International Journal

on Marine Navigation

and Safety of Sea Transportation

Volume 5

Number 4

December 2011

507

1 INTRODUCTION

A common problem is finding the shortest route

across the Earth surface between two positions. Such

trajectory is always a part of a geodesic (great circle,

great ellipse) on the modelling globe surface. The

geodesic is used by ship navigators attempting to

minimize distances and the radio operators with di-

rectional antennae used to look for a bearing yield-

ing the strongest signal. For many purposes, it is en-

tirely adequate to model the Earth as a sphere.

Actually, it is more nearly an oblate ellipsoid of rev-

olution. The earth’s flattening is quite small, about 1

part in 300, and navigation errors induced by assum-

ing the Earth is spherical do not exceed this, and so

for many purposes a spherical approximation may be

entirely adequate. On a sphere, the commonly used

coordinates are latitude and longitude, likewise on a

spheroid, however on a spheroid one has to be more

careful about what exactly one means by latitude

[Williams, 1996]. The spherical model is often used

in cartographic projections creating the frame of the

presented chart. The trajectory of the geodesic lines

and the loxodrome looks different depending on the

method of the projections given by the strict formu-

lae. Thus, many map projections are invaluable in

specialized applications.

The only conformal cylindrical projection, Mer-

cator’s device was a boon to navigators from the

16th-century until the present, despite suffering from

extreme distortion near the poles. We recall it has a

remarkable property: any straight line between two

points is a loxodrome or line of constant course on

the sphere. The Mercator loxodrome bears the same

angle from all meridians. Briefly, if one draws a

straight line connecting a journey’s starting and end-

ing points on a Mercator map, that line’s slope

yields the journey direction, and keeping a constant

bearing is enough to get to one’s destination.

A Mercator projection is not the only one used by

navigators, as the loxodrome does not usually coin-

cides with the geodesic. This projection was possi-

bly first used by Etzlaub (ca. 1511). However, it was

for sure only widely known after Mercator’s atlas of

1569. Mercator probably defined the graticule by

geometric construction. E. Wright formally present-

ed equations in 1599. Wright's work influenced,

among other persons, Dutch astronomer and mathe-

matician Willebrord Snellius, who introduced the

A Novel Approach to Loxodrome (Rhumb

Line), Orthodrome (Great Circle) and Geodesic

Line in ECDIS and Navigation in General

A. Weintrit & P. Kopacz

Gdynia Maritime University, Gdynia, Poland

ABSTRACT: We survey last reports and research results in the field of navigational calculations’ methods

applied in marine navigation that deserve to be collected together. Some of these results have often been re-

discovered as lemmas to other results. We present our approach to the subject and place special emphasis on

the geometrical base from a general point of view. The geometry of approximated structures implies the cal-

culus essentially, in particular the mathematical formulae in the algorithms applied in the navigational elec-

tronic devices and systems. The question we ask affects the range and point in applying the loxodrome

(rhumb line) in case the ECDIS equipped with the great circle (great ellipse) approximation algorithms of

given accuracy replaces the traditional nautical charts based on Mercator projection. We also cover the sub-

ject on approximating models for navigational purposes. Moreover, the navigation based on geodesic lines

and connected software of the ship’s devices (electronic chart, positioning and steering systems) gives a

strong argument to research and use geodesic-based methods for calculations instead of the loxodromic trajec-

tories in general.

508

word “loxodrome”; Adriaan Metius, the geometer

and astronomer from Holland; and the English

mathematician Richard Norwood, who calculated

the length of a degree on a great circle of the Earth

using a method proposed by Wright.

More commonly applied to large-scale maps, the

transverse aspect preserves every property of Merca-

tor’s projection, but since meridians are not straight

lines, it is better suited for topography than naviga-

tion. Equatorial, transverse and oblique maps offer

the same distortion pattern. The transverse aspect

with equations for the spherical case was presented

by Lambert in his seminal paper (1772). The ellip-

soidal case was developed, among others, by Carl

Gauss (ca. 1822) and Louis Krüger (ca. 1912). It is

frequently called the Gauss conformal or Gauss-

Krüger projection.

The vessel or aircraft can reach its destination fol-

lowing the fixed bearing along the whole trip disre-

garding some obvious factors like for instance

weather, fuel range, geographical obstructions.

However, that easy route would not be the most

economical choice in terms of distance. The two

paths almost coincide only in brief routes. Although

the rhumb line is much shorter on the Mercator map,

an azimuthal equidistant map tells a different story,

even though the geodesic does not map to a straight

line since it does not intercept the projection centre.

