Why did I say: "The Earth and its mathematics", because all of the math used in flight, or navigational simulators or SPS(games), are based on the Earth and it's dimensions ~ though the NAV simulation represents but a small fraction of the math used in flight simulators.
➀ Having nothing to do with the definition of Nautical Miles (“nMile”), from ???, we know that at the Earth's equatorial circumference is 21,604.2 [or 21,639 or ???] nMile. From High School Math and Geography, we know that a great circle is defined as any plane which passes through the center of the Earth (if the Earth were a sphere) [see figure 1]. We also know that the shortest distance between any two points of the surface of a spherical Earth must be a great circle route. Now, if the Earth were only a sphere.
➁ From High School Math, we know that a circle has 360ℴ (degrees), So if we divide 21604.2 by 360 or 21604.2n.m. /360 = 60.011 ≈ 60n.m at the equator. Remember that the closer you get to the poles , the longitudinal degrees converge, therefore, the change must be accounted for by using the cosine of the latitude.
|The following inverse pair of formulas describe the spherical relationships between the difference in distance (Δdist) along a parallel of latitude (lat) corresponding to a difference in longitude (Δlong), and vice-versa. (The formulas are simply stated here without explanation, but a full explanation of their derivation can be found in the book ‘Astro Navigation Demystified’ ).|
The traditional, historic definition of 1 nMile is 1 minute [= 1/60 degree] of latitude along the Greenwich meridian - thus, by this definition, 60 nMile = 1 degree of latitude; the current standard SI definition: 1 nMile is 1852 meters.
➂ Engineers and Cartographers do use degrees, as anyone who has looked at a map will recognize: latitude and longitude are denoted in degrees, minutes, and seconds; but, we will use radians for angle measurement So what is a radian?
Looking at a circle[figure 2], we get the definition of a Radian as the angle subtended at the center of the circle by an arc of the circumference of the circle whose length is equal to the radius of the circle.
∠XOY = 1 radian, where r = OX = OY = XY
Again from High School Math, we know the total circumference of a circle of radius, r, is 2 ∙ π ∙ r; thus, there are 2 ∙ π Radians in the circumference of a circle.
Since 2 ∙ π radians is equivalent to 360 degrees, a full circle, the relationships are :
- 1 radian = 180 / π degrees, and inversely
- 1 degree = π / 180 radians.
A radian can be written in a variety of ways: 1c, 2 radians, 3r, 4R, 5 ㎭.
The defined and some (approximate) derived relationships between degrees and radians:
360º = 2 ∙ π ㎭
1º = .01745 ㎭
1 ㎭ = 57.2957º
Keep Tuned In, More To Come
PS. Thank you, Paul, for all your help!
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