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Rectifying Latitude

Rectifying latitude is an auxiliary latitude that gives a sphere having correct distances along the meridians. It is denoted mu (or omega) and is given by

mu=(piM)/(2M_p).
(1)

M_p is evaluated for M at the north pole (phi=90 degrees), and M is given by

M=a(1-e^2)int_0^phi(dphi)/((1-e^2sin^2phi)^(3/2))
(2)
=a[int_0^phisqrt(1-e^2sin^2phi)dphi-(e^2sinphicosphi)/(sqrt(1-e^2sin^2phi))].
(3)

A series for M is

M=a[(1-1/4e^2-3/(64)e^4-5/(256)e^6-...)phi-(3/8e^2+3/(32)e^4+(45)/(1024)e^6+...)sin(2phi)+((15)/(256)e^4+(45)/(1024)e^6+...)sin(4phi)-((35)/(3072)e^6+...)sin(6phi)+...],
(4)

and a series for mu is

mu=phi-(3/2e_1-9/(16)e_1^3+...)sin(2phi)+((15)/(16)e_1^2-(15)/(32)e_1^4+...)sin(4phi)-((35)/(48)e_1^3-...)sin(6phi)+((315)/(512)e_1^4-...)sin(8phi)+...,
(5)

where

e_1=(1-sqrt(1-e^2))/(1+sqrt(1-e^2)).
(6)

The inverse formula is

phi=mu+(3/2e_1-(27)/(32)e_1^3+...)sin(2mu)+((21)/(16)e_1^2-(55)/(32)e_1^4+...)sin(4mu)+((151)/(96)e_1^3-...)sin(6mu)+((1097)/(512)e_1^4-...)sin(8mu)+....
(7)

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 Meridional Projections." Spec. Pub. No. 67. U. S. Coast and Geodetic Survey, pp. 125-128, 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. 16-17, 1987.

Referenced on Wolfram|Alpha

Rectifying Latitude

Cite this as:

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

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