International Journal
on Marine Navigation
and Safety of Sea Transportation
Volume 5
Number 4
December 2011
519
1 INTRODUCTION
Navigation on the surface of the Earth is possible in
two ways: by orthodrome and loxodrome. Ortho-
drome is a minor arc of the great circle bounded by
two positions, and corresponds to their distance on a
surface of the Earth, representing also the shortest
distance between these positions on the Earth as a
sphere. The ship, travelling in orthodromic oceanic
navigation, has her bow directed towards the port of
arrival all the time. The orthodorme is the curve of a
variable course – it intersects meridians at different
angles. When navigating by the orthodrome, course
should be constantly changed, which is unacceptable
from the navigational point of view. On the other
hand, loxodrome (rhumb line) intersects all meridi-
ans at the same angle, and it is more suitable in
maintaining the course. However, loxodromic path
is longer that the orthodromic one. Sailing by loxo-
drome, the bow of the ship will be directed toward
the final destination just before arrival. Due to the
mentioned facts, it is necessary to use the advantages
of both curves – the shorter path of the orthodrome
and the rhumb line conformity.
Orthodrome navigation is, as mentioned, incon-
venient. Therefore, only approximation of ortho-
drome navigation can be taken into account, reduc-
ing the number of course changes to an acceptable
number – always bearing in mind that if the number
of course alteration is greater, the navigation is clos-
er to the great circle. After defining elements for
course and distance determination on an orthodrome
curve, navigation between the derived points is car-
ried out in loxodromic courses.
Applying spherical trigonometry, the proposed
paper elaborates models of approximation for the or-
thodrome navigation with the secant method and the
tangent method. The secant method provides two
models of navigation. In the first model, the ortho-
drome is divided into desired waypoints – interposi-
tions between which the ship sails in loxodromic
courses. The second model of the method implies
the path between two positions divided into specific
intervals of unit distances, which then define other
elements of navigation (interposition coordinates
and loxodromic courses). In these two models, navi-
gation has been approximated with the secants of the
orthodrome curve on which the vessel sails. The
tangent method gives an approximation model by
determining the unit changes of orthodromic cours-
es, and defining the tangent on a curve, after which
other navigational elements needed for navigation
are performed.
2 IMPORTANT RELATIONS BETWEEN
ORTHODROMIC AND LOXODROMIC
DISTANCES FOR THE EARTH AS A SPHERE
As described above, the rhumb line, i.e. loxodrome,
represents a constant course, spiral-shaped curve,
asymptotically approaching the Pole. The ortho-
drome represents a variable course curve, the minor
arc of the great circle between two positions. For the
Earth as a sphere, between positions P
1
and P
2
, these
two distances are equal in two situations only (Fig-
ure 1):
1 if the positions are placed on the same meridian,
then Δλ=0, Δφ≠0,
Approximation Models of Orthodromic
Navigation
S. Kos & D. Brcic
University of Rijeka – Faculty of Maritime Studies, Croatia
ABSTRACT: The paper deals with two different approaches to orthodromic navigation approximation, the
secant method and the tangent method. Two ways of determination of orthodromic interposition coordinates
will be presented with the secant method. In the second, tangent method unit change of orthodromic course
(ΔK) will be used.
520
2 if the positions are placed on the Terrestrial Equa-
tor, then Δφ=0, Δλ≠0
In all other situations, the orthodrome distance is
always smaller, or D
O
≠ D
L
.
Maximum difference between distances D
O
and
D
L
occurs when φ
1
= φ
2
≠ 0, Δφ = 0, Δλ = 180º be-
tween P
1
and P
2
is applied. In this case, the function
extremum should be determined [Wippern, 1992]:
( )
( )
( )
( )
( )
0cos Δλ'f'
2sin Δλ f'
2πcos Δλ f
90 2cos Δλ f
<−=
+−=
+−=
−°−=
ϕ
ϕ
ϕϕ
ϕϕ
ϕ
ϕ
ϕ
ϕ
For φ from 0º to ± 90º the cosφ function is posi-
tive, so the second derivation f'
(φ)
<0. Therefore, the
function has an extremum, maximum:
( )
( )
( )
0
2
Δλ
4
1 Δλ'f'
2
sin1 Δλcos Δλ''f
Δλ
2
1
sin
02sin Δλ0f'
>−−=
−−=−=
−
=
=+−=
ϕ
ϕϕ
ϕ
ϕ
ϕ
ϕ
If Δλ = 180º = π, then φ = 39º 32' 24,8''
Figure 1: Orthodrome and Loxodrome relations on Earth as a
sphere
Source: Made by authors
From the completely theoretical point of view, it
follows that the strongest difference between ortho-
dromic and loxodromic distance appears if positions
P
1
and P
2
are placed on a geographic parallel of φ =
39º 32' 24,8'', and on anti-meridians, i.e. where Δλ =
180º. Then, the function value f
(φ)
amounts 2273,
5475...M [Benković et al]; this value represents the
difference between orthodromic and loxodromic
path. Maximum numerical saving of 2273,5 M in the
distance, expressed in percentage counts 37,5%
1
. In
most cases, the distance saving in percentages in
navigational practice reachs up to 10%.
