Ellipsoidal quadratic mean radius

An approximation for the average/mean radius of an ellipse's circumference, $$Er\,\!$$, is the elliptical quadratic mean: $$Er\approx Er_q=\sqrt{\frac{a^2+b^2}{2}}.\,\!$$
 * (where $$a\,\!$$ is the central, horizontal, transverse radius/semi-major axis
 * and $$b\,\!$$ is the central, vertical, conjugate radius/semi-minor axis)

As a meridian of an ellipsoid is an ellipse with the same circumference for a given set of $$a,b\,\!$$ values, its average/mean radius, $$M\!r\,\!$$, is also the same as for an ellipse, as is its quadratic mean approximation:
 * $$M\!r=Er\approx Er_q=M\!r_q=\sqrt{\frac{a^2+b^2}{2}};\,\!$$

Some use this approximation for the meridional average/mean radius and arcradius——as the average/mean radius of arc from $$a\,\!$$ to $$b\,\!$$ is the same as that of the radius——also for the average/mean arcradius of all of an ellipsoid's circumferences.

This would be appropriate if the arc paths of an ellipsoid were only north-south: They are not.

Great circles/ellipses
These arc paths——"great circles" on a sphere, "great ellipses" on an ellipsoid——exist at all different angles. This can be seen by tilting a globe down, so that the north pole is now on the equator: All of these (now) transverse (i.e., "crosswise" or perpendicular) meridians are great circles/ellipses.

Besides $$M\!r\,\!$$ and even $$a\,\!$$, another commonly used value attempting to approximate the average/mean (arc)radius of an ellipsoid is the authalic ("equal area") radius, $$Ar\,\!$$:
 * $$Ar\approx\frac{2a+b}{3};\,\!$$

While it may appear to be an——if not the——obvious choice as an average/mean, if one calculated random distances of varying lengths and extracted the average/mean arcradius of each distance, the cumulative average will be significantly greater than $$Ar\,\!$$, which reflects the surface area as a single, unified entity, rather than the differentiated sum of all of the differently angled circumferences averaged all together, that a great-circle/spherical arcradius attempts to satisfy.

So, the average/mean circumference of an ellipsoid is the average/mean circumference of each of the differently angled arc paths (an infinite number), all averaged together.

Ellipsoid axes
An ellipsoid is triaxial, with the polar Z-axis represented by the conjugate radius, $$b\,\!$$, while the two equatorial axes perpendicular to each other, X and Y, are defined with radii $$a_x\,\!$$ and $$a_y\,\!$$, respectively (conventionally, $$a_x,a_y,b\,\!$$ are denoted as $$a,b,c\,\!$$, but in most cases——such as Earth——the equator is considered spherical and the ellipsoid is considered a spheroid, with the equatorial radius being defined simply as $$a\,\!$$ and the polar as $$b\,\!$$——like an ellipse——thus $$a,b,c\,\!$$ triaxial notation can be confusing).

For an ellipsoid where $$a_x\,\!$$ does not equal $$a_y\,\!$$ (in which case the ellipsoid is considered scalene——"three unequal sides"), there is a third valuation of $$a\,\!$$ utilized, the geometric mean of $$a_x\,\!$$ and $$a_y\,\!$$, $$a_m\,\!$$: $$a_m=\sqrt{a_xa_y}.\,\!$$

The elliptical, or biaxial, quadratic mean arcradius for the (average) scalene meridional circumference uses $$a_m\,\!$$ as the equatorial radius:
 * $$M\!r_q=\sqrt{\frac{a^2_m+b^2}{2}}=\sqrt{\frac{a_xa_y+b^2}{2}};\,\!$$

In the scalene case, the equator, itself, is an ellipse (thus the conventional $$a,b\,\!$$ assignment for the X and Y axes), with its own quadratic mean:
 * $$a_q=\sqrt{\frac{a^2_x+a^2_y}{2}}.\,\!$$

Formulating the Ellipsoidal Quadratic Mean
The ellipsoidal, or triaxial, quadratic mean is the quadratic mean of $$M\!r_q\,\!$$ and $$a_q\,\!$$, $$Qr\,\!$$:
 * $$\begin{matrix}Qr&=&\sqrt{\frac{M\!r^2_q+a^2_q}{2}}&=&\sqrt{\frac{\frac{a^2_m+b^2}{2}+\frac{a^2_x+a^2_y}{2}}{2}}&=&\sqrt{\frac{a^2_x+a_xa_y+a^2_y+b^2}{4}},\\

&=&\sqrt{\frac{3a^2+b^2}{4}}&=&\frac{1}{2}\sqrt{3a^2+b^2};\end{matrix}\,\!$$ This particular average/mean arcradius is the natural choice as a general purpose approximation of an ellipsoid's great-circle/spherical radius, as it is between the two, more definitively defined, spherical valuations, one of which is based on geodetically defined great-ellipse arc paths, $$Sr\,\!$$, and the other on spherically delineated, elliptical great-circle arc paths ("transverse meridians"), $$T\!r\,\!$$ (The difference being, a geodetic arc turns vertical as it grows in length——albeit, minusculy at first——since a geodesic, by definition, is the shortest (arc)distance between two points and, in the case of complete circumferences on an ellipsoid, where $$a>b\,\!$$, a meridian is the shortest path, while the equator is the longest; thus $$Sr\,\!$$ is based on, and found by, averaging half or semi-circumferences).

