth radius (`meter`), ((a**(3/2) + b**(3/2)) / 2)**(2/3).
| Rs DEPRECATED, use property Rgeometric.
| rocPolar Get the polar meridional radius of curvature (`meter`), a**2 / b.
| volume Get the ellipsoid's volume (`meter**3`), 4 / 3 * PI * R3**3.
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 radius, semi-axis (`meter`). `b` - Optional, polar 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.

### circle4 (self, lat) |
Get the equatorial or a parallel circle of latitude. Arguments: `lat` - Geodetic latitude (`degrees90`, `str`). Returns: A Circle4Tuple`(radius, height, lat, beta)` instance. Raises: `RangeError` - Latitude `lat` outside valid range and rangerrors set to `True`. `TypeError` - Invalid `lat`. `ValueError` - Invalid `lat`. See Also: Definition of p and z under Parametric (or reduced) latitude and Karney's C++ CircleRadius and CircleHeight.

### degrees2m (self, deg, lat=0) |
Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. Arguments: `deg` - The angle (`degrees`). `lat` - Parallel latitude (`degrees90`, `str`). Returns: Distance (`meter`, same units as the equatorial and polar radii) or `0` for near-polar `lat`. 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` - Sine 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 a (geodetic) latitude, see `L`. 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 a (geodetic) latitude, see `L`. Arguments: `lat` - Geodetic latitude (`degrees90`). Returns: The meridional length at `lat`, negative on southern hemisphere (`meter`).

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

### m2radians (self, distance, lat=0) |
Convert a distance to an angle along the equator or along a parallel of (geodetic) latitude. Arguments: `distance` - Distance (`meter`). `lat` - Parallel latitude (`degrees90`, `str`). Returns: Angle (`radians`) or `INF` for near-polar `lat`. Raises: `RangeError` - Latitude `lat` outside valid range and rangerrors set to `True`. `ValueError` - Invalid `distance` or `lat`.

### radians2m (self, rad, lat=0) |
Convert an angle to the distance along the equator or along a parallel of (geodetic) latitude. Arguments: `rad` - The angle (`radians`). `lat` - Parallel latitude (`degrees90`, `str`). Returns: Distance (`meter`, same units as the equatorial and polar radii) or `0` for near-polar `lat`. Raises: `RangeError` - Latitude `lat` outside valid range and rangerrors set to `True`. `ValueError` - Invalid `rad` or `lat`.

### Rgeocentric (self, lat) |
Compute the geocentric earth radius of (geodetic) 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 of (geodetic) latitude. Arguments: `lat` - Latitude (`degrees90`). Returns: Approximate earth radius (`meter`). Raises: `RangeError` - Latitude `lat` outside valid range and rangerrors set to `True`. `TypeError` - Invalid `lat`. `ValueError` - Invalid `lat`. Note: `Rlat(90)` equals `Rpolar`. See Also: Method `Rparallel`.

### roc1_ (self, sa, ca=None) |
Compute the prime-vertical, normal radius of curvature of (geodetic) latitude, unscaled. Arguments: `sa` - Sine of the latitude (`float`, [-1.0..+1.0]). `ca` - Optional cosine of the latitude (`float`, [-1.0..+1.0]) to use an alternate formula. Returns: The prime-vertical radius of curvature (`float`). Note: The delta between both formulae with `Ellipsoids.WGS84` is less than 2 nanometer over the entire latitude range. See Also: Method roc2_ and class EcefYou.

### roc2 (self, lat, scaled=False) |
Compute the meridional and prime-vertical, normal radii of curvature of (geodetic) 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: Methods roc2_ and roc1_ and Local, flat earth approximation and meridional and prime vertical Radii of Curvature.

### roc2_ (self, phi, scaled=False) |
Compute the meridional and prime-vertical, normal radii of curvature of (geodetic) 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: Methods roc2 and roc1_ and Local, flat earth approximation and the meridional and prime vertical Radii of Curvature.

### rocBearing (self, lat, bearing) |
Compute the directional radius of curvature of (geodetic) 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 of (geodetic) 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 of (geodetic) 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 of (geodetic) latitude. Arguments: `lat` - Latitude (`degrees90`). Returns: Meridional radius of curvature (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Methods roc2 and roc2_ and Local, flat earth approximation and Radii of Curvature.