In Equator/Meridian sailing, as well as heading
close to the corresponding courses, the distance sav-
ing is minimal, given that the curves are more and
more closer. In other cases, that are headings in the
090º/270º sector, particularly when sailing on the
same parallel (with appropriate distance between po-
sitions), approximating the navigation could save up
to one day of navigation, which nowadays represents
an important element of the navigation venture.
3 APPROXIMATION OF ORTHODROMIC
NAVIGATION BY SECANT METHODS
3.1 The first secant method – Orthodrome
interposition division
In the first secant method the problem is approached
in a way that the orthodrome is divided into interpo-
sitions, between which the vessel sails in loxodromic
courses.
Interpositions differ in their longitude every 5º or
10º (mostly), while the division begins from the Ver-
tex of the orthodrome, under the condition that this
point is placed between the departure and arrival po-
sition. If Vertex is situated outside of the specific
positions, interpositions can be defined from the
point of departure, P
1
. In the following text, Vertex
interposition division is explained. In Figure 2. the
required relations between the elements are shown.
Figure 2: Determining the coordinates of orthodrome interposi-
tions
Source: Made by authors
1
Theoretically speaking, expressed saving can reach up to 57%, but it
has no practical importance for navigation, because these are very
short paths between positions on the parallel at the near Pole (e.g.
φ=88º).
521
V – Vertex of the orthodrome, defined by the coor-
dinates φ
V
i λ
V
Δλ
mt
– the selected difference of longitude for which
interpositions are required
M – orthodrome interposition, defined by the coor-
dinates φ
M
i λ
M
The navigator selects the interposition longitude:
mtVM
λλλ −=
The latitude is obtained by applying spherical
trigonometry for the right-angled triangle ΔP
N
MV
[Kos et al, 2010]:
In case that the Vertex lies outside positions P
1
and P
2
, the division begins from the point P
1
. Here,
the inclination of the orthodrome (i) should be de-
termined first. The inclination of orthodrome repre-
sents the angle at which orthodrome intersects the
Equator of the Earth, resulting in a right triangle of
the point of departure, P
1
.
)αsin (cos sin arc
αsin coscos
αsin )90( sincos
1
1
1
ϕ
ϕ
ϕ
=
=
−°=
i
i
i
The longitude of the intersection, λ
S
, is defined as
follows:
S1
Δλλλ +=
S
where
tgαsintgΔ
1S
ϕλ
−=
) tgαsin( tgarcΔλ
1S
ϕ
−=
2
In a right-angled triangle ΔSAM, equatorial leg (Δλ
S
+ Δλ
mt
) and the angle of inclination i are known. The
following relation are a result of this triangle (Figure
3) [Kos et al, 2010]:
)90( ctg ctg)]λλ(90[ cos
MmtS
ϕ
−°=∆+∆−° i
MmtS
tg ctg)λλ( sin
ϕ
i=∆+∆
i tg)ΔλΔλ( sintg
mtSM
+=
ϕ
] tg)ΔλΔλ( [sin tgarc
mtSM
i+=
ϕ
2
Some of mentioned elements perhaps require additional explanation,
mathematical derivation respectively. Bearing in mind the length limi-
tation of the paper, as well as the extensive nature of the matter, the
reader is referred to the additional literature [Kos et al, 2010].