Analysis and application through valuations for Earth
As Earth demonstrates, an ellipsoid is usually not smooth, as its surface peaks and dips with various sized mountains, valleys and even regionwide undulations.

As such, there is no single, definitive valuations of $$a\,\!$$ and $$b\,\!$$ for Earth, only defined "reference ellipsoids", some of which are intended for long distance, global applications, while others are tailored for specific regions.

In fact, the latitude and longitude coordinates that plot a position on an ellipsoid's surface are part of a geodetic datum, the values of which are determined by the defining reference ellipsoid model used (thus a given spot plotted will have varying coordinate values——the variance depending on the reference ellipsoids' difference, sometimes up to hundreds of meters, though usually less than a few——meaning, to find the theoretically precise distance between two points, the reference ellipsoid used must either be the same as the one the coordinates' datum is based on, or a reconciling adjustment made to X, Y, Z in the calculations).

With that in mind, a general purpose Earth spheroid, "GPES"——where differences between different reference ellipsoids/datums aren't important and/or the coordinates are either significantly rounded or their datum origin is mixed/unknown——can be defined as $$a=6378.135 \mbox{(km)},\,b=6356.750\,\!$$, resulting in $$M\!r\,(\!=\!)\,6367.44698883\,\!$$ (where "(=)" means "equal to rounded position"), $$M\!r_q\,(\!=\!)6367.45147766\,\,\!$$, $$Ar\,(\!=\!)\,6371.00507612\,\!$$, $$Sr\,(\!=\!)\,6372.79492383\,\!$$ , $$T\!r\,(\!=\!)\,6372.80165021\,\!$$ and $$Qr\,(\!=\!)\,6372.79547760\,\!$$. To better comprehend discrepancies that can arise by using mismatched geodetic datums, as well as reinforce the fact that there is no single, precise $$a\,\!$$ and $$b\,\!$$ valuation set, compare $$a\,\!$$, $$b\,\!$$,$$M\!r\,\!$$, $$Ar\,\!$$ and $$Qr\,\!$$ values for GPES, with the most up to date, GPS satellite created reference ellipsoid (1980-83), GRS-80/83, the South American/Australian (1966), SA/A, the Clarke (1866), Clarke, the Hayford/International (1924), Hayford, the Bessel (1841), Bessel, and the Plessis (1817), Plessis, all of which are still in use, in different areas (except when exact, all radii are rounded to .01 mm):


 * {|class="wikitable"

! Model !! $$a\,\!$$ !! $$b\,\!$$ !! $$M\!r\,\!$$ !! $$Ar\,\!$$ !! $$Qr\,\!$$
 * GPES||6378.135||6356.75||6367.44698883|| 6371.00507612 ||6372.79547760
 * GRS-80/83||6378.137||6356.75231420||6367.44914580|| 6371.00718090||6372.79755595
 * SA/A||6378.16||6356.77471920||6367.47184853|| 6371.02998249||6372.82040755
 * Clarke||6378.2064||6356.5838||6367.39968917|| 6370.99724063||6372.80762791
 * Hayford||6378.388||6356.91194613||6367.65450006|| 6371.22771134||6373.02577130
 * Bessel|| 6377.397155|| 6356.07896282||6366.74252023|| 6370.28951013||6372.07429334
 * Plessis||6376.523||6355.86293326||6366.19715711|| 6369.63482609||6371.36426393
 * }
 * Hayford||6378.388||6356.91194613||6367.65450006|| 6371.22771134||6373.02577130
 * Bessel|| 6377.397155|| 6356.07896282||6366.74252023|| 6370.28951013||6372.07429334
 * Plessis||6376.523||6355.86293326||6366.19715711|| 6369.63482609||6371.36426393
 * }
 * Plessis||6376.523||6355.86293326||6366.19715711|| 6369.63482609||6371.36426393
 * }
 * }

