Authalic Latitude
Authalic latitude is an auxiliary which results in a sphere with equal surface area relative to a spheroid. The authalic latitude is defined by
| [画像: beta=sin^(-1)(q/(q_p)), ] |
(1)
|
where
![画像: q=(1-e^2)[(sinphi)/(1-e^2sin^2phi)-1/(2e)ln((1-esinphi)/(1+esinphi))],](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/AuthalicLatitude/NumberedEquation2.svg){kind=link}
and q_p is q evaluated at the north pole (phi=90 degrees). Let R_q be the radius of the sphere having the same surface area as the ellipsoid, then
| [画像: R_q=asqrt((q_p)/2). ] |
(3)
|
The series for beta is
beta = phi-(1/3e^2+(31)/(180)e^4+(59)/(560)e^6+...)sin(2phi)+((17)/(360)e^4+(61)/(1260)e^6+...)sin(4phi)-((383)/(45360)e^6+...)sin(6phi)+....
(4)
The inverse formula is found from
| Deltaphi=((1-e^2sin^2phi)^2)/(2cosphi)[q/(1-e^2)-(sinphi)/(1-e^2sin^2phi)+1/(2e)ln((1-esinphi)/(1+esinphi))], |
(5)
|
where
| q=q_psinbeta |
(6)
|
and phi_0=sin^(-1)(q/2). This can be written in series form as
phi = beta+(1/3e^2+(31)/(180)e^4+(517)/(5040)e^6+...)sin(2beta)+((23)/(360)e^4+(251)/(3780)e^6+...)sin(4beta)+((761)/(45360)e^6+...)sin(6beta)+....
(7)
See also
Auxiliary Latitude, LatitudeExplore with Wolfram|Alpha
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References
Adams, O. S. "Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridional Projections." Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, 1921.Karney, C. F. F. "On Auxiliary latitudes." 21 May 2023. https://arxiv.org/abs/2212.05818.Snyder, J. P. Map Projections--A Working Manual. U. S. Geological Survey Professional Paper 1395. Washington, DC: U. S. Government Printing Office, p. 16, 1987.Referenced on Wolfram|Alpha
Authalic LatitudeCite this as:
Weisstein, Eric W. "Authalic Latitude." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AuthalicLatitude.html