On Jan 30, 6:19 pm, Taurio <email@example.com> wrote: > A meridian's radius of curvature at the equator equals b^2/a, an > ellipse's /semi-latus rectum/. > Does the polar radius of curvature, which equals a^2/b, have a > similar elliptical property equivalency? > Besides the equator, isn't the meridional curvature's radius the > only curvature radius that is geocentric?
Any ?radius of curvature? equals an equivalent linear radius. With a circle or sphere, curvature is constant, therefore radius of curvature is constant, equaling the linear, geocentric radius. Now consider an ellipse. The curvature at the x-axis/?equator? is curled inward, while at the y- axis/?pole? it?s flattened, meaning, if you took an infinitesimal snip of equatorial arc and made enough copies to form a circle, and did the same with a snip of polar arc, the composite polar circle would be bigger than the equatorial one: If you make an oblatum (i.e., an oblate spheroid) with the same equatorial and polar radii, the composite equatorial circle would be smaller than the equator, b^2/a, while the polar circle would be larger, a^2/b. Along the equator, which is a great circle, the horizontal radius of curvature equals the geocentric, equatorial radius, a. There are two angular extremes of curvature radii??the vertical, 0°, *Meridional*, M(Lat), and the perpendicular horizontal, 90°, *Normal*, N(Lat). The meridional equals an ellipse?s radii of curvature, thus M(0) = b^2/ a, while M(90°) = N(90°) = a^2/b. Now look at the radii graph for Earth:
Only M(?45.072°) and N(0) equal their respective geocentric radius, meaning, vertically/meridionally, only (geodetic/geographic) latitude 45.072° has a geocentric radius of curvature. If you look at fig.2 on pg.3 of Rod E. Deakin?s ?The Great Elliptic Arc on an Ellipsoid?,
you will see that the linear radius equivalent of N(P) extends below the geocenter, O, to H: N(P) = PH. As for the polar radius of curvature, a^2/b, I?m not sure if NH represents the linear radius equivalent. Of coarse, there is a *geodetic/geographic* radius of curvature in any direction at a given point, G, *IN THE GREAT ELLIPTIC PLANE*, which is the arcradius (not to be confused with the more recognized and cited Euler's radius of curvature in the normal section).