### rocPrimeVertical (self, lat) |
Compute the prime-vertical, normal radius of curvature of (geodetic) latitude, aka the transverse radius of curvature. Arguments: `lat` - Latitude (`degrees90`). Returns: Prime-vertical radius of curvature (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Methods roc2, roc2_ and roc1_ and Local, flat earth approximation and Radii of Curvature.

### rocTransverse (self, lat) |
Compute the prime-vertical, normal radius of curvature of (geodetic) latitude, aka the transverse radius of curvature. Arguments: `lat` - Latitude (`degrees90`). Returns: Prime-vertical radius of curvature (`meter`). Raises: `ValueError` - Invalid `lat`. See Also: Methods roc2, roc2_ and roc1_ and 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`). Overrides: named._Named.toStr

Property Details

### a Get the equatorial radius, semi-axis (`meter`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### equatoradius Get the equatorial radius, semi-axis (`meter`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### a2 Get the equatorial radius squared (`meter**2`), a**2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### a2_ Get the inverse of the equatorial radius squared (`meter**2`), 1 / a**2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### a_b Get the ratio equatorial over polar radius (`float`), a / b == 1 / (1 - f). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### a2_b Get the polar meridional radius of curvature (`meter`), a**2 / b. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) 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, see c2 and R2 or Rauthalic.

### a2_b2 Get the ratio equatorial over polar radius squared (`float`), (a / b)**2 == 1 / (1 - e**2) == 1 / (1 - e2) == 1 / e12. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### a_f Get the equatorial radius and flattening (a_f2Tuple). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### A Get the UTM meridional (or rectifying) radius (`meter`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### AlphaKs Get the Krüger Alpha series coefficients (KsOrder`-tuple`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### area Get the ellipsoid's surface area (`meter**2`), 4 * PI * c2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: areax, c2 and R2.

### areax Get the ellipsoid's surface area (`meter**2`), 4 * PI * c2x, more accurate for very oblate ellipsoids. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: area, c2x, R2x and GeodesicExact.

### b Get the polar radius, semi-axis (`meter`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### polaradius Get the polar radius, semi-axis (`meter`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### b_a Get the ratio polar over equatorial radius (`float`), b / a == 1 - f. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### b2 Get the polar radius squared (`float`), b**2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

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

### b2_a2 Get the ratio polar over equatorial radius squared (`float`), (b / a)**2 == (1 - f)**2 == 1 - e**2 == e12. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### BetaKs Get the Krüger Beta series coefficients (KsOrder`-tuple`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### c DEPRECATED, use property R2 or Rauthalic. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### c2 Get the authalic earth radius squared (`meter**2`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: c2x, area, R2, Rauthalic, Karney's equation 60 and C++ Ellipsoid.Area(), Authalic radius, Surface area and surface area.

### c2x Get the authalic earth radius squared (`meter**2`), more accurate for very oblate ellipsoids. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: c2, areax, R2x, Rauthalicx, GeodesicExact and Karney's comments at C++ attribute GeodesicExact._c2.

