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

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 ($H_1$)¹ 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 $d_n$.

From Figure 1, it is obvious that for a given distance between the observer or detector and the subject of observation or detection, the magnitude of $d_n$ decreases with increasing orthometric height $H_1$.

Figure 1. Analysis of the curvature (w/ elevated observer) of the (allegedly spheroidal) earth.

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The (Alleged) Linear (i.e., Line of Sight) Distance $d_h$ to the (Allegedly Spheroidal) Earth Horizon

In Figure 1, angle $\displaystyle{(∠)}\,\displaystyle{P_1(H_1)P_{horizon}O}$ is a right angle $\displaystyle{(∟)}$ since line segment $P_1(H_1)P_{horizon}$ (the magnitude of which is represented by $d_h$) is tangent to radial line $OP_{horizon}$. Hence, triangle $\displaystyle{(Δ)}\,P_1(H_1)P_{horizon}O$ is a right triangle $\displaystyle{(◿)}$.

Therefore, by Pythagoras’ theorem,

$$[P_1(H_1)P_{horizon}]^2+[OP_{horizon}]^2$$

$$=[OP_1(H_1)]^2,$$

and hence,

$$P_1(H_1)P_{horizon}=$$

$$\sqrt{[OP_1(H_1)]^2-[OP_{horizon}]^2}.$$

Whereas from Figure 1, $P_1(H_1)P_{horizon}=d_h$, $OP_1(H_1)=R_1+H_1$, and $OP_{horizon}=R_1$, where $R_1$ is the (alleged) mean radius of the (allegedly spheroidal) earth, then

$$d_h=\sqrt{(R_1+H_1)^2-R_1^2},$$

and hence,

$$d_h=\sqrt{2R_1H_1+H_1^2}.$$

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The (Alleged and Allegedly) Curvilinear (i.e., Surface) Distance $s_h$ to the (Allegedly Spheroidal) Earth Horizon

It is well known that arc length is equal to the radius times the angle (in radians) subtended by the arc. Hence, from Figure 1,

$$s_h=R_1𝜓,$$

where $𝜓=\displaystyle{∠}\,\displaystyle{P_1(H_1)OP_{horizon}}.$

Whereas (from above) triangle $\displaystyle{(Δ)}\,P_1(H_1)P_{horizon}O$ is a right triangle $\displaystyle{(◿)}$, then

$$\textrm{cos}\,𝜓=\left(\frac{R_1}{R_1+H_1}\right).$$

Hence,

$$s_{h}=$$

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

Note:The respective magnitudes of $d_h$ and $s_h$ are almost the same for the elevations or altitudes considered in the debate over the large-scale structure of the earth’s surface. Hence, on the web pages under the CALCULATORS folder, the calculated values for $d_h$ and $s_h$ are expressed to eight decimal places to indicate the minor but actual differences in their respective magnitudes.

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Derivation of the (Allegedly Spheroidal) Earth Curvature Equation W/ Elevated Observer

To repeat the concluding statement from the Introduction above: “From Figure 1, it is obvious that for a given distance 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., $P_2(φ_2,λ_2)$ or simply $P_2]$, the magnitude of $d_n$ decreases with increasing orthometric height $H_1$.” In other words, 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 $(P_2)$. 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 subject of observation or detection is therefore represented by the (allegedly curvilinear) distance or arc (⌒) $s-s_h$, i.e., arc (⌒) $P_{horizon}P_2$.

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 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 $R_1$, 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.² 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, 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 $P_2$ is again the position of the subject of observation or detection, 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, $$θ-𝜓=\;\scriptstyle{∠}\,\displaystyle{P_{horizon}OP_2.}$$

Hence,

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

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 ⌒$P_{horizon}P_2$ are represented by line $P_{horizon}α$ (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 (▭) $P_{horizon}αP_2β$, the line from the subject’s position $P_2$ with (and perpendicular to) line $OP_{horizon}$ intersects line $OP_{horizon}$ at $β$, lines $P_{horizon}β$ and $P_2β$ being equal and parallel to (⋕) lines $P_2α$ and $P_{horizon}α$ respectively; hence, triangle $P_2βO$ is a right triangle (◿).

In ◿$P_2βO$,

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

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

Hence,

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

and therefore,

$$Oβ=R_1\textrm{cos}\,(θ-𝜓).$$

But $R_1$ is also equal to $OP_{horizon}\,\textrm{(or}\,Oβ+P_{horizon}β)$, and whereas $P_{horizon}β⋕P_2α\,\textrm{(or}\,d_n)$, then $d_n=R_1-Oβ$.

Hence,

$$d_n=R_1-R_1\textrm{cos}\,(θ-𝜓),$$

or,

$$d_n=R_1[(1-\textrm{cos}\,(θ-𝜓)].$$

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 $P_1(H_1)$, the curvature equation in terms of the distance $d_n$ from $P_2$ normal to the observer or detector’s horizon line of sight, is:

$$d_{n}=$$

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

— FINIS —

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

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

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

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REVISION	0	1
DATE	2022-JUN-06	2023-NOV-09