Figure 3: The inclination of the orthodrome
Source: Made by authors
3.1.1 Loxodromic intercourse and distances deter-
mination
The loxodromic courses between the positions are
calculated from the loxodromic triangle [Benković
et al, 1986]:
=
=
M
M
Δ
Δλ
tgarcK
Δ
Δλ
tgK
ϕ
ϕ
The first course (K
1
), by which the orthodrome
navigation begins (in position P
1
), is calculated on
the basis of Δλ, that is, the longitude difference be-
tween P
1
and the first interposition, M
1
, and the
Mercator latitudes difference between the same
points. The second course (K
2
) in M
2
is calculated
on the basis of analogic Δλ and Δφ
M
points M
1
and
M
2
, etc. The Figure 4. shows graphic determination
of loxodromic courses and distances.
Besides loxodromic courses between two interpo-
sitions, to determine the distances, one needs to
know the latitude difference between the positions,
beginning at P
1
and M
1
, then M
1
and M
2
and so on to
the point of arrival P
2
;
cosK
Δ
D
L
ϕ
=
522
Figure 4: Orthodromic navigation approximation by the interposition division – The first secant method
Source: Made by authors
3.2 The second secant method – Division of the
orthodrome in unit distance intervals
This method is based on the theoretical assumption
that the orthodrome, which passes through two posi-
tions on the surface of the Earth as a sphere, is com-
posed of an infinite number of infinitively small
loxodromes [Kos, 1996], i.e.
∫
=
L
O
LO
ΔdD
It follows that the final greatness of the ortho-
drome passing through two positions that are suffi-
ciently distant from each other, can be replaced with
the infinitivelly small number of loxodromes, i.e.
dD
L
= dD
O
, respectively:
∫
=
O
D
O
OO
dDD
Given that the greatness of infinitively small lox-
odrome cannot be dimensionally defined, the loxo-
drome could be defined by the approximation of the
greatness of orthodromic unit distance intervals
(dD
O
), which is then approximately equal with the
loxodromic distance (dD
L
). In this way, the incon-
venient orthodrome navigation is replaced with the
loxodrome sailing. The intention is that the course
alternations are reduced to a navigationally accepta-
ble amount. The smaller the greatness of orthodrom-
ic unit distance, the minor the error of orthodromic
approximation. However, it requires more frequent
course alternation, which is in contradiction with
practical navigation. Therefore, it is proposed as fol-
lows:
- if two positions on Earth (approximated by the
shape of the sphere) are distant one from another ≤
30' = 30 M, the following approximation can be in-
troduced:
DL
≅
DO = 30'
Based on the above mentioned, the concept of unit
distance interval is introduced, and it is 30', i.e.
30 M.
The process of orthodromic navigation perform-
ing is as follows:
P
1
(φ
1
, λ
1
) – departure position coordinates
P
2
(φ
2
, λ
2
) – arrival position coordinates
Orthodromic distance between P
1
and P
2
is calcu-
lated, using the equation which is derived from the
nautical – positioning spherical triangle:
)Δλcos cos cossin (sin cosD
2121
1
O
ϕϕϕϕ
+=
−
°<∆<°−= 180λ0 ; λλΔλ
12
Δλ represents the difference between the longi-
tudes of departure and arrival positions.
Orthodromic distance (D
O
), expressed in degrees,
is then divided into orthodromic unit distances of
0,5º from the point of departure to the point of arri-
val.