Now consider value differences between (geographical) latitude, $$\phi,\,\!$$, and longitude, $$\lambda\,\!$$, coordinates for a given point defined by the North American Datum-1983 ("NAD83"), which uses GRS-80/83, and the converted coordinates for the same point, using the 1927 datum ("NAD27"), which is based on Clarke's ellipsoid. Using values rounded from the NADCON calculator, where $$\Delta\mbox{m}_\phi\,\!$$ is the distance between the latitudes, $$\Delta\mbox{m}_\lambda\,\!$$ the distance between the longitudes and $$\scriptstyle{\sqrt{\Delta\operatorname{m}^2_\phi+\Delta\operatorname{m}^2_\lambda}}\,\!$$ the Pythagorean distance between the two points (all expressed in meters):


 * {|class="wikitable"

! (NAD83) $$\phi,\lambda\,\!$$ !! (NAD27) $$\phi,\lambda\,\!$$ !! $$\Delta\mbox{m}_\phi\,\!$$ !! $$\Delta\mbox{m}_\lambda\,\!$$ !! $$\scriptstyle{\sqrt{\Delta\operatorname{m}^2_\phi+\Delta\operatorname{m}^2_\lambda}}\,\!$$ !! Given both the variation between $$Sr\,\!$$ (which, itself, has a complex and ambiguous formulation, due to the aforementioned fluid dynamics of great-ellipse delineation on an ellipsoid's surface) and $$T\!r\,\!$$ (which is the definitively defined and calculated average/mean elliptical great-circle radius), and the lack of a single, absolute set of $$a\,\!$$ and $$b\,\!$$ values, it should be obvious that, unlike for surface area, there is no single composite radius value to define an auxiliary sphere with respect to a uniform circumference——i.e., an exclusive "great-circle radius".
 * 30°,70° || 29.9996802°, 70.0007485° ||style="text-align: right;"| 35.5 ||style="text-align: right;"| 72.2 ||style="text-align: center;"|  80.5
 * 45,70 || 44.9999436, 70.0004927 ||style="text-align: right;"| 6.3 ||style="text-align: right;"|  38.8 ||style="text-align: center;"|  39.3
 * 30,90 || 29.9997979, 89.9999304 ||style="text-align: right;"| 22.4 ||style="text-align: right;"|  6.7 ||style="text-align: center;"|  23.4
 * 45,90 || 45.0000331, 89.9998688 ||style="text-align: right;"| 3.7 ||style="text-align: right;"|  10.3 ||style="text-align: center;"|  10.9
 * 30,110|| 29.9998677,109.9994041 ||style="text-align: right;"| 14.7 ||style="text-align: right;"| 57.5 ||style="text-align: center;"|  59.3
 * 45,110|| 45.0000605,109.9992597 ||style="text-align: right;"| 6.7 ||style="text-align: right;"|  58.4 ||style="text-align: center;"|  58.8
 * 30,130|| 29.9999417,129.9988331 ||style="text-align: right;"| 6.5 ||style="text-align: right;"| 112.6 ||style="text-align: center;"| 112.8
 * 45,130|| 45.0002047,129.9985160 ||style="text-align: right;"| 22.8 ||style="text-align: right;"| 117.0 ||style="text-align: center;"| 119.2
 * }
 * 30,90 || 29.9997979, 89.9999304 ||style="text-align: right;"| 22.4 ||style="text-align: right;"|  6.7 ||style="text-align: center;"|  23.4
 * 45,90 || 45.0000331, 89.9998688 ||style="text-align: right;"| 3.7 ||style="text-align: right;"|  10.3 ||style="text-align: center;"|  10.9
 * 30,110|| 29.9998677,109.9994041 ||style="text-align: right;"| 14.7 ||style="text-align: right;"| 57.5 ||style="text-align: center;"|  59.3
 * 45,110|| 45.0000605,109.9992597 ||style="text-align: right;"| 6.7 ||style="text-align: right;"|  58.4 ||style="text-align: center;"|  58.8
 * 30,130|| 29.9999417,129.9988331 ||style="text-align: right;"| 6.5 ||style="text-align: right;"| 112.6 ||style="text-align: center;"| 112.8
 * 45,130|| 45.0002047,129.9985160 ||style="text-align: right;"| 22.8 ||style="text-align: right;"| 117.0 ||style="text-align: center;"| 119.2
 * }
 * 45,110|| 45.0000605,109.9992597 ||style="text-align: right;"| 6.7 ||style="text-align: right;"|  58.4 ||style="text-align: center;"|  58.8
 * 30,130|| 29.9999417,129.9988331 ||style="text-align: right;"| 6.5 ||style="text-align: right;"| 112.6 ||style="text-align: center;"| 112.8
 * 45,130|| 45.0002047,129.9985160 ||style="text-align: right;"| 22.8 ||style="text-align: right;"| 117.0 ||style="text-align: center;"| 119.2
 * }
 * 30,130|| 29.9999417,129.9988331 ||style="text-align: right;"| 6.5 ||style="text-align: right;"| 112.6 ||style="text-align: center;"| 112.8
 * 45,130|| 45.0002047,129.9985160 ||style="text-align: right;"| 22.8 ||style="text-align: right;"| 117.0 ||style="text-align: center;"| 119.2
 * }
 * 45,130|| 45.0002047,129.9985160 ||style="text-align: right;"| 22.8 ||style="text-align: right;"| 117.0 ||style="text-align: center;"| 119.2
 * }

Instead, the best one can do is accept an approximation of best fit, using $$Sr\,\!$$ and $$T\!r\,\!$$ as boundaries.

