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 â€“ 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 â€“ 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 â€“ 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 â€“ 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 Î”Î»=0, Î”Ï†â‰ 0,

Approximation Models of Orthodromic

Navigation

S. Kos & D. Brcic

University of Rijeka â€“ 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

(Î”K) will be used.

520

2 if the positions are placed on the Terrestrial Equa-

tor, then Î”Ï†=0, Î”Î»â‰ 0

In all other situations, the orthodrome distance is

always smaller, or D

O

â‰ D

L

.

Maximum difference between distances D

O

and

D

L

occurs when Ï†

1

= Ï†

2

â‰ 0, Î”Ï† = 0, Î”Î» = 180Âº be-

tween P

1

and P

2

is applied. In this case, the function

extremum should be determined [Wippern, 1992]:

( )

( )

( )

( )

( )

0cos Î”Î»'f'

2sin Î”Î» f'

2Ï€cos Î”Î» f

90 2cos Î”Î» f

<âˆ’=

+âˆ’=

+âˆ’=

âˆ’Â°âˆ’=

Ï•

Ï•

Ï•Ï•

Ï•Ï•

Ï•

Ï•

Ï•

Ï•

For Ï† from 0Âº to Â± 90Âº the cosÏ† function is posi-

tive, so the second derivation f'

(Ï†)

<0. Therefore, the

function has an extremum, maximum:

( )

( )

( )

0

2

Î”Î»

4

1 Î”Î»'f'

2

sin1 Î”Î»cos Î”Î»''f

Î”Î»

2

1

sin

02sin Î”Î»0f'

>âˆ’âˆ’=

âˆ’âˆ’=âˆ’=

ï£·

ï£¸

ï£¶

ï£¬

ï£

ï£«

âˆ’

=

=+âˆ’=

Ï•

Ï•Ï•

Ï•

Ï•

Ï•

Ï•

If Î”Î» = 180Âº = Ï€, then Ï† = 39Âº 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 Ï† =

39Âº 32' 24,8'', and on anti-meridians, i.e. where Î”Î» =

180Âº. Then, the function value f

(Ï†)

amounts 2273,

5475...M [BenkoviÄ‡ 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

090Âº/270Âº 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 â€“ 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 5Âº or

10Âº (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.

Ï†=88Âº).

521

V â€“ Vertex of the orthodrome, defined by the coor-

dinates Ï†

V

i Î»

V

Î”Î»

mt

â€“ the selected difference of longitude for which

interpositions are required

M â€“ orthodrome interposition, defined by the coor-

dinates Ï†

M

i Î»

M

The navigator selects the interposition longitude:

mtVM

Î»Î»Î» âˆ’=

The latitude is obtained by applying spherical

trigonometry for the right-angled triangle Î”P

N

MV

[Kos et al, 2010]:

) tgÎ»(cosÎ” tgarc

tgÎ»cosÎ”tg

tgctgÎ»cosÎ”

)(90 ctg )](90[90 ctgÎ”Î» cos

VmtM

VmtM

MVmt

MVmt

Ï•Ï•

Ï•Ï•

Ï•Ï•

Ï•Ï•

=

=

=

âˆ’Â°âˆ’Â°âˆ’Â°=

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

.

)Î±sin (cos sin arc

Î±sin coscos

Î±sin )90( sincos

1

1

1

Ï•

Ï•

Ï•

=

=

âˆ’Â°=

i

i

i

The longitude of the intersection, Î»

S

, is defined as

follows:

S1

Î”Î»Î»Î» +=

S

where

tgÎ±sintgÎ”

1S

Ï•Î»

âˆ’=

) tgÎ±sin( tgarcÎ”Î»

1S

Ï•

âˆ’=

2

In a right-angled triangle Î”SAM, equatorial leg (Î”Î»

S

+ Î”Î»

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)]Î»Î»(90[ cos

MmtS

Ï•

âˆ’Â°=âˆ†+âˆ†âˆ’Â° i

MmtS

tg ctg)Î»Î»( sin

Ï•

i=âˆ†+âˆ†

i tg)Î”Î»Î”Î»( sintg

mtSM

+=

Ï•

] tg)Î”Î»Î”Î»( [sin tgarc

mtSM

i+=

Ï•

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 [BenkoviÄ‡

et al, 1986]:

ï£·

ï£·

ï£¸

ï£¶

ï£¬

ï£¬

ï£

ï£«

=

=

M

M

Î”

Î”Î»

tgarcK

Î”

Î”Î»

tgK

Ï•

Ï•

The first course (K

1

), by which the orthodrome

navigation begins (in position P

1

), is calculated on

the basis of Î”Î», 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 Î”Î» and Î”Ï†

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

Î”

D

L

Ï•

=

522

Figure 4: Orthodromic navigation approximation by the interposition division â€“ The first secant method

Source: Made by authors

3.2 The second secant method â€“ 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.

âˆ«

=

L

O

LO

Î”dD

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:

âˆ«

=

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 â‰¤

30' = 30 M, the following approximation can be in-

troduced:

DL

â‰…

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

(Ï†

1

, Î»

1

) â€“ departure position coordinates

P

2

(Ï†

2

, Î»

2

) â€“ arrival position coordinates

Orthodromic distance between P

1

and P

2

is calcu-

lated, using the equation which is derived from the

nautical â€“ positioning spherical triangle:

)Î”Î»cos cos cossin (sin cosD

2121

1

O

Ï•Ï•Ï•Ï•

+=

âˆ’

Â°<âˆ†<Â°âˆ’= 180Î»0 ; Î»Î»Î”Î»

12

Î”Î» 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

0,5Âº from the point of departure to the point of arri-

val.