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.

220px-Sphere_halve.png
figure 1

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.

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’ ).
  • Δdist = Δlong x Cos(lat), and inversely
  • Δlong = Δdist ÷ Cos(lat)
a-radian.gif
figure 2

 

 

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.

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

~~ Cris    

19MAY2016

PS. Thank you, Paul, for all your help!

 

 

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