- 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”, Lat_{v}, 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”, Lat
_{v}=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), Lat
_{v}=0, where sin(Lat_{p}) = cos(AP)sin(TL_{p}), thus Lat=Lat(AP,TL) and Lat(0,TL) = TL; - All other perspectives between the conjugate and transverse, 0<Lat
_{v}<90° are considered “oblique graticule perspectives”.]

- 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(Lat_{v}(90°,0)) = Vr{90°,0},

= Vr(Lat_{v}(AP_{v},0)) = Vr{AP_{v},0},

= Vr(Lat_{v}(90°,TL_{v})) = 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, AP
_{v}is the “master vertex AP” that the Lat_{v}(AP_{v},TL_{v}) “pole” travels along, and a given Vr{AP_{v},TL_{v}} is the average radius of the defined graticule perspective, equaling the average Cr between Cr(0) and Cr(AP_{v}), though AP differentiation is by local azimuth at Lat_{v}, AP(Az_{v}), from 0 to 90°, where AP_{v}= AP(90°)), Vr{AP_{v},0}―>Vr{AP_{v},90°}, averaged together, Pr(AP_{v}), 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.

_{x}and a

_{y}, are different), rather than Vr{AP

_{v},0}, Tr = Tr

_{xy}= Vr{90°,0}―>Vr{90°,90°} = Pr(90°), resulting in a 4D radius average, and Ur assumes a longitude variable, Ur(Long

_{v}), and the scalene Ur = Ur

_{xy}= 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=)

~Kaimbridge~