While $$T\!r\,\!$$ is the definitive average/mean elliptical great-circle radius (albeit, involving an iterated integral), $$Sr\,\!$$ is the one based on geodetic distance, which is usually what a great-circle radius seeks to represent.

As such, the ideal approximation should be midway between or closer to $$Sr\,\!$$ than $$T\!r\,\!$$.

Using the values for GPES, different averagings of $$Sr\,\!$$ and $$T\!r\,\!$$ work out as so:
 * $$\sqrt{Sr{\cdot}T\!r}\,(\!=\!)\,\frac{Sr+T\!r}{2}\,(\!=\!)\,\sqrt{\frac{Sr^2+T\!r^2}{2}}\,(\!=\!)\,\sqrt[xp]{\frac{Sr^{xp}+T\!r^{xp}}{2}}\,(\!=\!)\,6372.79828702;\,\!$$
 * $$\sqrt[xp]{\frac{2Sr^{xp}+T\!r^{xp}}{3}}\,(\!=\!)\,6372.79716596;\quad

\sqrt[xp]{\frac{Sr^{xp}+2T\!r^{xp}}{3}}\,(\!=\!)\,6372.79940808;\,\!$$
 * $$\sqrt[xp]{\frac{10Sr^{xp}+T\!r^{xp}}{11}}\,(\!=\!)\,6372.795535;\quad

\sqrt[xp]{\frac{12Sr^{xp}+T\!r^{xp}}{13}}\,(\!=\!)\,6372.795441;\,\!$$
 * $$\sqrt[xp]{\frac{11Sr^{xp}+T\!r^{xp}}{12}}\,(\!=\!)\,\mathbf{6372.79548}4;\,\!$$

Different averagings involving $$a\,\!$$, $$b\,\!$$ and $$Mr\,\!$$ provide these results:
 * $$\begin{matrix}\sqrt[4]{a^3b}\!\!\!&(\!=\!)&\!\!\!6372.78201486;\quad\quad\;\;\;\frac{3a+b}{4}\!\!\!&=&\!\!\!\mathbf{6372.78875};\quad\;\;\\

\sqrt{\frac{3a^2+b^2}{4}}\!\!\!&(\!=\!)&\!\!\!\mathbf{6372.79547}760;\quad \sqrt[3]{\frac{3a^3+b^3}{4}}\!\!\!&(\!=\!)&\!\!\!6372.80219765;\end{matrix}\,\!$$
 * $$\begin{matrix}\sqrt{Mr{\cdot}a}\!\!\!&(\!=\!)&\!\!\!\mathbf{6372.78875}377;\quad\quad\;\;\;\frac{Mr+a}{2}\!\!\!&=&\!\!\!6372.79099442;\\

\sqrt{\frac{Mr^2+a^2}{2}}\!\!\!&(\!=\!)&\!\!\!6372.79323507;\quad \sqrt[3]{\frac{Mr^3+a^3}{2}}\!\!\!&(\!=\!)&\!\!\!\mathbf{6372.79547}571;\end{matrix}\,\!$$ From these different specimens, two relationships stand out:
 * $$\sqrt{Mr{\cdot}a}(\!=\!)\frac{3a+b}{4}=6372.78875;\,\!$$
 * $$\begin{matrix}\sqrt[xp]{\frac{11Sr^{xp}+T\!r^{xp}}{12}}\!\!\!&(\!=\!)&\!\!\!

\sqrt[3]{\frac{Mr^3+a^3}{2}}\,\,(\!=\!)\,\,\sqrt{\frac{3a^2+b^2}{4}}\,\!\!\!&(\!=\!)&\!\!\!6372.79548,\\ &\approx&\!\!\!\!\!\!2\sqrt{Mr{\cdot}a}-\sqrt[4]{a^3b}\!\!\!\!\!\!\!\!\!&(\!=\!)&\!\!\!6372.79549;\end{matrix}\,\!$$ In both cases, there is a remarkable coproximity between the different models of the respective result. It also appears to be the only two such whole integer exponent relationships, suggesting that these two results represent approximations of an inherent elliptic quantity (in this case, likely $$Sr\,\!$$).

Given all of these considerations, it would seem that the ellipsoidal quadratic mean provides the ideal (approximative) great-circle radius for an ellipsoid.