PURPOSE

AND

OVERVIEW

OF THE

(ALLEGEDLY

SPHEROIDAL)

EARTH

CURVATURE

ANALYSIS

Purpose of the (Allegedly Spheroidal) Earth Curvature Analysis

One of the main objectives of this website is to facilitate independent or collaborative research into long-range, terrestrial observations for the purpose of subjecting the current (spheroidal) model of the large-scale structure of the earth’s surface to worldwide scientific scrutiny.

The purpose of the curvature analysis developed over several web pages of the CURVATURE section is really twofold: (a) to impart to readers the information necessary to fully understand and utilize the practical research tools provided in the CALCULATORS section, and (b) to use the essential geometry of the existing heliocentric or spheroidal earth paradigm to demonstrate (in the CALCULATORS section based on data associated with direct field observations) its inherent and absolute fallaciousness.

The analysis is centered on determining the magnitude of the (allegedly spheroidal) earth curvature between the observer (i.e., either (I) an observer or optical instrument, or (II) a radar transmitter or radio frequency transceiver) and the observed or detected (i.e., either (I) a distant island, promontory, mountain, or structure, or (II) an aircraft or radio frequency transmitter. In terms of practical field observations, the operative component of that curvature is the (linear) distance from the observed normal to the observer’s horizon line of sight; the expressions for that distance, as well as for the (alleged) linear and (alleged and allegedly) curvilinear distances to the horizon have been translated into Javascript for application in the CALCULATORS section.

Overview of the (Allegedly Spheroidal) Earth Curvature Analysis

Our web page titled, (Allegedly Spheroidal) Earth Mensuration, provides the reader with a technical understanding of the two parameters essential for calculations determining the (alleged) curvature of the (allegedly spheroidal) earth, specifically, (a) the (alleged) mean radius of the (allegedly spheroidal) earth based on World Geodetic System (WGS) 1984, and (b) several definitions of orthometric (or geoidal) height from several leading geodesy texts and a diagrammatic description of the concept.

The fundamental geometrical construct for assessing the (alleged) visibility limitations particular to the spheroidal model of the earth’s large-scale structure is depicted on our web titled, (Allegedly Spheroidal) Earth Curvature Geometry. The linear component of the increasing curvature associated with increasing distances between the observer and the observed is described in terms of a series of parallel lines normal to a ray tangent to the (allegedly spheroidal) earth at the observer’s point.

The derivation of the equation to calculate the linear component \(d_n\) of the curvature between the observer at the (theoretically) mean sea level (MSL) position \(P_1\) and the observed at the (theoretically) MSL position \(P_2\) is provided on our web page titled, Basic (Allegedly Spheroidal) Earth Curvature Equation, \(d_n\) being represented by the line from the observer at \(P_2\) normal to the ray from the observer that is tangent to the observer's horizon and coplanar with the observed, said line intersecting said ray at point \(α\). The derived equation for \(d_n\) is as follows:

\[d_n=R_1\left[1-\textrm{cos}\left(\frac{s} {R_1}\textrm{rad}\right)\right],\]

where,

\(R_1\) is the (alleged) mean radius of the (allegedly spheroidal) earth,1 and

\(s\) is the (allegedly curvilinear) distance between the observer at \(P_1\) and the observed at \(P_2\).

The derivation of the equation to calculate the linear component \(d_n\) of the curvature between the observer at the orthometric height position \(P_1(H_1)\) and the observed at the (theoretically) MSL position \(P_2\) is provided on our web page titled, (Allegedly Spheroidal) Earth Curvature Equation W/ Elevated Observer. The reader is shown diagrammatically the extent to which the (alleged) relative curvature (as represented by \(d_n\)) decreases with the increasing height or elevation of the observer. Whereas the observer is at orthometric height position \(P_1(H_1)\), equations for both the (alleged) linear distance \(d_h\) to the horizon and the (alleged and allegedlly) curvilinear distance \(s_h\) to the horizon are also derived. The derived equations equations for \(d_h\), \(s_h\), and \(d_n\) are as follows:

\[d_h=\sqrt{2R_1H_1+H_1^2},\]

where,

\(H_1\) is the orthometric height2 of the observer at \(P_1\);

\[s_{h}=\]

\[R_1 \bigg\{{\left[\textrm{cos}\,^{-1}\left(\frac{R_1}{R_1+H_1}\right)\right]\textrm{rad}\bigg\}};\]

and

\[d_{n}=\]

\[R_1\bigg\{{1- \textrm{cos}\,\left[\left(\frac{s}{R_1}\textrm{rad}\right)-\\\textrm{cos}\,^{-1}\left(\frac{R_1}{R_1+H_1}\right)\textrm{rad}\,\right]\bigg\}.}\]