523
Figure 5: Division of the orthodrome in unit distance intervals of 0,5º
≅
30 M – The second secant method
Source: Made by authors
3.2.1 Orthodromic interposition coordinates deter-
mination
Applying spherical trigonometry, the interposi-
tion coordinates can be determined in different
ways. The following equation can be derived from
the nautical – positioning spherical triangle:
−
=
−
O1
O12
1
Dsin cos
Dcos sinsin
cosK'
ϕ
ϕϕ
The Initial Orthodromic Course (K
OP
) is deter-
mined by the spherical angle K' with the following
relations:
K
OP
= K' if Δλ > 0
K
OP
= 360º − K' if Δλ < 0
The following relations can be derived from the
spherical triangle P
1
P
N
P
2
shown in Figure 5:
( )
OP11
1
m
Kcos cos 5,0sinsin 5,0cos sin
ϕϕϕ
°+°=
−
−°
=
−
m1
m1
1
cos cos
sin sin5,0cos
cosx
ϕϕ
ϕϕ
where:
x = Δλ
m
− angle in terrestrial Pole enclosed by the
meridians of two adjacent interpositions
Δλ
m
> 0 – eastward navigation (E)
Δλ
m
< 0 – westward navigation (W)
Interposition coordinates P
m
(φ
m
, λ
m
= λ
1
+ x)
°
°−
=
−
m
m1
1
cos 5,0sin
sin 5,0cossin
cosy
ϕ
ϕϕ
y – spherical triangle in the respective orthodromic
interposition
−°=
±°=
y360K
y180K
0
0
K
0
– orthodromic course in interposition P
m
, which
depends on the hemisphere on which the ship is sail-
ing (N or S) and sailing direction (E or W)
The coordinates of other orthodromic interposi-
tions from P
m1
to P
mn
can be determined with the fol-
lowing equations:
( )
−
−°
==
°+°=
−
−
−
−−−
−
mn1mn
mn1mn
1
mnn
1on
1mn1mn
1
mn
coscos
sin
sin5,0cos
cosΔλx
Kcos sin 5,0sinsin 5,0cos sin
ϕϕ
ϕϕ
ϕϕϕ
( )
n1mnmnmnmn
xλλ,P +=
−
ϕ
°
°−
=
−
−
mn
mn1mn
1
n
cos sin0,5
sin cos0,5sin
cosy
ϕ
ϕϕ
−°=
±°=
non
non
y360K
y180K
K
on
– orthodromic course in interposition P
mn
524
3.2.2 Loxodromic course determination
From one interposition to another, the ship sails
in unaltered loxodromic course (K
L
), calculated by
the equation derived from the III. Loxodromic Tri-
angle (the Course Triangle) [Kos, 1996]:
Mm
m
Δ
Δλ
tgK
ϕ
=
where:
Δλ
m
= λ
mn
− λ
mn-1
– longitude difference between
two adjacent orthodromic interpositions, expressed
in angular minutes
Δφ
Mm
= φ
Mmn
− φ
Mmn-1
– Mercator latitudes differ-
ence between two adjacent orthodromic interposi-
tions, expressed in angular minutes
If the shape of the Earth is approximated by the
shape of the sphere, then:
]...['
2
45 tg 67898log7915,70446
mn
Mm
+°=
ϕ
ϕ
If the shape of the Earth is approximated by the
shape of the biaxial rotation ellipsoid, then [Ben-
ković et al, 1986]:
]...['
sin e1
sin e1
2
45 tg 67898log7915,70446
2
e
mn
mnmn
Mm
+
−
+°=
ϕ
ϕϕ
ϕ
where:
e – the first numerical eccentricity of the ellipsoid
K – the angle in III. loxodromic triangle
K
L
– general loxodromic navigation course
The following quadrant navigation cases are pos-
sible, which then define loxodromic courses (Figure
6) [Wippern, 1982]:
1 I. navigation quadrant; Δλ
m
> 0, Δφ
Mm
> 0, K
L
=360º + K = K
2 II. navigation quadrant; Δλ
m
> 0, Δφ
Mm
< 0,
K
L
=180º + K
3 III. navigation quadrant; Δλ
m
< 0, Δφ
Mm
< 0,
K
L
=180º + K
4 IV. navigation quadrant; Δλ
m
< 0, Δφ
Mm
> 0,
K
L
=360º + K
Figure 6: Four possibilities of Quadrant Navigation
Source: Made by authors
From the initial position P
1
to the interposition P
m
the ship sails in loxodromic course K
L
. Then, be-
tween P
m
and P
m1
in course K
L1
, between P
m1
and
P
m2
in K
L2
, ... from the interposition P
mn-1
to P
mn
the
ship sails in loxodromic course K
Ln
, and finally,
from P
mn
to the arrival position P
2
in the last loxo-
dromic course. The length of the last stage of navi-
gation between P
mn
and P
2
is always ≤ 30 M.
If, while navigating, the ship is not placed on the
planned orthodromic path
3
, new orthodrome is cal-
culated from exact current position towards the posi-
tion of arrival, and the procedure is then repeated,
dividing the new orthodrome in unit distance inter-
vals of 0,5º, and calculating navigation elements
again [Kos, 1996].