Since there is a trade-off: following the geodesic

would imply constant changes of direction (those are

changes from the current compass bearing and are

only apparent: on the sphere, the trajectory is as

straight as it can be). Following the rhumb line

would waste time and fuel. So a navigator could fol-

low a hybrid procedure [Snyder, 1987]:

− trace the geodesic on an azimuthal equidistant or

gnomonic map,

− break the geodesic in segments,

− plot each segment onto a Mercator map,

− use a protractor and read the bearings for each

segment,

− navigate each segment separately following its

corresponding constant bearing.

2 GEODESIC APPROACH

For curved or more complicated modelling surfaces

the metric can be used to compute the distance be-

tween two points by integration. The distance gener-

ally means the shortest distance between two points.

Roughly speaking, the distance between two points

is the length of the path connecting them. Most often

the research and calculus in navigational literature

are considered on the spherical or spheroidal models

of Earth because of practical reasons. The flow of

geodesics on the ellipsoid of revolution (spheroid)

differs from the geodesics on the sphere. There are

known different geodesics on the same surface with

the same metric considered. However geodesic re-

fers to the metric what is usually not taken into con-

sideration in the navigational lectures. And there are

different flows of geodesics on the same surface

when different metrics are applied. That means we

can obtain geometrically different results in naviga-

tional aspect if we change the researched modelling

object with its geometrical and physical features

(Kopacz, 2006).

Let us focus on two essential notions creating the

base for the various fields of the mathematical re-

search: the metric and topology. A metric space is a

set with a global distance function (the metric) that,

for every two points in, gives the distance between

them as a nonnegative real number.

Definition 1. A function g: X × X → [0,

∞)

is called

a metric (or distance) in X if

(1) g(x, y) = 0 iff x = y (positivity);

(2) g(x, y) = g(y, x) for every x, y ∈ X (symmetry);

(3) g(x, y) ≤ g(x, z) + g(z, y) for every x, y, z ∈ X

(triangle inequality).

Metric as a nonnegative function describes

the ”distance” between neighbouring points for a

given set. When viewed as a tensor, the metric is

called a metric tensor. We can define a metric in

each non-empty set (X ≠ ∅). The notion of metric

has been introduced by M. Frechet in 1906. Formal-

ly the pair (X, g) where g is a metric in a set X is

called a metric space. Fig. 1 points out the essential

role played by the metric in geodesic approach to the

subject.

Figure 1. Geometrical basis in geodesic analysis [Kopacz, 2006]

Making one step further we can generalize the

metric space to the topological space.

Definition 2. Let X ≠

∅

be a set and P(X) the power

set of X, i.e.

(){: }PX U U X= ⊂

. Let

Ω

⊂ P(X) be a

collection of its subsets such that:

509

(1)

()UU

ιι

∈Ι

∀ ∈Ι ∈Ω ⇒ ∈Ω



(the union of a

collection of sets, which are elements of

Ω

, belongs

Ω

);

(2)

,UV U V∈Ω⇒ ∩ ∈Ω

(the intersection of a fi-

nite collection of sets, which are elements of

Ω, belongs to

Ω

);

(3)

∅,

X∈

Ω

(the empty set

∅

and the whole set X

belong to

Ω

Then

− Ω is called a topological structure or just a topol-

ogy in X;

− the pair (X, Ω) is called a topological space;

− an element of X is called a point of this topologi-

cal space;

− an element of Ω is called an open set of the topo-

logical space (X, Ω).

The conditions in the definition presented above

are called the axioms of topological structure. A to-

pology, that is a metric topology, means that one can

define a suitable metric that induces it. Additionally

we assume here that although the metric exists, it

may be unknown. In a metric space (X, g) the family

of sets

Ω

{ : (, ) }

U X Bx U

∈>

Ω= ⊂ ∀ ∃ ⊂

satisfies the above mentioned axioms of topology.

That means (X, g) is a topological space and thus,

each metric space is a topological space. There are

sufficient criteria on the topology that assure the ex-

istence of such a metric even if this is not explicitly

given. An example of an existence theorem of this

kind is due to Urysohn who proved that a regular T

space whose topology has a countable basis is me-

trizable [Kelley, 1955]. Conversely, a metrizable

space is always T

and regular but the condition on

the basis has to be weakened since in general, it is

only true that the topology has a basis which is

formed by countably many locally finite families of

open sets. Special metrizability criteria are known

for Hausdorff spaces (T

-spaces). A compact

Hausdorff space is metrizable if and only if the set

of all elements is a zero set [Willard, 1970]. The

continuous image of a compact metric space in a

Hausdorff space is metrizable. This implies in par-

ticular that a distance can be defined on every path

in T

-space.

