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In the Manual of Land Surveying Instructions, 2009 Edition, why is the formula for radius of parallel written as
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?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ R = a/[Tan(Phi) * sqrt(1 - e^2 * sin(Phi)^2)]
Instead of?ÿ ?ÿ
?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ ?ÿ R = [a* cos(Phi)]/sqrt[1 - e^2 * sin(Phi)^2]?ÿ ?
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Where
R = radius of the parallel at latitude Phi, and
Phi = parallel of the base latitude
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The formulas represent two different things. To see what the second formula represents, look at auxiliary line p on page 4 in this publication:?ÿ http://clynchg3c.com/Technote/geodesy/radiigeo.pdf
To see what the first formula represents, look at the cone in figure 2.1a in this publication:?ÿ https://geodesy.noaa.gov/library/pdfs/NOAA_Manual_NOS_NGS_0005.pdf ?ÿ Instead of being a secant, though, the PLSS line is tangent to the ellipsoid at the parallel of interest.
Two such tangents to the same parallel meet at an angle on the ellipsoid's axis at some point above the ellipsoid. The angle between these tangents is the convergence angle. It takes a little 3d thinking to see why these are called radii in the first place and then, in the second place, why they are radii of parallels, but the effort is well worth the time spent.
For Javascript code that calculates offsets from these tangents, look here:?ÿ https://surveyorconnect.com/community/education-training/offset-and-azimuth-to-tangent-to-parallel/
Note the use of the first formula in the Javascript code.
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