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Conformal Latitude

Conformal latitude is an auxiliary latitude defined by

chi=2tan^(-1){tan(1/4pi+1/2phi)[(1-esinphi)/(1+esinphi)]^(e/2)}-1/2pi
(1)
=2tan^(-1)[(1+sinphi)/(1-sinphi)((1-esinphi)/(1+esinphi))^e]^(1/2)-1/2pi
(2)
=phi-(1/2e^2+5/(24)e^4+3/(32)e^6+(281)/(5760)e^8+...)sin(2phi)+(5/(48)e^4+7/(80)e^6+(697)/(11520)e^8+...)sin(4phi)-((13)/(480)e^6+(461)/(13440)e^8+...)sin(6phi)+((1237)/(161280)e^8+...)sin(8phi)+....
(3)

The inverse is obtained by iterating the equation

phi=2tan^(-1)[tan(1/4pi+1/2chi)((1+esinphi)/(1-esinphi))^(e/2)]-1/2pi
(4)

using phi=chi as the first trial.

A series form is

phi=chi+(1/2e^2+5/(24)e^4+1/(12)e^6+(13)/(360)e^8+...)sin(2chi)+(7/(48)e^4+(29)/(240)e^6+(811)/(11520)e^8+...)sin(4chi)+(7/(120)e^6+(81)/(1120)e^8+...)sin(6chi)+((4279)/(161280)e^8+...)sin(8chi)+...
(5)

The conformal latitude was called the isometric latitude by Adams (1921), but this term is now used to refer to a different quantity.

See also

Auxiliary Latitude, Latitude

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References

Adams, O. S. "Latitude Developments Connected with Geodesy and Cartography with Tables, Including a Table for Lambert Equal-Area Meridianal Projections." Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, pp. 18 and 84-85, 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, pp. 15-16, 1987.

Referenced on Wolfram|Alpha

Conformal Latitude

Cite this as:

Weisstein, Eric W. "Conformal Latitude." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConformalLatitude.html

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