Radius of Curvature (applications) - Elliptic, Latitudinal Components

Elliptic, Latitudinal Components

The radius extremes of an oblate spheroid are the equatorial radius, or semi-major axis, a, and the polar radius, or semi-minor axis, b. The "ellipticalness" of any ellipsoid, like any ellipse, is measured in different ways (e.g., eccentricity and flattening), any and all of which are trigonometric functions of its angular eccentricity, :

The primary parameter utilized in identifying a point's vertical position is its latitude, . A latitude can be expressed either directly or from the arcsine of a trigonometric product, the arguments (i.e., a function's "input") of the factors being the arc path (which defines, and is the azimuth at the equator of, a given great circle, or its elliptical counterpart) and the transverse colatitude, which is a corresponding, vertical latitude ring that defines a point along an arc path/great circle. The relationship can be remembered by the terms' initial letter, L-A-T:

Therefore, along a north-south arc path (which equals 0°), the primary quadrant form of latitude equals the transverse colatitude's at a given point. As most introductory discussions of curvature and their radius identify position in terms of latitude, this article will too, with only the added inclusion of a "0" placeholder for more advanced discussions where the arc path is actively utilized: There are two types of latitude commonly employed in these discussions, the planetographic (or planetodetic; for Earth, the customized terms are "geographic" and "geodetic"), or, and reduced latitudes, or :

The calculation of elliptic quantities usually involves different elliptic integrals, the most basic unit integrands being, its inverse, and its complement, :

\begin{matrix} E'&=&E'(\theta)&=&\sqrt{1-(\sin(\theta)\sin(\alpha))^2}=\sqrt{\cos^2(\alpha)+(\cos(\theta)\sin(\alpha))^2},{\color{white}.}\\ &=&E'\{0,\theta\}&=&\sqrt{1-(\cos(0)\sin(\theta)\sin(\alpha))^2};\qquad&&&&\\ n'&=&\frac{1}{E'}&=&n'(\theta)=\frac{1}{E'(\theta)}=n'\{0,\theta\}=\frac{1}{E'\{0,\theta\}};\qquad&&&&\\ C'&=&C'(\theta)&=&\sqrt{1-(\cos(\theta)\sin(\alpha))^2}=\sqrt{\cos^2(\alpha)+(\sin(\theta)\sin(\alpha))^2},{\color{white}.}\\ &=&C'\{0,\theta\}&=&\sqrt{\cos^2(\alpha)+(\cos(0)\sin(\theta)\sin(\alpha))^2};\qquad&&&&\\ \end{matrix}\,\!
\begin{matrix}{}_{\color{white}.} R=R(\beta)=R\{0,\beta\}&=&aE'\{0,\beta\} &=&\sqrt{(a\cos(\beta))^2+(b\sin(\beta))^2},{\color{white}.} \\ &=&ar'\{0,\phi\}&=&a\sqrt{1-\cos^2(\alpha)(n'^2\{0,\phi\}-1)};{\color{white}.}\\{}^{\color{white}.}\end{matrix}\,\!
\begin{matrix}{}_{\color{white}.} S=S(\beta)=S\{0,\beta\}&=&aC'\{0,\beta\} &=&\sqrt{(a\sin(\beta))^2+(b\cos(\beta))^2},{\color{white}.}\\ &=&as'\{0,\phi\}&=&a\cos(\alpha)n'\{0,\phi)=bn'\{0,\phi\}.{\color{white}.}\\{}^{\color{white}.}\end{matrix}\,\!

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