Geodesic line . In Mathematics , Algebra , Geometry and more specifically, Metric space , Analytical geometry and Topology , it is said to be the smallest surface line connecting two unequal points on the surface of a sphere , thus replacing the Cartesian idea of the line by the arc of the geodetic line connecting the points in question.
This type of locus is very useful, since planets and stars usually have a spherical shape, so the determination of more realistic distances on their surfaces can be solved by defining the orthodrome or geodesic line.
Summary
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- 1
- 2 Calculation of the length of the orthodrome.
- 3 See also.
- 4
Definitions.
Let be a solid sphere of center O and radius r as shown in the following figure:
and two points A = (x 0 ; y 0 ) and B = (x 1 ; y 1 ) located on the surface of the sphere, the smallest arc AB that connects both points on the sphere is said to be the geodetic line u orthodroma (in red color).
Calculation of the length of the orthodrome.
Let the perfect sphere be the center at the origin of coordinates O and radius r and two surface points and where are the lengths of their respective points with respect to the x axis ; are the corresponding latitudes of A and B .
Translated into Euclidean coordinates in space, the points and ; the amplitude of the arc corresponding to that segment is:
Assuming that the arcsine function returns the amplitude of the angle in radians , the length of the arc between two points given their spherical coordinates remains:
