Global Distances

by Nikolai V. Shokhirev

Geodesic

A geodesic is "the closest path between two points on the surface of some object". It depends on the shape of this object.

Sphere

A sphere is defined as "the set of points an equal distance (called the radius) from a single point called the center". The shortest path is the arc of the great circle that containing the two points. A great circle is the intersection of a plane and a sphere where the plane also passes through the center of the sphere.

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).

Spherical Polar Coordinates

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 .

http://www.astro.cf.ac.uk/undergrad/module/PX3104/tp3/node8.html

Geographic coordinates

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/ ).

DMS

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".

Ellipsoid

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 .

Earth

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.

Radius length

In the polar coordinates the ellipsoid equation is ( ):

The equatorial radius is a ( in the above equation) and the polar radius is b ( or )

Earth dimensions

(from http://www.geocities.com/Athens/Olympus/4844/cosmo.html )

CONSTANT	NUMERIC VALUE
Equatorial Radius	3963.19245606 mi	6378.14000000 km
Polar Radius	3949.90462476 mi	6356.75528816 km
Mean Radius	3958.73926185 mi	6370.97327862 km

Simplified calculations

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:

Geographic coordinates of towns and cities (except USA) http://www.calle.com/world/index.html
Geographic locations (USA, Canada, other Countries): http://www.bcca.org/misc/qiblih/latlong.html

Program

The program "Global Distance" for Windows can be downloaded here.

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