Calculation of the (alleged) distance \(d_{n(K = 5/4)}\)¹ (\(K=5/4\) meaning w/ maximum tropo[spheric] optical refraction) of a subject of observation (i.e., the highest elevation of the island, promontory, or mountain, or the top of the marine or shoreline structure, or marine vessel, or the top of the land structure at elevation) below the (alleged) spheroidal earth horizon—more specifically, (allegedly) below and normal to the observer’s or optical instrument’s (alleged) horizon line of sight.²

Nota Bene

As stated in the first three paragraph of our web page titled, Introduction to the (Allegedly Spheroidal) Earth Calculators:

The purpose of the (allegedly spheroidal) earth calculators is to demonstrate to readers carrying out and and analyzing measurable planar earth field observations in their own respective regions (or further abroad), the fallacy of the (allegedly spheroidal) earth and hence, the heliocentric paradigm. To that end, readers are encouraged to acquire high quality cameras and other optical equipment to document such observations. (Obviously, professional photographers, film makers, and surveyors will already have a plethora of equipment at their disposal.)

In the case of radar or radio freqency (RF) observations, e.g., radars or radio frequency tranceivers in contact with aircraft or radio frequency transmitters, the equipment involved typically operates in highly regulated, specialized circumstances, being outside the purview of most people. The exception to this of course, would be the licensed amateur radio operators, typically referred to as “hams” (a pejorative term stemming from nineteenth century telegraphy).

Readers are strongly advised to discuss such observations with people who have to work in safety critical, real-world industries or environments where errors are unacceptable; for example, those who work in civil enginering, aviation, shipping, and telecommunications. Discussions with others (including academics) adhering to the existing heliocentric paradigm but not otherwise responsible in the aforesaid safety-critical fields, however, are unlikely to be productive.

Such calculations are of paramount importance in REVEALING THE PLANAR LARGE-SCALE STRUCTURE OF THE EARTH’S SURFACE.

To determine whether the subject of observation is even (allegedly) over the (alleged) spheroidal earth horizon, it is necessary to calculate the (alleged) distance to the (alleged) spheroidal earth horizon. Both the (alleged) linear (i.e., line of sight) distance \(d_{h(K = 5/4)}\) and the (alleged and allegedly curvilinear) surface (i.e., geodesic) distance \(s_{h(K = 5/4)}\) are calculated, however, it is the (alleged and allegedly curvilinear) surface distance \(s_{h(K = 5/4)}\) that is operative in respect of calculating the (alleged) distance \(d_{n(K = 5/4)}\) of a subject of observation (allegedly) below and normal to the observer’s or optical instrument’s (alleged) horizon line of sight. Nevertheless, for the distances involved in practical field work, the calculated values for \(d_{h(K = 5/4)}\) and \(s_{h(K = 5/4)}\) are nearly the same as will be demonstrated in CALCULATION A below.

EQUATION A-1: \( d_{h(K)}\)

The (alleged) linear (i.e., line of sight) distance \(d_{h(K)}\) to the (alleged) spheroidal earth horizon is calculated using the following expression:

\[d_{h(K)}=\sqrt{2KR_1H_1+H_1^2}\]

where,

\(R_1\) is the (alleged) mean radius or Mean Radius of the Three Semi-axes (\(R_1\)) of the earth \(=6371008.7714\;\textrm{m}\),³ or (expressed in imperial units):\[R_1=\frac{6,371,008.7714 \textrm{ m}}{0.3048 \textrm{ m ft}^{-1}},\]

\(H_1\) is the orthometric height of the observer or optical instrument,⁴ and

\(K\) is the Effective Earth Radius Factor (EERF);⁵ if expressed in terms of the coefficient of tropo[spheric] refraction \(k\), then \[K=\frac{1}{1-k}.\]

EQUATION A-2: \( s_{h(K)}\)

The (alleged and allegedly curvilinear) surface (i.e., geodesic) distance \(s_{h(K)}\) to the (alleged) spheroidal earth horizon is calculated using the following expression:

\[s_{h(K)}=\]

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

CALCULATION A: \( d_{h(K=5/4)}\) and \(s_{h(K=5/4)}\)

EQUATION B: \( d_{n(K)}\)

Finally, the (alleged) distance \(d_{n(K)}\) (w/o tropo[spheric] refraction) of a subject of observation (i.e., the highest elevation of the island, promontory, or mountain, or the top of the marine or shoreline structure, or marine vessel, or the top of the land structure at elevation) below the (alleged) spheroidal earth horizon—more specifically, (allegedly) below and normal to the observer’s or optical instrument’s (alleged) horizon line of sight—is calculated using the following expression:⁶

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

where,

\(s\) is the (allegedly curvilinear) distance of the mean sea level (MSL) or geodetic reference point for the observer or optical instrument from the mean sea level (MSL) or geodetic reference point for the subject of observation, i.e., the island, promontory, or mountain, or the the marine or shoreline structure, or marine vessel, or the land structure at elevation, and

\(H_2\) is the orthometric height of the highest elevation of the island, promontory, or mountain, or the top of the marine or shoreline structure, or marine vessel, or the top of the land structure at elevation.

CALCULATION B: \( d_{n(K=5/4)}\)

— FINIS —