### e Get the (1st) eccentricity (`float`), sqrt(1 - (b / a)**2)), see a_b2e. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### eccentricity Get the (1st) eccentricity (`float`), sqrt(1 - (b / a)**2)), see a_b2e. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### e2 Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see a_b2e2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### eccentricity2 Get the (1st) eccentricity squared (`float`), f * (2 - f) == 1 - (b / a)**2, see a_b2e2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### e2abs Get `abs` value of the (1st) eccentricity squared (`float`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### e12 Get 1 less (1st) eccentricity squared (`float`), 1 - e**2 == (1 - f)**2 == b**2 / a**2, see b2_a2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### e22 Get the 2nd eccentricity squared (`float`), e2 / (1 - e2) == e2 / (1 - f)**2 == (a / b)**2 - 1, see a_b2e22. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### eccentricity2nd2 Get the 2nd eccentricity squared (`float`), e2 / (1 - e2) == e2 / (1 - f)**2 == (a / b)**2 - 1, see a_b2e22. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### e22abs Get `abs` value of the (2nd) eccentricity squared (`float`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### e32 Get the 3rd eccentricity squared (`float`), e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2), see a_b2e32. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### eccentricity3rd2 Get the 3rd eccentricity squared (`float`), e2 / (2 - e2) == (a**2 - b**2) / (a**2 + b**2), see a_b2e32. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### e32abs Get `abs` value of the (3rd) eccentricity squared (`float`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### e4 Get the (1st) eccentricity to 4th power (`float`), e**4 == e2**2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### es Get the (1st) eccentricity signed (`float`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### es_c Get (1 - f) * exp(es_atanh(1)) (`float`), b_a * exp(es_atanh(1)). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### f Get the flattening (`float`), (a - b) / a, `0` for spherical, negative for prolate. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### flattening Get the flattening (`float`), (a - b) / a, `0` for spherical, negative for prolate. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### f_ Get the inverse flattening (`float`), 1 / f == a / (a - b), `0` for spherical, see a_b2f_. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### f2 Get the 2nd flattening (`float`), (a - b) / b == f / (1 - f), `0` for spherical, see a_b2f2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### flattening2nd Get the 2nd flattening (`float`), (a - b) / b == f / (1 - f), `0` for spherical, see a_b2f2. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### geodesic Get this ellipsoid's wrapped Karney Geodesic, provided the geographiclib package is installed. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### geodesicx Get this ellipsoid's GeodesicExact. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### geodsolve Get this ellipsoid's wrapped utility GeodSolve, provided the path to the GeodSolve executable is specified with env variable `PYGEODESY_GEODSOLVE`. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### isEllipsoidal Is this model ellipsoidal (`bool`)? Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### isOblate Is this ellipsoid oblate (`bool`)? Prolate or spherical otherwise. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### isProlate Is this ellipsoid prolate (`bool`)? Oblate or spherical otherwise. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### isSpherical Is this model spherical (`bool`)? Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### KsOrder Property to get and set the Krüger series' order (`int`), see properties `AlphaKs`, `BetaKs`. 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: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### Mabcd Get the OSGR meridional coefficients (`4-Tuple`), `Airy130` only. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### majoradius DEPRECATED, use property a or Requatorial. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### minoradius DEPRECATED, use property b or `Rpolar`. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### n Get the 3rd flattening (`float`), f / (2 - f) == (a - b) / (a + b), see a_b2n. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### flattening3rd Get the 3rd flattening (`float`), f / (2 - f) == (a - b) / (a + b), see a_b2n. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### quarteradius DEPRECATED, use property L or Llat. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

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

### Rmean Get the mean earth radius per IUGG (`meter`), (2 * a + b) / 3. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: Earth radius and method `Rgeometric`.

### R2 Get the authalic earth radius (`meter`), sqrt(c2). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: R2x, c2, area and Earth radius.

### Rauthalic Get the authalic earth radius (`meter`), sqrt(c2). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: R2x, c2, area and Earth radius.

### R2x Get the authalic earth radius (`meter`), sqrt(c2x). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: R2, c2x and areax.

### Rauthalicx Get the authalic earth radius (`meter`), sqrt(c2x). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: R2, c2x and areax.

### R3 Get the volumetric earth radius (`meter`), (a * a * b)**(1/3). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: Earth radius and volume.

### Rvolumetric Get the volumetric earth radius (`meter`), (a * a * b)**(1/3). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: Earth radius and volume.

### Rbiaxial Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### Requatorial Get the equatorial radius, semi-axis (`meter`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### Rgeometric Get the geometric mean earth radius (`meter`), sqrt(a * b). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: R1.

### Rpolar Get the polar radius, semi-axis (`meter`). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### Rquadratic Get the biaxial, quadratic mean earth radius (`meter`), sqrt((a**2 + b**2) / 2). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### Rr DEPRECATED, use property Rrectifying. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### Rrectifying Get the rectifying earth radius (`meter`), ((a**(3/2) + b**(3/2)) / 2)**(2/3). Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: Earth radius.

### Rs DEPRECATED, use property Rgeometric. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst)

### rocPolar Get the polar meridional radius of curvature (`meter`), a**2 / b. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) 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, see c2 and R2 or Rauthalic.

### volume Get the ellipsoid's volume (`meter**3`), 4 / 3 * PI * R3**3. Get method: _fget(inst) Set method: _fset_error(inst, val) - Throws an `AttributeError`, always. Delete Method: _fdel(inst) See Also: R3.