Editing Earth radius (section)

====Principal radii of curvature====

There are two [[Principal curvature|principal radii of curvature]]: along the meridional and prime-vertical [[normal section]]s.  

=====Meridional=====
In particular, the ''Earth's [[meridian (geography)|meridional]] radius of curvature'' (in the north–south direction) at {{mvar|φ}} is<ref name="Jekeli">{{Cite book|author=Christopher Jekeli|title=Geometric Reference Systems in Geodesy|url=https://kb.osu.edu/bitstream/handle/1811/77986/Geom_Ref_Sys_Geodesy_2016.pdf|access-date=13 May 2023|year=2016|publisher=Ohio State University, Columbus, Ohio}}</ref>
:<math>M(\varphi)=\frac{(ab)^2}{\big((a\cos\varphi)^2+(b\sin\varphi)^2\big)^\frac32}
=\frac{a(1-e^2)}{(1-e^2\sin^2\varphi)^\frac32}
=\frac{1-e^2}{a^2} N(\varphi)^3,</math>
where <math>e</math> is the [[Eccentricity (mathematics)|eccentricity]] of the earth.  This is the radius that [[Eratosthenes#Measurement of the Earth|Eratosthenes]] measured in his [[arc measurement]].

=====Prime vertical=====
[[Image:Geodetic latitude and the length of Normal.svg|thumb|The length PQ, called the ''prime vertical radius'', is <math>N(\phi).</math> The length IQ is equal to <math>e^2 N(\phi).</math> <math>R = (X, Y, Z).</math>]]

If one point had appeared due east of the other, one finds the approximate curvature in the east–west direction.<ref group=lower-alpha name=curvprim>East–west directions can be misleading. Point B, which appears due east from A, will be closer to the equator than A. Thus the curvature found this way is smaller than the curvature of a circle of constant latitude, except at the equator. West can be exchanged for east in this discussion.</ref>

This ''Earth's [[prime vertical|prime-vertical]] radius of curvature'', also called the ''Earth's transverse radius of curvature'', is defined perpendicular ([[orthogonal]]) to {{mvar|M}} at geodetic latitude {{mvar|φ}}<ref group=lower-alpha>{{mvar|N}} is defined as the radius of curvature in the plane that is normal to both the surface of the ellipsoid at, and the meridian passing through, the specific point of interest.</ref> and is<ref name="Jekeli"/>
:<math>N(\varphi)=\frac{a^2}{\sqrt{(a\cos\varphi)^2+(b\sin\varphi)^2}}
=\frac{a}{\sqrt{1-e^2\sin^2\varphi}}.</math>

''N'' can also be interpreted geometrically as the [[normal distance]] from the ellipsoid surface to the polar axis.<ref name="Bowring">{{cite journal
|title=Notes on the curvature in the prime vertical section
|first=B. R.| last=Bowring |journal=Survey Review |volume=29
|issue=226 |date=October 1987 |pages=195–196|doi=10.1179/sre.1987.29.226.195|bibcode=1987SurRv..29..195B }}
<!--https://www.tandfonline.com/doi/abs/10.1179/sre.1987.29.226.195--></ref>
The radius of a [[parallel of latitude]] is given by <math>p=N\cos(\varphi)</math>.<ref>{{cite book |last=Bomford |first=G. |title=Geodesy |title-link=Geodesy (book) |publisher=Oxford University Press |year=1952 |author-link=Guy Bomford}}</ref><ref>{{Cite book|author=Christopher Jekeli|title=Geometric Reference Systems in Geodesy|url=https://kb.osu.edu/bitstream/handle/1811/77986/Geom_Ref_Sys_Geodesy_2016.pdf|access-date=13 May 2023|year=2016|publisher=Ohio State University, Columbus, Ohio}}</ref>

=====Polar and equatorial radius of curvature=====
The ''Earth's meridional radius of curvature at the equator'' equals the meridian's [[semi-latus rectum]]:

:<math>M(0\text{°})=\frac{b^2}{a}=6,335.439 \text{ km.}</math>

The ''Earth's prime-vertical radius of curvature at the equator'' equals the equatorial radius, <math>N(0\text{°})=a</math> 

The ''Earth's polar radius of curvature'' (either meridional or prime-vertical) is <math>M(90\text{°})=N(90\text{°})=\frac{a^2}{b}=6,399.594 \text{ km.}</math>

=====Derivation=====
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The principal curvatures are the roots of Equation (125) in:<ref name="Lass">{{cite book
|last=Lass |first=Harry |title=Vector and Tensor Analysis |url=https://archive.org/details/vectortensoranal00hlas |url-access=limited |date=1950
|publisher=McGraw Hill Book Company, Inc. |pages=[https://archive.org/details/vectortensoranal00hlas/page/n83 71]–77|isbn=9780070365209 }}</ref>

:<math>(E G - F^2) \kappa^2 - (e G + g E - 2 f F) \kappa + (e g - f^2) = 0
= \det(A - \kappa B),</math>

where in the [[first fundamental form]] for a surface (Equation (112) in<ref name="Lass"></ref>):

:<math>ds^2 = \sum_{ij} a_{ij} dw^i dw^j = E \,d\varphi^2 + 2 F \,d\varphi \,d\lambda + G \,d\lambda^2,</math>

''E'', ''F'', and ''G'' are elements of the [[metric tensor]]:

:<math> A = a_{ij} = \sum_\nu \frac{\partial r^\nu}{\partial w^i} \frac{\partial r^\nu}{\partial w^j}
= \begin{bmatrix} E & F \\ F & G \end{bmatrix},</math>

<math>r = [r^1, r^2, r^3]^T = [x, y, z]^T</math>, <math>w^1 = \varphi</math>, <math>w^2 = \lambda,</math>

in the [[second fundamental form]] for a surface (Equation (123) in<ref name="Lass"></ref>):

:<math> 2 D = \sum_{ij} b_{ij} dw^i dw^j =  e \,d\varphi^2 + 2 f \,d\varphi \,d\lambda + g \,d\lambda^2,</math>

''e'', ''f'', and ''g'' are elements of the shape tensor:

:<math>B = b_{ij} = \sum_\nu n^\nu \frac{\partial ^2 r^\nu}{\partial w^i \partial w^j}
= \begin{bmatrix} e & f \\ f & g \end{bmatrix},</math>

<math>n = \frac{N}{|N|}</math> is the unit normal to the surface at <math>r</math>, and because
<math>\frac{\partial r}{\partial \varphi}</math> and <math>\frac{\partial r}{\partial \lambda}</math> are tangents to the surface,

:<math>N = \frac{\partial r}{\partial \varphi} \times \frac{\partial r}{\partial \lambda}</math>
is normal to the surface at <math>r</math>.

With <math>F = f = 0</math> for an oblate spheroid, the curvatures are

:<math>\kappa_1 = \frac{g}{G}</math> and <math>\kappa_2 = \frac{e}{E},</math>

and the principal radii of curvature are

:<math>R_1 = \frac{1}{\kappa_1}</math> and <math>R_2 = \frac{1}{\kappa_2}.</math>

The first and second radii of curvature correspond, respectively, to the Earth's meridional and prime-vertical radii of curvature.

Geometrically, the second fundamental form gives the distance from <math>r + dr</math> to the plane tangent at <math>r</math>.
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