Radius Averaging Dimensions

Higher-dimensional geometry (previously "Polyshapes").

Radius Averaging Dimensions

Postby Kaimbridge » Fri Apr 08, 2016 5:47 am

  • The radius of a circle/ellipse is considered 1D;
  • A circumference of a sphere/ellipsoid (i.e., a great circle/ellipse), is considered 2D;
  • A graticule perspective of a sphere/ellipsoid is
    considered 3D;
    [where a graticule is the “coordinate webbing” (i.e., “latitude”/“longitude”) of a sphere/ellipsoid and its perspective is the angle that it is constructed, measured by its “vertex latitude”, Latv, which is where the meridians/longitudes merge/hinge at a point (the “pole” or vertex):
  • The “upright” perspective that latitude/longitude measures is the “conjugate graticule perspective”, Latv=90°;
  • If the vertex/pole is pulled down 90° to the equator, the perspective is perpendicular or "transverse" to the conjugate and is known as the “transverse graticule perspective” (and the transverse meridians/angled great circles can be considered “arc path”s, “AP”s, with corresponding transverse (co‐)latitudes, “TL”s), Latv=0, where sin(Latp) = cos(AP)sin(TLp), thus Lat=Lat(AP,TL) and Lat(0,TL) = TL;
  • All other perspectives between the conjugate and transverse, 0<Latv<90° are considered “oblique graticule perspectives”.]
Given this, would it be right to say that (on an ellipsoid),
  • The average radius of a great ellipse, Cr, is a 2D radius average;
  • The average radius of the circumferences measured at the equator from the north-south, meridional great ellipse, Cr(AP=0), to the east-west, equatorial great circle, Cr(AP=90°)—i.e., the average radius of the transverse graticule perspective,
      Tr = Vr(0) = Vr(Latv(90°,0))   = Vr{90°,0},
                 = Vr(Latv(APv,0))   = Vr{APv,0},
                 = Vr(Latv(90°,TLv)) = Vr{90°,0};
    is a 3D radius average;
    (In this example, we have Vr{0,0°} + Vr{0,90°} + c.Vr{0,50°})
  • The average radius of all of the different graticule perspectives along a given circumference (here, APv is the “master vertex AP” that the Latv(APv,TLv) “pole” travels along, and a given Vr{APv,TLv} is the average radius of the defined graticule perspective, equaling the average Cr between Cr(0) and Cr(APv), though AP differentiation is by local azimuth at Latv, AP(Azv), from 0 to 90°, where APv = AP(90°)), Vr{APv,0}―>Vr{APv,90°}, averaged together, Pr(APv), is a 4D radius average;
  • And, in culmination, the average radius of all of the different graticule perspectives (“Vr”s) along all of the different great circles/ellipses (the “unified average radius”, Ur), Pr(0)―>Pr(90°), is the 5D radius average.
However, in the case of a scalene ellipsoid (where the two equatorial radii/axes, ax and ay, are different), rather than Vr{APv,0}, Tr = Trxy = Vr{90°,0}―>Vr{90°,90°} = Pr(90°), resulting in a 4D radius average, and Ur assumes a longitude variable, Ur(Longv), and the scalene Ur = Urxy = Ur(0)―>Ur(90°), resulting in a 6D radius average.
Would this dimensional assessment be correct?
(FWIW, I do have a small, work-in-progress—at this point, non-scalene—Python Ur calculating program, if anyone is interested P=)

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