d = r · α [rad] = 6385,112 MatheAss - Distances on the sphere

MatheAss 9.0 - Geometry 3D

Distances on the sphere

The distance between two points P1 and P2 on a sphere is calculated by combining some functions of MatheAss.

The GPS data (latitude and longitude) of the two points are entered. Together with the radius of the sphere, they give the polar coordinates of the points.

These are converted into their Cartesian coordinates with the corresponding function of MatheAss . The result supplies the coordinates of their position vectors in a coordinate system with the center of the sphere as the origin.

With its scalar product one obtains the angle α between the two vectors and finally, as the product of α in radians together with the spherical radius, the length of the circular arc on the sphere.

Example:

As the crow flies between Alexanderplatz in Berlin and City Hall in New York.
The earth is idealized as a sphere with a radius of 6371 km.


GPS decimal
¯¯¯¯¯¯¯¯¯¯¯
  Berlin : 52.523403,  13.411400
New York : 40.714268, -74.005974

GPS dms
¯¯¯¯¯¯¯
  Berlin : 52°31'24.2508"N, 13°24'41.0400"E
New York : 40°42'51.3648"N, 74° 0'21.5064"W

Polar coordinates
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  Berlin : (6371 | 13.411400° | 52.523403°)
New York : (6371 |-74,005974° | 40,714268°)

Cartesian coordinates
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
  Berlin : (3770.6450 | 899.08721 | 5056.0379)
New York : (1330,5796 |-4642,1091 | 4155,7216)

Position vectors
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
   ->  ⎧   3770,645048 ⎫   ->  ⎧ 1330,57957098 ⎫
   a = ⎪ 899,087213119 ⎪   b = ⎪-4642,10910614 ⎪
       ⎩ 5056.03788605 ⎭       ⎩ 4155.72160425 ⎭

               ->  ->
   α = arccos( a · b / r2 ) 
     = 1,002215 [rad] = 57,422692°

Distance
¯¯¯¯¯¯¯¯
   d = r · α [rad] = 6385,112

Entering GPS data

GPS data can be entered in decimal degrees as well as in degrees, minutes and seconds.
Both representations are output.

The program is therefore also suitable for converting decimal degrees into degrees, minutes and seconds (dms) and vice versa.

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