Package pygeodesy :: Module datum :: 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

Ellipsoid with major and minor radius, semi-axis, (inverse) flattening and other pre-computed, frequently used attributes.

Instance Methods
 
__init__(self, a, b, f_, name='')
New Ellipsoid.
 
__eq__(self, other)
Compare this and an other ellipsoid.
 
degrees2m(self, deg, lat=0)
Convert angle to distance along the equator or along a parallel at an other 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.
 
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 the 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.
 
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
Inverse flattening: a / (a - b) = 1 / f
  _a2_ = None
(1 / a**2) # for ellipsiodalNvector.Cartesian.toNvector
Properties
  a
Get the major (equatorial) radius, semi-axis (meter).
  a_b
Get ratio a / b (float).
  a2
Get the major radius squared (float), a**2.
  a2_
Get the inverse of the major radius squared (float), 1 / a**2.
  a2_b
Get the polar meridional radius of curvature (meter), a**2 / b, see rocPolar.
  area
Get the ellipsoid's surface area (meter**2), 4 * PI * R2**2.
  A
Get the UTM meridional radius (meter).
  AlphaKs
Get the Krüger Alpha series coefficients (KsOrder-tuple).
  b
Get the minor (polar) radius, semi-axis (meter).
  b_a
Get ratio b / a (float), 1 - f.
  b2
Get the minor radius squared (float), b**2.
  b2_a
Get the equatorial meridional radius of curvature (meter), b**2 / a.
  b2_a2
Get ratio b / a) squared (float), (1 - f)**2.
  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)).
  e12
Get 1 - e**2 (float).
  e2
Get the (1st) Eccentricity squared (float), f * (2 - f) == 1 - (b / a)**2.
  e22
Get the 2nd Eccentricity squared (float), e2 / (1 - e2) == (a / b)**2 - 1.
  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.
  f_
Get the Inverse flattening (float), 1 / f or a / (a - b).
  f2
Get the 2nd Flattening (float), (a - b) / b.
  geodesic
Get this ellipsoid's wrapped Karney Geodesic, provided the geographiclib package is installed.
  isEllipsoidal
Check whether this model is ellipsoidal (bool).
  isSpherical
Check whether this model is spherical (bool).
  KsOrder
Property to get and set the Krüger series' order (int).
  Mabcd
Get the OSGR meridional coefficients (4-Tuple), Airy130 only.
  n
Get the 3rd Flattening (float), f / (2 - f) == (a - b) / (a + b).
  R1
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).
  R3
Get the volumetric earth radius (meter), (a * a * b)**(1/3).
  Rr
Get the rectifying earth radius (meter), ((a**(3/2) + b**(3/2)) / 2)**(2/3).
  Rs
Get another mean earth radius (meter), sqrt(a * b).
  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

Inherited from object: __class__

Method Details

__init__ (self, a, b, f_, name='')
(Constructor)

 

New Ellipsoid.

Arguments:
  • a - Semi-major, equatorial axis (meter).
  • b - Semi-minor, polar 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__

__eq__ (self, other)
(Equality operator)

 

Compare this and an other ellipsoid.

Arguments:
Returns:
True if equal, False otherwise.

degrees2m (self, deg, lat=0)

 

Convert angle to distance along the equator or along a parallel at an other 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 Also: Function flatLocal/hubeny.

ecef (self, Ecef=None)

 

Return ECEF converter.

Arguments:
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.

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 the 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.

rocPrimeVertical (self, lat)

 

Compute the prime-vertical, normal radius of curvature at the given latitude.

Arguments:
  • lat - Latitude (degrees90).
Returns:
Prime-vertical radius of curvature (meter).
Raises:
  • ValueError - Invalid lat.

toStr (self, prec=8)

 

Return this ellipsoid as a text string.

Arguments:
  • prec - Optional number of decimals, unstripped (int).
Returns:
Ellipsoid attributes (str).
Raises:
Overrides: named._Named.toStr

Property Details

a

Get the major (equatorial) radius, semi-axis (meter).

Get method:
a(self) - Get the major (equatorial) radius, semi-axis (meter).
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

Get the major radius squared (float), a**2.

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

a2_

Get the inverse of the major radius squared (float), 1 / a**2.

Get method:
a2_(self) - Get the inverse of the major radius squared (float), 1 / a**2.
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.

area

Get the ellipsoid's surface area (meter**2), 4 * PI * R2**2.

Get method:
area(self) - Get the ellipsoid's surface area (meter**2), 4 * PI * R2**2.
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.

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 minor (polar) radius, semi-axis (meter).

Get method:
b(self) - Get the minor (polar) 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 minor radius squared (float), b**2.

Get method:
b2(self) - Get the 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.

Get method:
b2_a(self) - Get the equatorial meridional radius of curvature (meter), b**2 / a.
Set method:
immutable(inst, value) - Throws an AttributeError, always.

See Also: Radii of Curvature.

b2_a2

Get ratio b / a) squared (float), (1 - f)**2.

Get method:
b2_a2(self) - Get ratio b / a) squared (float), (1 - f)**2.
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:

e

Get the (1st) Eccentricity (float), sqrt(1 - (b / a)**2)).

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

e12

Get 1 - e**2 (float).

Get method:
e12(self) - Get 1 - e**2 (float).
Set method:
immutable(inst, value) - Throws an AttributeError, always.

e2

Get the (1st) Eccentricity squared (float), f * (2 - f) == 1 - (b / a)**2.

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

e22

Get the 2nd Eccentricity squared (float), e2 / (1 - e2) == (a / b)**2 - 1.

Get method:
e22(self) - Get the 2nd Eccentricity squared (float), e2 / (1 - e2) == (a / b)**2 - 1.
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.

Get method:
f(self) - Get the Flattening (float), (a - b) / a.
Set method:
immutable(inst, value) - Throws an AttributeError, always.

f_

Get the Inverse flattening (float), 1 / f or a / (a - b).

Get method:
f_(self) - Get the Inverse flattening (float), 1 / f or a / (a - b).
Set method:
immutable(inst, value) - Throws an AttributeError, always.

f2

Get the 2nd Flattening (float), (a - b) / b.

Get method:
f2(self) - Get the 2nd Flattening (float), (a - b) / b.
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

Check whether this model is ellipsoidal (bool).

Get method:
isEllipsoidal(self) - Check whether this model is ellipsoidal (bool).
Set method:
immutable(inst, value) - Throws an AttributeError, always.

isSpherical

Check whether this model is spherical (bool).

Get method:
isSpherical(self) - Check whether this model is 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.

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.

n

Get the 3rd Flattening (float), f / (2 - f) == (a - b) / (a + b).

Get method:
n(self) - Get the 3rd Flattening (float), f / (2 - f) == (a - b) / (a + b).
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.

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 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.

Rr

Get the rectifying earth radius (meter), ((a**(3/2) + b**(3/2)) / 2)**(2/3).

Get method:
Rr(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.

Rs

Get another mean earth radius (meter), sqrt(a * b).

Get method:
Rs(self) - Get another mean earth radius (meter), sqrt(a * b).
Set method:
immutable(inst, value) - Throws an AttributeError, always.

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.