BASIC

(ALLEGEDLY

SPHEROIDAL)

EARTH

CURVATURE

EQUATION

Introduction to the Basic (Allegedly Spheroidal) Earth Curvature Equation

On the (allegedly spheroidal1) surface of the earth (see Figure 1), consider the mean sea level (MSL)2 geographic position \(P_1(φ_1,λ_1)\), henceforth referred to as \(P_1\), of an observer or detector — the Greek letters \(φ\) (phi) and \(λ\) (lambda) referring to the position’s latitude and longitude respectively (a succinct description of which is offered by Torge and Müller).3 In relation to position \(P_1\), consider a distant subject at (MSL) geographic position \(P_2(φ_2,λ_2)\), henceforth referred to as \(P_2\). The (alleged) distance \(d_n\) is represented by the line from \(P_2(φ_2,λ_2)\) below and normal to the observer or detector’s horizon line of sight, i.e., the ray fom the observer or detector that is tangent to the observer or detector’s horizon and coplanar with \(P_2(φ_2,λ_2)\), said line intersecting said ray at point \(α\). The basic curvature equation — in terms of the extent to which \(P_2\) is (allegedly) hidden from \(P_1\) by the (alleged) curvature of the earth — is essentially the equation for determining the (alleged) magnitude of \(d_n\).


O (Center of the allegedly spheroidal earth) NOTE: OBJECT SIZES AND DISTANCES HAVE BEEN SCALED FOR ILLUSTRATIVE PURPOSES. θ Allegedly spheroidal earth P 1 ( φ 1 , λ 1 ) (Mean sea level position of observer or detector) P horizon (Observer or detector's horizon point coincident with mean sea level position of observer or detector at zero elevation) Ray from observer or detector tangent to observer or detector's horizon and coplanar with subject of observation or detection P 2 ( φ 2 , λ 2 ) (Mean sea level position of subject of observation or detection) d n α β R 1 (Alleged mean radius of the allegedly spheroidal earth) s

Figure 1. Analysis of the basic curvature of the (allegedly spheroidal) earth.


Derivation of the Basic (Allegedly Spheroidal) Earth Curvature Equation

Obviously, the (alleged) distance \(d_n\) is proportional to the amount of (alleged) curvature of the (allegedly spheroidal) earth between the observer or detector and the subject of observation or detection as represented by the (allegedly curvilinear) distance \(s\), i.e., arc (⌒) \(P_1P_2\). Expressed in terms of angular measurement — for this derivation, radians (rad) — the amount of curvature (on any spheroid) is directionally dependent and inversely proportional to a function of its three semi-axes; in the case of the (allegedly spheroidal) earth, the amount of curvature is inversely proportional to \(R_1\), where \(R_1\) is the (alleged) mean radius of the earth, specifically, the (alleged) Mean Radius of the Three Semi-axes of the (allegedly spheroidal) earth.4 Hence, the (allegedly curvilinear) distance \(s\) subtends the angle \(θ\), the magnitude of which is,

\[θ=\frac{s}{R_1}\textrm{rad}.\]

Referring again to Figure 1, if \(O\) is the center of the (allegedly) spheroidal earth, and \(P_1\) is (again) the position of the observer or detector — at a distance \(R_1\) from \(O\), whereby \(R_1\) is represented by line \(OP_1\), and if \(P_2\) is again the position of the subject of observation or detection, whereby \(R_1\) is represented by line \(OP_2\), then the angle (\(\scriptstyle{∠}\)) \(θ\) subtended by ⌒\(s\), i.e., ⌒\(P_1P_2\), can also be expresed as, \[θ=\;\scriptstyle{∠}\,\displaystyle{P_1OP_2.}\]

Hence,

\[θ=\;\scriptstyle{∠}\,\displaystyle{P_1OP_2=\frac{s}{R_1}\textrm{rad}.}\]

Referring once again to Figure 1, and as indicated previously, \(α\) is the intersection of the line from the subject’s position \(P_2\) with (and normal to) the observer or detector’s horizon line of sight. Hence, the two components of ⌒\(P_1P_2\) are represented by line \(P_1α\) (collinear with the observer or detector’s horizon line of sight) and line \(P_2α\) (normal to the observer or detector’s horizon line of sight). In rectangle (▭) \(P_1αP_2β\), the line from the subject’s position \(P_2\) with (and perpendicular to) line \(P_1O\) intersects line \(P_1O\) at \(β\), lines \(P_1β\) and \(P_2β\) being equal and parallel to (⋕) lines \(P_2α\) and \(P_1α\) respectively; hence, triangle \(P_2βO\) is a right triangle (◿).

In ◿\(P_2Oβ\),

\[\textrm{cos}\,\scriptstyle{∠}\,\displaystyle βOP_2\,\textrm{(or cos}\,θ)=\]

\[\dfrac{Oβ}{OP_2}\,\left(\textrm{or}\;\dfrac{Oβ}{R_1}\right).\]

Hence,

\[R_1=\frac{Oβ}{\textrm{cos}\,θ},\]

and therefore,

\[Oβ=R_1\textrm{cos}\,θ.\]

But \(R_1\) is also equal to \(OP_1\,\textrm{(or}\,Oβ+P_1β)\), and whereas \(P_1β⋕P_2α\,\textrm{(or}\,d_n)\), then \(d_n=R_1-Oβ\).

Hence,

\[d_n=R_1-R_1\textrm{cos}\,θ,\]

or,

\[d_n=R_1(1-\textrm{cos}\,θ).\]

And whereas,

\[θ=\frac{s}{R_1}\textrm{rad},\]

then, the basic curvature equation in terms of the distance \(d_n\) from \(P_2\) normal to the observer or detector’s horizon line of sight, is:

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


— FINIS —


The curvature equation is further developed to include the orthometric height of the observer or detector on our web page titled, (Allegedly Spheroidal) Earth Curvature Equation W/ Elevated Observer.


  1. Also referred to as ellipsoidal.↩️

  2. In this case, the observer is considered to be theoretically at mean sea level (MSL). Clearly, that condition is unrealistic as the observer or detector’s viewpoint would be at least several feet above MSL. The abstraction to theoretical MSL, however, is necessary to provide a simple analysis of the basic geometry of the currently accepted paradigm. Positional elevation is factored into the analyses presented on our web pages titled, (Allegedly Spheroidal) Earth Curvature Equation W/ Elevated Observer and (Allegedly Spheroidal) Earth Curvature Equation W/ Elevated Observer and Subject. For a technical description of elevation or orthometric height (\(H\)), see (Allegedly Spheroidal) Earth Mensuration.↩️

  3. Wolfgang Torge and Jürgen Müller, Geodesy, Fourth Edition (Berlin: De Gruyter, 2012), p. 93 (in reference to their Fig. 4.2, p. 92). They characterize latitude and longitude as follows:

    The system of geodetic surface coordinates is defined by the ellipsoidal latitude \(φ\) and longitude \(λ\) (also geodetic latitude and longitude). [Latitude] \(φ\) is the angle measured in the meridian plane between the equatorial plane \((\bar{X},\bar{Y}\textrm{-plane})\) of the ellipsoid and the surface normal at \(P\). Longitude \(λ\) is the angle measured in the equatorial plane between the zero meridian \((\bar{X}\textrm{-axis})\) and the meridian plane of \(P\). Here, \(φ\) is positive northwards and negative southwards, and \(λ\) is postive reckoned towards the east. [...]↩️

  4. See our web page titled, (Allegedly Spheroidal) Earth Mensuration.↩️


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DATE 2022-MAY-16 2022-MAY-29