Figure 2. Flows of geodesics (distance functions) on locally

modelling surfaces of differing curvatures

The mathematical formulae used in approxima-

tion of the navigational calculations are being stud-

ied and are based on spherical (spheroidal) model.

However if we consider different shape of the sur-

face the formulae change considerably. The exam-

ples of the flows of geodesics on locally modelling

surfaces of differing curvature are presented graph-

ically in Fig. 2. Let us imagine that the vessels do

not sail on spheroidal earth but locally torus - shaped

planet. In this case the flow of geodesics and men-

tioned rhumb line or used charts are based on other

mathematical expressions due to different geomet-

rical object considered. The torus is topologically

more simple than the sphere, yet geometrically it is a

very complicated manifold indeed.

Figure 3. The geodesics on a torus T

= S

x S

The round torus metric is most easily constructed

via its embedding in a Euclidean space of one higher

dimension.

Taking into consideration the main theoretical as-

pects of the subject above mentioned as well as the

practical ones influencing the base and components

of the navigational algorithm to be applied we col-

510

lect all of them together what has been shown in

Figure 4.

Figure 4. Navigational calculations’ algorithm guidelines

The notion of geodesics makes sense not only for

surfaces in R

but also for abstract surfaces and more

generally (Riemannian) manifolds. We also refer to

[Funar, Gadgil, 2001] where the notion of a topolog-

ical geodesic in a 3-manifold have been introduced.

Geodesics in Riemannian manifolds with metrics of

negative sectional curvature play an essential role in

geometry. It is shown there that, in the case of 3-

dimensional manifolds, many crucial properties of

geodesics follow from a purely topological charac-

terization in terms of knotting as well as proved

basic existence and uniqueness results for topologi-

cal geodesics under suitable hypotheses on the fun-

damental group. For further reading we send the

reader to the wide literature on Riemannian and

Finsler geometry and topology, in particular the ge-

odesic research.

3 PLANE MODEL

The surface of revolution as the Earth’s model -

sphere S

or the spheroid is locally approximated by

the Euclidean plane tangent in a given position.

Generally, we approximate locally the curved sur-

face by the Euclidean plane. For some applications

such approximation is allowed and sufficient for

practical need of research. That is satisfactory if we

do not exceed the required accuracy of provided cal-

culations. Hence the boundary conditions of apply-

ing the Euclidean plane or spherical geometry ought

to be strictly defined. The mathematical components

of the plane Euclidean geometry applied in naviga-

tional device are widely known and there is a com-

mon Euclidean metric used in the calculus as the dis-

tance function. We emphasize that the geodesics

may look different even on the plane if different

metrics are considered. For the practical reasons and

the ease of use there is Euclidean plane tangent to

the modelled surface used in many applications, for

instance in dynamic positioning (DP) software. The

plane model enables the satisfactory accuracy in a

local approximation. In the local terrain geodesic re-

search the area can be considered flat if it is inside

the circle of a radius of ca. 15.5 km. This corre-

sponds to the area of spherical circle which diameter

equals ca. 17’ of the great circle [Kopacz, 2010].

Practically such an approximation allows the direct

geodesic measurements without considering the cur-

vature of the modelled Earth surface and presenting

the results on the plane in the appropriate scale. In

the global modelling of the Earth’s surface (geodesy,

cartography, navigation, astronomy) the Euclidean

geometry becomes not sufficient for the geometric

description and the calculus coming from it. Thus,

the limits of application of the approximation meth-

ods based on the flat Euclidean geometry must be

clearly determined [Kopacz, 2010].

In a field of flat chart projections scale distortions

on a chart can be shown by means of ratio of the

scale at a given point to the true scale (a scale factor

- SF). Scale distortions exist at locations where the

scale factor differs from 1. For instance, a scale fac-

tor at a given point on the map is equal to 0.99960

signifies that 1000 m on the reference surface of the

Earth will actually measure 999.6 m on the chart.

This is a contraction of 40 cm per 1 km.

Figure 5. Scale distortions on a tangent (a) and a secant (b)

map surface [Knippers, 2009]

Distortions increase as the distance from the cen-

tral point (tangent plane) or closed line(s) of inter-

section increases. Scale distortions for tangent and

secant map surfaces are illustrated in the Fig. 5. On a

secant map projection - the application of a scale

factor of less than 1.0000 to the central point or the

central meridian has the effect of making the projec-

tion secant - the overall distortions are less than on

one that uses a tangent map surface. Most countries

have derived there map coordinate system from a

projection with a secant map surface for this reason

[Knippers, 2009].