The derivation of the equation to calculate the linear component \(d_n\) of the curvature between the observer at the orthometric height position \(P_1(H_1)\) and the observed at the orthometric height position \(P_2(H_2)\) is provided on our web page titled, (Allegedly Spheroidal) Earth Curvature Equation W/ Elevated Observer and Subject. The reader is shown diagrammatically the extent to which the (alleged) relative curvature (as represented by \(d_n\)) decreases with the increasing height or elevation of the observed. The equation is as follows:

\[d_{n}=R_1-{}\]\[ \bigg\{{(R_1-H_2)\times \textrm{cos}\,\left[\left(\frac{s}{R_1}\textrm{rad}\right)-\\\textrm{cos}\,^{-1}\left(\frac{R_1}{R_1+H_1}\right)\textrm{rad}\,\right]\bigg\},}\]

where,

\(H_2\) is the orthometric height of the observed at \(P_2\).

Also explained to the reader is that the magnitude of \(d_n\) is again lowered by considering tropo[spheric] refraction in terms of a multiplicative factor applied the the (alleged) mean radius \(R_1\) of the earth, effectively reducing curvature, thereby flattening (to a certain extent) the earth’s surface. \(K\) is the Effective Earth Radius Factor (EERF);3 if expressed in terms of the coefficient of tropo[spheric] refraction \(k\), then \[K=\frac{1}{1-k}.\]

Hence, the final expression for \(d_n\) in terms of \(K\) is as follows:

\[d_{n(K)}=KR_1-{}\]\[ \bigg\{{(KR_1-H_2)\times \textrm{cos}\,\left[\left(\frac{s}{KR_1}\textrm{rad}\right)-\\\textrm{cos}\,^{-1}\left(\frac{KR_1}{KR_1+H_1}\right)\textrm{rad}\,\right]\bigg\}.}\]

Similarly, the final expressions for the (alleged) linear distance \(d_h\) and (alleged and allegedly) curvilinear distance \(s_h\) to the horizon (derived on the prior web page4) are provided as follows:

\[d_h=\sqrt{2K_1R_1H_1+H_1^2}.\]

and

\[s_{h}=\]

\[KR_1 \bigg\{{\left[\textrm{cos}\,^{-1}\left(\frac{KR_1}{KR_1+H_1}\right)\right]\textrm{rad}\bigg\}}.\]


— FINIS —


The reader is advised to proceed to our web page titled, (Allegedly Spheroidal) Earth Mensuration The reader is also advised to check out our web pages titled, Refraction Overview and Introduction to the (Allegedly Spheroidal) Earth Calculators.


  1. For a referenced definition of \(R_1\), see our web page titled, (Allegedly Spheroidal) Earth Mensuration.↩️

  2. For a description of orthometric (or geoidal) height, see again our web page titled, (Allegedly Spheroidal) Earth Mensuration.↩️

  3. For a derivation of the (alleged) Effective Earth Radius Factor \(K\), see Effective Earth Radii for Optical and Radar (or RF) Refraction in the Tropo[sphere]. The following \(K\) factors are used in this website: \(K=1\) (w/o tropo[spheric] refraction), \(K=7/6\) (w/ mean tropo[spheric] optical refraction), \(K=5/4\) (w/ maximum tropo[spheric] optical refraction), \(K=4/3\) (w/ mean tropo[spheric] radar or radio frequency refraction), and \(K=1.45\) (w/ maximum tropo[spheric] radar or radio frequency refraction).↩️

  4. See again, our web page titled, (Allegedly Spheroidal) Earth Curvature Equation w/ Elevated Observer.↩️


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REVISION 0 1
DATE 2022-JUL-09 2022-NOV-09