by Nikolai V. Shokhirev
The length of this arc is . Here a is the angle of the arch, r is the radius of the sphere, and f is the conversion factor that depends on the angle unit of measure. The factor f = 0.0174532925199433 per degree and f = 1 if the angle is measured in radians ( radian, 3.141592653589793238462643).
For the description of a point on a sphere surface are used the spherical polar coordinates ( ). They are related to the ordinary (x, y, z) coordinates:
so that .
In mathematics the angles vary in the following regions: .
The angle is called "the Longitude". The positive values are marked by "E" after the angle (e.g 118.10 E, "E" stands for "East") and the negative values are marked by "W" (e.g. 46.30 W, "W" stands for "West"). The curves with the same longitude are called "the meridians". The zero meridian miraculously passes through Greenwich (Britain).
The second geographic angle is "the Latitude": . The zero latitude corresponds to the equator. The positive values of the latitude are marked with "N" ("N" stands for "North") and the negative values are marked by "S" ("S" stands for "South"). (see a nice picture in http://astronomy.swin.edu.au/~pbourke/geometry/sphere/ ).
Very often angles are measured not in degrees and degree fractions (e.g. 30.5083333333. . . ), but in degrees, minutes and seconds (DMS). One degree = 60 minutes (denoted ' ), one minute = 60 seconds ("). For example: 30030'30".
Its surface is described by the equation:
It means that along the x-axis (when ) the surface is at x = . The surface crosses the y-axis at and the z-axis at .
Its shape is modeled as a flattened ellipsoid: . Here it is assumed that the z-axis is Earth's rotation axis and the xy-plane is the equatorial plane.
In the polar coordinates the ellipsoid equation is ( ):
The equatorial radius is a ( in the above equation) and the polar radius is b ( or )
(from http://www.geocities.com/Athens/Olympus/4844/cosmo.html )
|Equatorial Radius||3963.19245606 mi||6378.14000000 km|
|Polar Radius||3949.90462476 mi||6356.75528816 km|
|Mean Radius||3958.73926185 mi||6370.97327862 km|
For the calculation with the accuracy within several miles one can neglect the Earth's Flattening. The equatorial radius is only 0.11% larger than the Mean radius. The Polar radius is 0.22% shorter than the Mean radius.
For example, the angle between Los Angeles and New York is 35.6360. Using the Mean radius we get the following distance: L = 3962.5 km = 2462.2 mi
The Equatorial radius gives the distance estimation 2.8 mi larger, while the Polar radius gives the distance estimation 5.5 mi smaller. These variations are negligible in comparison with the sizes of the towns. The distance also depends on the definition of the center of a town. In the above calculations the distance between Los Angeles airport (33° 56' N, 118° 24' W) and New York Central Park (40° 47' N, 73° 58' W) was estimated.
The Global coordinates can be found e.g. in:
The program "Global Distance" for Windows can be downloaded here.
© Nikolai Shokhirev, 2001.
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