511

The curved Earth is navigated using flat maps or

charts, collected in an atlas. Similarly, in a calculus

on manifolds a differentiable manifold can be de-

scribed using mathematical maps, called coordinate

charts, collected in a mathematical atlas. It is not

generally possible to describe a manifold with just

one chart, because the global structure of the mani-

fold is different from the simple structure of the

charts. For example, no single flat map can properly

represent the entire Earth. When a manifold is con-

structed from multiple overlapping charts, the re-

gions where they overlap carry information essential

to understanding the global structure. In the case of a

differentiable manifold, an atlas allows to do calcu-

lus on manifolds. The atlas containing all possible

charts consistent with a given atlas is called the max-

imal atlas. Unlike an ordinary atlas, the maximal at-

las of a given atlas is unique. Though it is useful for

definitions, it is a very abstract object and not used

directly for example in calculations. Charts in an at-

las may overlap and a single point of a manifold

may be represented in several charts. If two charts

overlap, parts of them represent the same region of

the manifold. Given two overlapping charts, a transi-

tion function can be defined which goes from an

open ball in R

to the manifold and then back to an-

other (or perhaps the same) open ball in R

. The re-

sultant map is called a change of coordinates, a co-

ordinate transformation, a transition function or a

transition map.

4 SPHERICAL AND SPHEROIDAL MODEL

As the Earth’s global model an oblate spheroid is

used providing the navigational calculations i.e. dis-

tances and angles or the sphere for the ease of use. A

sphere, spheroid or a torus surface are examples of

2-dimensional manifolds. Manifolds are important

objects in mathematics and physics because they al-

low more complicated structures to be expressed and

understood in terms of the relatively well understood

properties of simpler spaces. The study of manifolds

combines many important areas of mathematics: it

generalizes concepts such as curves and surfaces as

well as ideas from linear algebra and topology. Cer-

tain special classes of manifolds also have additional

algebraic structure. They may behave like groups,

for instance. To measure distances and angles on

manifolds, the manifold must be Riemannian. We

recall that a Riemannian manifold is an analytic

manifold in which each tangent space is equipped

with an inner product in a manner which varies

smoothly from point to point. This allows one to de-

fine various notions such as length, angles, areas (or

volumes), curvature, gradients of functions and di-

vergence of vector fields. More general geometric

structure a Finsler manifold allows the definition of

distance, but not of angle. It is an analytic manifold

in which each tangent space is equipped with a

norm, in a manner which varies smoothly from point

to point. This norm can be extended to a metric, de-

fining the length of a curve; but it cannot in general

be used to define an inner product. Any Riemannian

manifold is a Finsler manifold. Manifold theory has

come to focus exclusively on these intrinsic proper-

ties (or invariants), while largely ignoring the extrin-

sic properties of the ambient space.

Triaxial ellipsoid as the 2-dimensional sub-

manifold M in R

is defined by the equation

= 0

where

2 22

222

(, ,) 1

xyz

abc

Φ =++−

Let N be the non-vanishing normal vector field on

M. Then

123

22 2

( , , ) 0,5

xyz

N x y z grad e e e

abc

= Φ= + +

where the {e

, e

} is the canonical basis of the

vector space R

. The Gaussian curvature K of the

modelling triaxial ellipsoid equals

222

abc N

The Gaussian curvature is the determinant of the

shape operator. For the sphere a=b=c=r and then

,NK

= =

where r denotes a radius of the

modelling sphere. Thus, we conclude here that the

curvature affects the geometry of the locally approx-

imating surfaces essentially and in particular their

geodesic trajectories.

2-dimensional sphere S

is widely considered to

model globally the surface of the Earth. As a calcu-

lating tool the spherical trigonometry is used which

states the base for the comparison analysis and algo-

rithms implemented in the software of navigational

aids e.g. receivers of the positioning systems, EC-

DIS. The surface of the Earth may be taken mathe-

matically as a sphere instead of ellipsoid for maps at

smaller scales. In practice, maps at scale 1:5 000 000

or smaller can use the mathematically simpler sphere

without the risk of large distortions. At larger scales,

the more complicated mathematics of ellipsoids are

needed to prevent these distortions in the map. A

sphere can be derived from the certain ellipsoid cor-

responding either to the semi-major or semi-minor

axis, or average of both axes or can have equal vol-

ume or equal surface than the ellipsoid [Knippers,

2009].

512

Figure 6. Geodesics on 2-dimensional modelling manifolds of

positive curvature a) sphere, b) spheroid (ellipsoid of revolu-

tion), c) triaxial ellipsoid.

