(ALLEGEDLY

SPHEROIDAL)

EARTH

CURVATURE

GEOMETRY

The methodology for determining whether a distant land-based or airborne subject1 is visible to an observer or optical instrument, or detectable by a radar or radio frequency (RF) transceiver — based on the current spheroidal (i.e., heliocentric) model of the earth2 — is essentially that of a straightforward trigonometric analysis that assumes a spherical earth, its mean radius being equal to the Mean Radius of the Three Semi-axes (\(R_1\)) of the (alleged) oblate ellipsoid of revolution or oblate spheroid.3

A section of the (allegedly spheroidal4) earth is depicted in Figure 1, with point \(P_0\) representing the mean sea level (MSL) position of an observer or detector at the (so-called) North Pole.5 (Whereas the consideration of any elevation of the observer or detector is not essential for illustrating the basic geometry of subject visibility or detectability on a curved surface, the horizon point (\(P_{horizon}\)) is necessarily coincident with the (mean sea level) position of \(P_0\) as shown in Figure 1.6)


A B C D O (Center of the allegedly spheroidal earth) P 4 (Subject at equator) NOTE: OBJECT SIZES AND DISTANCES HAVE BEEN SCALED FOR ILLUSTRATIVE PURPOSES. 90° 45° 60° 30° Equatorial radius (a) Polar radius (b) Allegedly spheroidal earth P 3 P 2 P 1 P 0 (Mean sea level position of observer or detector at so-called North Pole) 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 collinear with subject of observation or detection

Figure 1. Basic spheroidal earth curvature geometry.


The trigonometric analysis begins with the (allegedly curvilinear) distance \(s_i\), i.e., arc (⌒) \(P_1P_i \) \((i=1,2,3,4)\) — as shown in Figure 1 — of the distant subject from the observer or detector.7

More specifically, the analysis focuses on calculating the distance \(d_{normal}\) (referred to as \(d_n\)), or (in terms of the present example) \(d_{n_i}\) \((i=1,2,3,4)\) represented by the component of \(s_i\) (allegedly) below and normal to the observer or detector’s horizon line of sight (i.e., the ray in Figure 1). Hence, for (allegedly curvilinear) distances \(s_1,\, s_2,\,s_3,\,s_4\) represented by ⌒\(\,P_0P_1,\) ⌒\(\,P_0P_2,\) ⌒\(\,P_0P_3,\) ⌒\(\,P_0P_4\) respectively, component distances \(d_1,\, d_2,\, d_3,\, d_4\) are represented by lines \(P_1A,\,P_2B,\,P_3C,\,P_4D\) respectively.

The greatest arc length in Figure 1 is clearly ⌒\(\,P_0P_4\) representing a quarter of the (alleged) polar circumference of the (allegedly spheroidal) earth. In this case, the length of corresponding component \(d_4\) (allegedly) below and normal to the observer or detector’s (zero elevation) horizon line of sight, is simply the equivalent of the (alleged) polar radius \(b\), line \(P_4D\) being tangent to the (allegedly spheroidal) earth at the equator.

Obviously, the distances involved in the debate over the large-scale structure of the earth’s surface are a small fraction of what is being illustrated in Figure 1 to convey the essential priniple.

Finally, the reader is advised that the terms a curvature drop (of so many feet or meters) or over the curve (by so many feet or meters) that may be used in this website and elsewhere, relate to what has been described as \(d_n\) above.


— FINIS —


Further to the above depiction of (allegedly spheroidal earth) curvature, the detailed trigonometric analysis deriving the basic curvature equation is presented on our web page titled, Basic (Allegedly Spheroidal) Earth Curvature Equation.


  1. For visible light, the subject is typically a distant mountain, island, promontory, structure or marine vessel; for radar illumination or radio frequency (RF) communication, the subject is typically an aircraft or RF transmitter.↩️

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

  3. Loc. cit.↩️

  4. Figure 1 depicts an allegedly spheroidal earth given that Equatorial Radius (\(a\)) and Polar Radius (\(b\)), each of which have a different value, are specifically identified. Subsequent diagrams on our web pages under the CURVATURE folder supporting the derivation of equations relating to the (alleged) curvature of the earth, depict an allegedly spherical earth, identifying a single-valued Mean Radius of the Three Semi-axes (\(R_1\)).↩️

  5. Whereas the large-scale structure of the earth’s surface is that of a circular plane, the so-called North Pole is in reality the center of that circular plane and is referred to as the North Center. Obviously, the (alleged) South Pole cannot exist on a circular plane, the southern extremum of the earth’s surface being referred to as the Southern Circumference, sometimes referred to as the Great Southern Circumference.↩️

  6. Positional elevation is factored into the analyses presented on our web pages titled, W/ Elevated Observer and W/ Elevated Observer and Subject. For a technical description of elevation or orthometric height (\(H\)), see (Allegedly Spheroidal) Earth Mensuration.↩️

  7. For an observer or detector at elevation, the (allegedly curvilinear) distance would be \(s\,^\prime\) (where typically, \(s\,^\prime < s\)), i.e., arc (⌒) \(P_{horizon},P_n \) \((n=1,2,3,4)\) — as shown in Figure 1 — of the distant subject from the observer or detector’s horizon. But to illustrate the basic methodology, the elevation is zero, i.e., mean sea level, whereby \(P_0\) and \(P_{horizon}\) are coincident.↩️


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DATE 2022-APR-29