the shortest distance between two points on the earth's surface is minute of arc.
the shortest distance between two points on the earth's surface is minute of arc.
the shortest distance between two points on the earth's surface is?
minute of arc
a tunnel between points on earth is defined by a line through three-dimensional space between the points of interest.









































the great circle chord length may be calculated as follows for the corresponding unit sphere:









































the tunnel distance between points on the surface of a spherical earth is









































formula_48. for short distances (formula_49), this underestimates the great circle distance by formula_50.









































section::::ellipsoidal-surface formulae.









































an ellipsoid approximates the surface of the earth much better than a









































sphere or a flat surface does. the shortest distance along the surface









































of an ellipsoid between two points on the surface is along the
geographical distance
for example, consider the measurement of distance between two places on the surface of the earth. this is a simple, familiar example of spherical geometry. because the surface of the earth is two-dimensional, points on the surface of the earth can be specified by two coordinates — for example, the latitude and longitude. specification of a metric requires that one first specify the coordinates used. in our simple example of the surface of the earth, we could choose any kind of coordinate system we wish, for example latitude and longitude, or x-y-z cartesian coordinates. once we have chosen a specific coordinate system, the numerical values of the coordinates of any two points are uniquely determined, and based upon the properties of the space being discussed, the appropriate metric is mathematically established too. on the curved surface of the earth, we can see this effect in long-haul airline flights where the distance between two points is measured based upon a great circle, rather than the straight line one might plot on a two-dimensional map of the earth's surface. in general, such shortest-distance paths are called "geodesics". in euclidean geometry, the geodesic is a straight line, while in non-euclidean geometry such as on the earth's surface, this is not the case. indeed, even the shortest-distance great circle path is always longer than the euclidean straight line path which passes through the interior of the earth. the difference between the straight line path and the shortest-distance great circle path is due to the curvature of the earth's surface. while there is always an effect due to this curvature, at short distances the effect is small enough to be unnoticeable.
expansion of the universe