(ALLEGEDLY

SPHEROIDAL)

EARTH

MENSURATION

 
 


I. The (Alleged) Mean Radius of the (Allegedly Spheroidal) Earth

The (alleged) equatorial radius or Semi-major Axis (\(a\)) of the earth is \(6378137.0\;\textrm{m}\).1 The (alleged) polar or Semi-minor Axis (\(b\)) of the earth is \(6356752.3142\;\textrm{m}\).2

Whereas the earth is currently modelled as an ellipsoid of revolution, specifically an oblate ellipsoid of revolution or oblate spheroid, the third semi-axis is considered to be equal to the semi-major axis. Hence, the (alleged) mean radius or Mean Radius of the Three Semi-axes (\(R_1\)) of the earth is \((2a+b)/3\) or \(6371008.7714\;\textrm{m}\).3

For purposes of the analyses developed for this website, the (alleged) geodesic distance between any two sets of coordinates is approximated by the (alleged) mean great circle distance.

For the development of JavaScript calculators (see for example, (Allegedly Spheroidal) Earth Calculator I: w/o Tropo[spheric] Refraction) that are convenient for field work using the imperial system of measurement,4 the (alleged) mean radius of the earth (in meters) will be converted to feet.5 Hence, the expression used in calculations involving the (alleged) mean radius of the earth (in feet) will be: \[R_1=\frac{6,371,008.7714 \textrm{ m}}{0.3048 \textrm{ m ft}^{-1}}.\]


II. Orthometric (or Geoidal) Height

Whereas an observer or optical instrument sighting a distant island, promontory, structure, or marine vessel, is necessarily at some elevation or orthometric height (\(H\)) above the mean sea level surface of the earth, the magnitude of the (alleged) obscuration by the (alleged) curvature of the earth is purportedly reduced. For a complete analysis, see (Allegedly Spheroidal) Earth Curvature Equation w/ Elevated Observer, however, the basic geometry of the matter is depicted in Figure 1 as follows:


<rdf:RDF data-preserve-html-node="true"><cc:Work data-preserve-html-node="true" rdf:about=""><dc:format data-preserve-html-node="true">image/svg+xml</dc:format><dc:type data-preserve-html-node="true" rdf:resource="http://purl.org/dc/dcmitype/StillImage" /><dc:title data-preserve-html-node="true"></dc:title></cc:Work></rdf:RDF><sodipodi:namedview data-preserve-html-node="true" pagecolor="#ffffff" bordercolor="#666666" borderopacity="1" objecttolerance="10" gridtolerance="10" guidetolerance="10" inkscape:pageopacity="0" inkscape:pageshadow="2" inkscape:window-width="1600" inkscape:window-height="900" id="namedview3674" showgrid="false" inkscape:zoom="1.3782552" inkscape:cx="392.89449" inkscape:cy="331.39826" inkscape:window-x="0" inkscape:window-y="0" inkscape:window-maximized="1" inkscape:current-layer="svg3361" /> P horizon ( Alleged horizon point of observer or optical instrument) P 2 ( φ 2 , λ 2 ) ( Mean sea level position of island, promontory, structure, or marine vessel) Alleged line of sight ray of observer or optical instrument P 1 ( φ 1 , λ 1 , H ) ( Orthometric height position of observer or optical instrument) P 1 ( φ 1 , λ 1 ) ( Mean sea level position of observer or optical instrument) 90° P intersection ( Alleged intersection of line from mean sea level position of island, promontory, structure or marine vessel, normal to alleged line of sight ray of observer or optical instrument) Geoid

Figure 1. The (allegedly spheroidal earth) mean sea level position of an island, promontory, structure, or marine vessel in relation to the orthometric height, (alleged) horizon point, and (alleged) line-of sight of an observer or optical instrument, where \(P\), \(φ\), \(λ\), and \(H\) refer to point, latitude, longitude, and orthometric height respectively.6


The concept of orthometric height is relatively straightforward in terms of the existing (spheroidal) geodetic paradigm (and would also be easily transferable to the correct (planar) paradigm since it is based on mean sea level and therefore essentially independent of the spheroidal paradigm). Bomford (1980) offers the following preliminary definitions in explaining this concept:

The position of points being defined by latitude and longitude on a prescribed spheroid,[7] […] the rational definition of the height of any point is its distance above the spheroid measured along the spheroidal normal. Such heights are called spheroidal heights. Spheroidal heights are required for some purposes, but the geoid or mean sea-level surface has great significance, and a more generally useful height is the distance[8] above the geoid. Such heights are called geoidal heights and, unless otherwise stated, the height of a point implies it geoidal height. For common use it is essential that the zero height contour should lie close to mean sea-level, and spheroidal heights are not an acceptable basis for topo-graphical contouring […]9,10

Bomford (1980) subsequently defines orthometric height as follows:

The height of a point above the geoid, measured in metres or other linear units, is known as its orthometric (geoidal) height, and these are the heights generally quoted.11

Vaníček and Krakiwsky (1986) define orthometric height as follows:

The orthometric height \(H_i^o\) of a point \(P_i\) is defined as the geometrical distance between the geoid and the point, measured along the plumb line of \(P_i\) [...]12

More recently, Torge and Müller (2012) define orthometric height as follows:

The orthometric height \(H\) is defind as the linear distance between the surface point and the geoid, reckoned along the curved plumb line […]. This definition corresponds to the common understanding of “heights above sea level”. […]13

The geoid is not exactly coincident with mean sea level because there are minor variations (\(± 2.0 \;\textrm{m}\)) in sea surface elevation worldwide relative to the geoid.14 These variations are referred to as sea surface topography (SST) or dynamic ocean topography (DOT).15 The geoidal height reference surface is established by assigning a geopotential surface value for large regions (typically by international agreement).16 Coastal sea level elevations are thereby determined relative to an agreed geopotential surface. For example, the agreed (USA and Canada) geopotential surface (\(W_0= 62,636,856.0\;\textrm{m}^{2}\;\textrm{s}^{-2}\)) is approximately \(17\;\textrm{cm}\) below the coastal Pacific sea level and \(39\;\textrm{cm}\) above the coastal Atlantic sea level.17

The following diagram illustrates the concept of orthometric height, \(H\):

<rdf:RDF><cc:Work rdf:about=""><dc:format>image/svg+xml</dc:format><dc:type rdf:resource="http://purl.org/dc/dcmitype/StillImage" /><dc:title></dc:title></cc:Work></rdf:RDF><sodipodi:namedview pagecolor="#ffffff" bordercolor="#666666" borderopacity="1" objecttolerance="10" gridtolerance="10" guidetolerance="10" inkscape:pageopacity="0" inkscape:pageshadow="2" inkscape:window-width="1920" inkscape:window-height="1017" id="namedview3437" showgrid="false" inkscape:zoom="1.0963542" inkscape:cx="512" inkscape:cy="384" inkscape:window-x="-8" inkscape:window-y="-8" inkscape:window-maximized="1" inkscape:current-layer="svg3336" /> Terrain Alleged spheroid Geoid (Orthometric height) H

Figure 2. Exaggerated depiction of orthometric height of terrain in relation to the geoid and the (alleged) spheroid.


— FINIS —


The reader is advised to proceed to our web page titled, (Allegedly Spheroidal) Earth Curvature Geometry.


  1. National Geospatial-Intelligence Agency (NGA) Standardization Document (Office of Geomatics), Department of Defense World Geodetic System 1984: Its Definition and Relationships with Local Geodetic Systems (NGA.STND.0036_1.0.0_WGS84), Version 1.0.0, 2014-07-08, Table B.1 WGS 84 Defining Parameters.↩️

  2. Ibid., Table 3.5 WGS 84 Ellipsoid Derived Geometric Constants, page 3-9.↩️

  3. Ibid.↩️

  4. Metric system units will be included in a future revision to this website.↩️

  5. The international yard was defined as 0.9144 meter effective July 1, 1959. See Can. J. Phys. 1959. 37:84-84.(http://www.nrcresearchpress.com/doi/pdf/10.1139/p59-014). By extension, the foot is 0.3048 meter. Hence the meter is 1/0.3048 feet.↩️

  6. Obviously, sizes and distances have been scaled for illustrative purposes.↩️

  7. Bomford uses the term spheroid. The term ellipsoid in lieu of spheroid appears to have become more widespread in recent decades, possibly resulting from the de facto standard based on the reference ellipsoid defined by World Geodetic System (WGS) 1984. The reader should be aware that the terms ellipsoid, spheroid, oblate ellipsoid or ellipsoid of revolution, and oblate spheroid, are commonly used interchangeably in the literature, notwithstanding the more general nature of the terms ellipsoid and spheroid.↩️

  8. Here (in reference to distance), Bomford adds the following statement as a footnote: “Whether it is measured along the vertical or the normal is immaterial.”↩️

  9. Guy Bomford, Geodesy, Fourth Edition (Oxford: Clarendon Press, 1980), p. 197.↩️

  10. Whereas the large-scale structure of the earth is planar, the spheroidal or ellipsoidal height is a physically meaningless concept. It is therefore not surprising that geodesists attach primary significance to the geoidal height based on the physically meaningful concept of mean sea level. The geoid’s large-scale spheroidal superimposition implied and depicted above in Figure 1 and Figure 2 respectively, is necessarily elucidatory in respect of the current heliocentric model and is consequently incorporated here without prejudice.↩️

  11. Ibid.↩️

  12. Petr Vaníček and Edward Krakiwsky, Geodesy: The Concepts, Second Edition (Amsterdam: Elsevier, 1986), p. 370.↩️

  13. Wolfgang Torge and Jürgen Müller, Geodesy, Fourth Edition (Berlin: De Gruyter, 2012), p. 83 (in reference to their Fig. 3.14 on p. 82).↩️

  14. Natural Resources Canada online document titled, Height Reference System Modernization, p. 8, Mean Sea Level (MSL) (https://www.nrcan.gc.ca/sites/nrcan/files/files/pdf/Height_reference_system_modernization_(EN).pdf).↩️

  15. Loc. cit.↩️

  16. Concerning geopotential, while this website disputes the heliocentric precept of gravity, as concluded in the analysis of The Cavendish Experiment (1798), preceptive but otherwise sophistical attributes thereof will be tolerated (without prejudice) insofar as their elucidation is necessary to debate the heliocentric model.↩️

  17. Natural Resources Canada, loc. cit.↩️



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