Orthodromy


The orthodromy indicates the the shortest way between two points of a Sphère, i.e. the arc of Grand circle which passes by these two points. For the navigators an orthodromic road thus indicates the shortest road on the surface of the terrestrial sphere between two points.

On a chart in Projection of Mercator, orthodromy is not represented by a straight line but by a curved line. Indeed, on a chart in projection of Mercator which preserves the angles but not the distances, only the Loxodromie will be represented by a straight line.

On a chart in gnomonic Projection, orthodromy is represented by a line; the charts in gnomonic projection are used for navigation in high latitudes.

On a short distance, one can confuse orthodromy and loxodromic curve. The distinction becomes important at the time of the oceanic crossings for courses E-W (and conversely) and especially to the high latitudes (see the formula giving the profit of orthodromy on the loxodromic curve).

It will be noted that the curve of orthodromy on the Mercator chart is open towards the equator (either curved towards the pole, north in the northern, southern hémishère in the southern hemisphere). This means that for East-West crossing (and conversely) one will approach the pole. The point of inflecting of orthodromy is called the vertex. The determination of the latitude of the vertex (maximum latitude attack) is a size interesting to determine to prepare a maritime circumpolar crossing (in the southern hemisphere consequently, for example of Tasmanie to Cape Horn) where it is important not to gain too much in latitude because of the danger of the ices and the ice-barrier. The road chosen then will break up into a section of orthodromy until the extreme latitude which one does not want to exceed, then a section of loxodromic curve to this latitude and finally another section of orthodromy to go up until destination.

Calculs of orthodromy between has (\ varphi_A, G_A) and B (\ varphi_B, G_B) with angles in degrees and distance in marine miles:

  • orthodromic distance : M \, :

M = 60 \ arccos (\ sin \ varphi_A \ sin \ varphi_B + \ cos \ varphi_A \ cos \ varphi_B \ cos (G_B - G_A))\,

  • profit outdistances some compared to the loxodromic curve : m - M \, :

m - M = \ frac {m^3} {24. (3438) ^2} \ sin^ 2 R _v \ tan^2 \ varphi_m \,
with:
m \, the loxodromic distance
M \, the orthodromic distance
R_v \, the true road loxodromic
\ varphi_m \, the average latitude ( \ varphi_m = \ frac {\ varphi_A + \ varphi_B} {2} \, )


  • initial road R_o \, (angle of the initial section of road):
\ operatorname {cotan} R_o = \ frac {\ sin \ varphi_A} {\ tan (G_B - G_A)} - \ frac {\ cos \ varphi_A \ tan \ varphi_B} {\ sin (G_B - G_A)}


  • latitude of the vertex \ varphi_v \, :
\ cos \ varphi_v = \ sin R_o \ cos \ varphi_A \,

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