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Introduction to the (Allegedly Spheroidal) Earth Curvature Equation w/ Elevated Observer and Subject

The following introductory statement was provided on our preceding web page¹ with respect to the (Allegedly Spheroidal) Earth Curvature Equation w/ Elevated Observer (the subject of observation being at mean sea level):

On the (allegedly spheroidal) surface of the earth (see Figure 1), note the mean sea level (MSL) geographic position $P_1(φ_1,λ_1)$ of an observer or detector (henceforth referred to as $P_1$), where the Greek letters $φ$ (phi) and $λ$ (lambda) refer to the position’s latitude and longitude respectively. But whereas the observer or detector is at some elevation or orthometric height ($H_1$)² above MSL, then note also the orthographic height position $P_1(φ_1,λ_1, H_1)$ of the observer or detector, henceforth referred to as $P_1(H_1)$. In relation to position $P_1(H_1)$, 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 $P_2$ 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 $P_2$, said line intersecting said ray at point $α$. The curvature equation w/ elevated observer — in terms of the extent to which $P_2$ is (allegedly) hidden from $P_1(H_1)$ by the (alleged) curvature of the earth — is essentially the equation for determining the (alleged) magnitude of $d_n$.

On our preceding web page,³ it was determined that for an observer or detector at orthographic height position $P_1(H_1)$, the curvature equation in terms of the distance $d_n$ from the mean sea level (MSL) position of the subject of observation or detection $P_2$ normal to the observer or detector’s horizon line of sight, is:

$$d_{n}=$$

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

where,

$R_1$ is the (alleged) mean radius or Mean Radius of the Three Semi-axes ($R_1$) of the (allegedly spheroidal) earth,⁴

$s$ is the (allegedly curvilinear) distance between the MSL position of the observer or detector $P_1$ and the MSL position of the subject of observation or detection $P_2$, and

$H_1$ is the orthographic height of the observer or detetctor at orthographic height position $P_1(H_1)$.

Whereas the subject of observation or detection (or a part thereof) is typically at some height or elevation, then a more general equation would include $H_2$, i.e., the orthographic height of the subject of observation or detection at orthographic height position $P_2(H_2)$; see Figure 1 as well as the relevant detail depicted in Figure 2. The reader will notice that the distance $d_n$ is thereby reduced by a component of $H_2$, the specific equation thereof being derived below.

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Figure 1. Analysis of the curvature (w/ elevated observer and subject) of the (allegedly spheroidal) earth. (Note: Object sizes and distances have been scaled for illustrative purposes.)

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Figure 2. Re-annotated enlargement of subject of observation area, $P_{2}(φ_{2},λ_{2},H_{2})$ from Figure 1. (Note: Object sizes and distances have been scaled for illustrative purposes.)

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Derivation of the (Allegedly Spheroidal) Earth Curvature Equation w/ Elevated Observer and Subject

From Figure 1 and Figure 2 above, it is obvious that for a given (allegedly curvilinear) distance $s$ between the observer or detector $[$i.e., $P_1(φ_1,λ_1, H_1)]$ or simply $P_1(H_1)]$ and the subject of observation or detection $[$i.e., $P_2(φ_2,λ_2,H_2)$ or simply $P_2(H_2)]$, the magnitude of $d_n$ decreases with increasing orthometric heights $H_1$ and $H_2$. In other words: (a) the greater the observer or detector’s orthometric height $H_1$, the closer the observer or detector’s horizon point $(P_{horizon})$ is to the subject of observation or detection at $P_2(H_2)$, and (b) the greater the subject of observation or detection’s orthometric height $H_2$, the closer the subject of observation or detection is to the observer or detector’s horizon line of sight. Whereas the observer or detector at $P_1(H_1)$ has a direct line of sight to the observer or detector’s horizon point $(P_{horizon})$, the effective amount of (alleged) curvature of the (allegedly spheroidal) earth between the observer or detector and the (MSL position) of the subject of observation or detection is therefore reduced, being represented by the (allegedly curvilinear) distance or arc (⌒) $s-s_h$, i.e., arc (⌒) $P_{horizon}P_2$. Whereas the subject of observation or detection at the orthometric height position $P_2(H_2)$ is obviously closer to the observer or detector’s horizon line of sight than the corresponding MSL position $P_2,$ the effective amount of (alleged) curvature of the (allegedly spheroidal) earth between the observer or detector and the (orthometric height position) of the subject of observation or detection is therefore further reduced, being represented by the (allegedly curvilinear) segment $P_{horizon}u$ of the (allegedly spheroidal) earth, where $u$ is the intersection of the line from $P_2(H_2)$ parallel to the observer or detector’s horizon line of sight with the (allegdly spheroidal) earth, and $v$ is the intersection of the line from $u$ below and normal to the observer or detector’s horizon line of sight. Hence, in rectangle (▭) $vα\,'P_2(H_2)u$, $uv=P_2(H_2)α\,'=d_n$.⁵

