DEVELOPMENT: DISTANCE FORMULAE; 



SECTION 3. DISTANCE COMPUTATIONS AND CONVERSIONS; AZIMUTHS 



If we are given two points Pi((;6i, A.,), PjC^zj ^2) o" 'he ellipsoid of reference as shown in 

 Figure 10, we may compute distances and azimuths according to known or given elements. That 

 is we may compute the geographic coordinates of the point PjCtjijjAj) if we know the geographic 

 coordinates of Pi (^,,Ai) the distance between Pj and Pj, and the azimuth from P, to Pj. This is 

 the direct problem and the one most important in Geodesy relative to establishing triangulation 

 control nets. If the coordinates of both Pi and Pj are given, the distance between them and the 

 azimuths can be computed. This is the inverse problem, and the one concerned primarily in 

 electronic positioning systems as Loran. 



Since there are several possible curves connecting the points Pi and Pj on the ellipsoid 

 along which distances would differ very little, for instance — the geodesic, the normal sections, 

 the great elliptic arc, the curve of alinement, etc. — criteria for selection would be simplicity in 

 computations relative to required accuracy. Also to be considered are other useful geometric 

 quantities associated with the configuration and expressible in terms of common computational 

 parameters. (See Figure 11). 



The shortest distance is always the geodesic or the geodetic line between Pi and Pj. It is 

 usually a space curve (that is it has a first and second curvature at each point). For instance on 

 the reference ellipsoid, the equator and the meridians are the only plane geodesies, [8]. 



Now in Figure 10, the point Po(0o' Ag) is the vertex of the great elliptic arc, that is Po is 

 the point where the great elliptic arc is orthogonal to a meridian. The goedesic, or geodetic line, 

 between Pi and Pj also has a vertex where it is orthogonal to a meridian. Since the geodesic is 

 a space curve and climbs nearer to the ellipsoid pole, T^, than any of the other representative 

 curves (if Pi and P-. were ends of a diameter of the equator, the geodesic would be the elliptic 

 meridian through Pi and Pj since it is shorter than the equator), the vertex of the geodesic is 

 closer to To than is Po. Unfortunately the geographic coordinates of the geodesic vertex cannot 

 be expressed simply in terms of the geographic coordinates of Pi and P^, hence an approximation 

 scheme, usually iterative, is used. [9] The computations are usually quite lengthy for long 

 lines. Many schemes and formulae have been devised to approximate the geodesic and studies 

 have been made comparing them. [21] The geodetic line is of most interest to the geodesist 

 proper, since he is primarily concerned with closure on a particular ellipsoid of reference of large 

 arcs and areas of triangulation, hence the geodesic or geodetic line and geodetic azimuths on the 

 ellipsoid are consonant with his mathematical model. 



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