Wikipedia:Reference desk/Archives/Mathematics/2020 July 28

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July 28

Geocentric vs great elliptic

Is the geocentric radius of curvature the same as the great elliptic radius of curvature? — Preceding unsigned comment added by 198.212.199.39 (talk) 20:30, 28 July 2020 (UTC)

I am not familiar with either of these terms, and a literature search comes up empty-handed. Where did you encounter them?  --Lambiam 23:28, 28 July 2020 (UTC)

I think the answer is no, there are two separate values. See: Earth_radius#Radii_of_curvature. 2601:648:8202:96B0:0:0:0:5B74 (talk) 04:15, 29 July 2020 (UTC)

That article mentions the concept of geocentric radius, which is not a radius of curvature. It coincides with the prime vertical radius of curvature on the equator, but not elsewhere. The prime vertical radius of curvature coincides with the meridional radius of curvature at the poles. There are many great ellipses going through a given point on the spheroid approximating the Earth's surface, all of which are (by definition) geocentric.  --Lambiam 07:21, 29 July 2020 (UTC)

I believe the proper term is "geocentric radius of curvature in the great elliptic plane".

The nomenclature seems a bit weird, but I think its equation is found here ("$G\{{\widehat {\mathrm {A} }},{\widehat {\sigma }}\}$"):

The series expansion below it appears to be an extension of Andoyer's approximation.  50.195.27.137 (talk) 00:41, 30 July 2020 (UTC)

I have reduced the size of the image a bit. This raises more questions for me than it answers. Is the meaning of "parageodetic" something like "almost geodetic"? Given a specified elliptic arc and a point on that arc, the notion of radius of curvature at that point is well defined, independent of whether the plane in which the arc lies is geocentric (contains the centre of a preferred ellipsoid) or not. Does it add something then to use "geocentric radius of curvature" instead of simply "radius of curvature" when applied to a great elliptic arc – which by definition is geocentric? So by whose authority or by what convention is this "the proper term"?  --Lambiam 05:57, 30 July 2020 (UTC)

I have no idea what "parageodetic" means, maybe "like geodetic" (similar to "paramilitary", "paralegal" or "parathyroid" being related but not synonymous to military, legal, thyroid), or mqybe just "greqt ellipic"?

Regarding the descriptive term, p or M is fully the "meridional radius of curvature in the normal section" (though usually just referenced as "radius of curvature in the meridional plane") and v or N is the "prime vertical radius of curvature in the normal section",  as implied here:

There are two principal radii of curvature: along the meridional and prime-vertical normal sections.

I personally don't care one way or the other, I just think "geocentric radius of curvature in the great elliptic plane" sounds like the full, technically descriptive name.

If you want to keep it concise, I would probably, informally just refer to it as the "radius of arc".  50.195.27.137 (talk) 00:52, 31 July 2020 (UTC)

I think the OP would not have asked this question unless they encountered these two terms in some context. The terms for the two principal radii of curvature for a point on a spheroid are well established and commonly used, so the more plausible possibility IMO is that these terms were introduced to refer to other radii of curvature, of which, perhaps, $M(\varphi )$ and $N(\varphi )$ are two instances. Or perhaps the context is a more general one of ellipsoids.  --Lambiam 11:46, 31 July 2020 (UTC)