4 APPROXIMATION OF ORTHODROMIC
NAVIGATION BY THE TANGENT METHOD
– ORTHODROME DIVISION IN UNIT
COURSE ALTERATIONS
Instead of secants determined by the interpositions
(the first secant method), or the unit distance inter-
vals (the second secant method), the navigation is
here approximated by the tangent lines of the ortho-
drome, i.e. unit orthodromic course alterations (ΔK)
are derived as follows [Zorović et al, 2010]:
− the Initial Orthodromic Course in position P
1
(K
OP
) and the Final Orthodromic Course (K
OK
) in
position P
2
are calculated. The following values
are then calculated:
3
For example, by the ship's drift due to the sea currents, the wind,
waves, the collision avoidance, etc.
525
( )
ΔK
KK
x
OPOK
−
=
[ ]
M
x
D
D
O
X
=
for
( )
OPOK
O
X
KK
D
D1ΔK
−
=→°=
where:
x – total amount of orthodromic course alteration
ΔK – 1º, 2º, 3º... arbitrarily selected orthodromic
unit course alteration value
D
O
– orthodromic distance between positions P
1
and P
2
K
OP
– the initial orthodromic course in departure
position P
1
K
OK
– the final orthodromic course in arrival posi-
tion P
2
D
X
– unit orthodromic distance
Figure 7: Approximation of orthodromic navigation by the tangent method
Source: Made by authors
5 CONCLUSION
From a theoretical point of view, it is not possible to
navigate on great circle. The orthodrome navigation
is the shortest, while the loxodrome is acceptable
nautically, given that the navigation here is obtained
in constant, general loxodromic course (with the
longer distance travelled) [Bowditch, 1984]. Using
the combination of this navigation curves, the prob-
lem is solved in a way that the features of the ortho-
drome as a shortest distance between two points on
Earth are maximally utilized. The proposed goal of
navigation is thus fulfilled from the practical point
of view, given that the great circle is divided in unit
values of the specific elements, depending of the
method used, on which the navigation is then carried
out in loxodromic courses, and loxodromic distances
respectively.
Three approximation models of orthodromic nav-
igation have been elaborated in the paper. The first
model determines the orthodromic interpositions,
which can be calculated from the orthodrome vertex
or the initial position, depending on the position of
the Vertex, whether it is placed inside the position of
departure and arrival or not. With this method, the
ship sails in unequal distance intervals. The second
model implies the division of the orthodrome in unit
distance intervals with the amount of 0,5º
≅
30 M.
Hereby, the orthodromic interposition coordinates
are determined, and the ship sails in constant loxo-
dromic courses between them. This method is the
most accurate of all of the three elaborated. In the
third method, the orthodromic navigation is approx-
imated by the determination of the orthodromic unit
course alterations ΔK. Here, it is first necessary to
calculate the unit course alterations, after which unit
orthodromic distances are defined, expressed in nau-
tical miles, representing the navigation of the vessel
in the specific course, in a way that the required al-
teration ΔK would appear.
The extent to which the navigation will be ortho-
drome – like, depends on several parametres – con-
sidering a specific navigation case, and taking navi-
gation courses and distances between two positions
into account. In the Equator and Meridian sailing,
the orthodrome and the loxodrome overlap – their
distances are equal. This also applies to smaller dis-
tances between positions, where there are no dis-
526
crepancies between these curves. However, in cer-
tain cases, the difference between these two distanc-
es reaches noticeable values, and then, by approxi-
mating the orthodrome, the time spent in navigation
can be significantly reduced.
ACKNOWLEDGEMENTS
The authors acknowledge the support of research
project ''Research into the correlations of maritime-
transport elements in marine traffic'' (112-1121722-
3066) funded by the Ministry of Science, Education
and Sports of the Republic of Croatia.
REFERENCES
[1] Benković, F. And others: Terestrička i elektronska navi-
gacija, Hidrografski institut JRM, Split, 1986.
[2] Bowditch, N.: The American Practical Navigator, Vol I,
US Defence Mapping Agency, Bethesda, 1984.
[3] Kos, S.: Aproksimacija plovidbe po ortodromi, Zbornik ra-
dova Pomorskog fakulteta, Rijeka, 1996.
[4] Kos, S., Zorović, D., Vranić D.: Terestrička i elektronička
navigacija, Pomorski fakultet u Rijeci, Rijeka, 2010.
[5] Wippern, K.C.C.: On Loxodromic Navigation, The Journal
of Navigation, Royal Institute of Navigation, 45, Cam-
bridge, 1992.
[5] Zorović, D. And others: Vademecum Maritimus, Pomorski
fakultet u Rijeci, Rijeka, 2002.