The (alleged) distance $d_n$ is proportional to the amount of (alleged) curvature of the (allegedly spheroidal) earth between the observer or detector’s horizon point $(P_{horizon})$ and the (MSL) position of the subject of observation or detection as represented by the (allegedly curvilinear) distance arc (⌒) $s-s_h$, i.e., arc (⌒) $P_{horizon}P_2$. 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 $R_1$. Hence, the (allegedly curvilinear) distance $s-s_h$ subtends the angle $θ-𝜓$, the magnitude of which is,

$$θ-𝜓=\frac{s-s_h}{R_1}\textrm{rad}.$$

Referring again to Figure 1 and Figure 2, if $O$ is the center of the (allegedly) spheroidal earth, and $P_1(H_1)$ is (again) the orthometric height position of the observer or detector — at a distance $R_1+H_1$ from $O$, whereby $R_1$ is represented by lines $OP_1$ and $OP_{horizon}$, and if $P_2(H_2)$ is (again) the orthometric height position of the subject of observation or detection — at a distance $R_1+H_2$ from $O$, whereby $R_1$ is also represented by line $OP_2$, then the angle ($\scriptstyle{∠}$) $θ-𝜓$ subtended by ⌒ ($s-s_h$), i.e., ⌒ $P_{horizon}P_2$, can also be expresed as, $$θ-𝜓=\;\scriptstyle{∠}\,\displaystyle{P_{horizon}OP_2(H_2).}$$

Hence,

$$\scriptstyle{∠}\,\displaystyle{P_{horizon}OP_2(H_2)\\\ =\frac{s-s_h}{R_1}\textrm{rad}.}$$

Referring once again to Figure 1 and Figure 2, $α$ is the intersection of the line from the subject’s MSL position $P_2$ with (and normal to) the observer or detector’s horizon line of sight, whereas $α\,'$ is the intersection of the line from the subject’s orthographic height position $P_2(H_2)$ with (and normal to) the observer or detector’s horizon line of sight. In rectangle (▭) $P_{horizon}α\,'P_2(H_2)β\,'$ (see Figure 1), the line from the subject’s position $P_2(H_2)$ with (and perpendicular to) line $OP_{horizon}$ intersects line $OP_{horizon}$ at $β\,'$, lines $P_{horizon}β\,'$ and $P_2(H_2)β\,'$ being equal and parallel to (⋕) lines $P_2(H_2)α\,'$ and $P_{horizon}α\,'$ respectively; hence, triangle $P_2(H_2)β\,'O$ is a right triangle (◿).

In ◿$P_2(H_2)β\,'O$,

$$\textrm{cos}\,\scriptstyle{∠}\,\displaystyle β\,'OP_2(H_2)\,\textrm{or cos}\,(θ-𝜓)$$

$$=\dfrac{Oβ\,'}{OP_2(H_2)}\,\left(\textrm{or}\;\dfrac{Oβ\,'}{R_1+H_2}\right).$$

Hence,

$$R_1+H_2=\frac{Oβ\,'}{\textrm{cos}\,(θ-𝜓)},$$

and therefore,

$$Oβ\,'=(R_1+H_2)\,\textrm{cos}\,(θ-𝜓).$$

But $R_1$ is also equal to $OP_{horizon}\,\textrm{(or}\,Oβ\,'+P_{horizon}β\,')$, and whereas $P_{horizon}β\,'⋕P_2(H_2)α\,'\,\textrm{(or}\,d_n)$, then $d_n=R_1-Oβ\,'$.

Hence,

$$d_n=$$

$$R_1-[(R_1+H_2)\,\textrm{cos}\,(θ-𝜓)],$$

And whereas,

$$θ=\frac{s}{R_1}\textrm{rad},$$

and

$$𝜓=\textrm{cos}^{-1}\left(\frac{R_1}{R_1+H_1}\right)\textrm{rad},$$

then, for an observer or detector at orthographic height position $P_1(H_1)$ and a subject of observation or detection at orthometric height $P_2(H_2)$, the curvature equation in terms of the distance $d_n$ from $P_2(H_2)$ normal to the observer or detector’s horizon line of sight, is:

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

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Tropo[spheric] Refraction and the Effective Earth Radius Factor (EERF)

Within the spheroidal earth paradigm, modifying the above equation in respect of tropo[spheric] refraction has the effect of ostensibly lowering the magnitude of $d_n$ for any given distance between the observer and the observed.

Hence, the purpose of the REFRACTION section is to (a) present to the reader, the underlying rationale for (spheroidal earth) geodesists using (alleged) tropo[spheric] refraction to explain (in many cases) the visibilty of subjects of observation that should otherwise be below the horizon (i.e., below and normal to the observer’s horizon line of sight), and (b) give the benefit of the doubt to the spheroidal earth paradigm by allowing readers to assume the impact of maximum tropo[spheric] refraction on their field observation calculations.⁶

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);⁷ 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:

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

Similarly, the final expressions for the (alleged) linear distance $d_h$ and (alleged and allegedly) curvilinear distance $s_h$ to the horizon (derived on our previous web page⁸) are provided as follows:

$$d_h=\sqrt{(KR_1+H_1)^2-KR_1^2} ,$$

and

$$s_{h}=$$

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

— FINIS —

For further information concerning tropo[spheric] refraction and rationalizations for the effective earth radius factors used in this website, the reader is referred to our REFRACTION section, beginning with our web page titled, Refraction Overview The above final expression for $d_h$, $s_h$, and $d_n$ are utilized for practical field calculations in our CALCULATORS section; see our webpage titled, Introduction to the (Allegedly Spheroidal) Earth Calculators.

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

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

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

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

5. From Figure 1 and Figure 2, the reader will notice the increased line of sight distance $P_1(H_1)α$ to $P_1(H_1)α\,'$. The magnitude of that increase is not determined nor relevant to this aspect of the paradigmatic argument, said aspect being focused exclusively on determining the magnitude of $d_n$.↩️

6. See our web page titled, Refraction Overview.↩️

7. 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).↩️

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

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REVISION	0	1
DATE	2022-JUL-03	2022-NOV-09