We recall the great circle is the equivalent of the

Euclidean straight line, it has the finite distance and

it is closed. The geodesics starting from a given po-

sition on three main modelling surfaces (2-

dimensional modelling manifolds of positive curva-

ture), i.e. sphere, spheroid and triaxial ellipsoid are

presented in Fig. 6. The disadvantage of orthodromic

sailing is bound with continuous course alteration.

That is why the loxodromic line is mainly sailed on-

ly or mainly used in the approximation of the great

circle sailing. Thus the combination of these two

lines create the base for planned and monitored tra-

jectories while at sea.

The general question we ask affects the range and

point of usage of the rhumb line in case the ECDIS

systems equipped with the great circle / great ellipse

approximation algorithms of given accuracy replaces

the traditional paper charts based on Mercator pro-

jection. Moreover, the navigation based on geodesic

lines and connected software of the ship’s device

(electronic chart, positioning and steering systems)

gives a strong argument to use this method for calcu-

lations instead of the loxodromic one in general.

Although the basic solutions for navigational pur-

poses have already been known and widely used

there are still the new approaches and efforts made

to the subject. The examples of the spherical and

spheroidal approach have been found recently in the

literature reviewed further in the article. The main

efforts affect the optimization and approximation

methods which potentially may give the practical

benefits for the navigators.

As we mentioned above the shortest path between

two points on a smooth surface is called a geodesic

curve on the surface. On a flat surface the geodesics

are the straight lines, on a sphere they are the great

circles. Remarkably the path taken by a particle slid-

ing without friction on a surface will always be a

geodesic. This is because a defining characteristic of

a geodesic is that at each point on its path, the local

centre of curvature always lies in the direction of the

surface normal, i.e. in the direction of any con-

strained force required to keep the particle on the

surface. There are thus no forces in the local tangent

plane of the surface to deflect the particle from its

geodesic path. There is a general procedure, using

the calculus of variations, to find the equation for

geodesics given the metric of the surface [Williams,

1996]. Obviously the Earth is not an exact ellipsoid

and deviations from this shape are continually eval-

uated. The geoid is the name given to the shape that

the Earth would assume if it were all measured at

mean sea level. This is an undulating surface that

varies not more than about a hundred meters above

or below a well-fitting ellipsoid, a variation far less

than the ellipsoid varies from the sphere. The choice

of the reference ellipsoid used for various regions of

the Earth has been influenced by the local geoid, but

large-scale map projections are designed to fit the

reference ellipsoid, not the geoid. The selection of

constants defining the shape of the reference ellip-

soid has been a major concern of geodesists since

the early 18th century. Two geometric constants are

sufficient to define the ellipsoid itself e.g. the semi-

major axis and the eccentricity. In addition, recent

satellite-measured reference ellipsoids are defined

by the semimajor axis, geocentric gravitational con-

stant and dynamical form factor which may be con-

verted to flattening with formulas from physics.

Between 1799 and 1951 there were 26 determina-

tions of dimensions of the Earth. There are over a

513

dozen other principal ellipsoids, however, which are

still used by one or more countries. The different

dimensions do not only result from varying accuracy

in the geodetic measurements (the measurements of

locations on the Earth), but the curvature of the

Earth’s surface (geoid) is not uniform due to irregu-

larities in the gravity field. Until recently, ellipsoids

were only fitted to the Earth’s shape over a particu-

lar country or continent. The polar axis of the refer-

ence ellipsoid for such a region, therefore, normally

does not coincide with the axis of the actual Earth,

although it is assumed to be parallel. The same ap-

plies to the two equatorial planes. The discrepancy

between centres is usually a few hundred meters at

most. Satellite-determined coordinate systems are

considered geocentric. Ellipsoids for the latter sys-

tems represent the entire Earth more accurately than

ellipsoids determined from ground measurements,

but they do not generally give the best fit for a par-

ticular region. The reference ellipsoids used prior to

those determined by satellite are related to an initial

point of reference on the surface to produce a datum,

the name given to a smooth mathematical surface

that closely fits the mean sea-level surface through-

out the area of interest. The initial point is assigned a

latitude, longitude, elevation above the ellipsoid, and

azimuth to some point. Satellite data have provided

geodesists with new measurements to define the best

Earth-fitting ellipsoid and for relating existing coor-

dinate systems to the Earth’s centre of mass. For the

mapping of other planets and natural satellites, Mars

is treated as an ellipsoid. Other bodies are taken as

spheres, although some irregular satellites have been

treated as triaxial ellipsoids and are mapped ortho-

graphically [Snyder, 1987].

5 ECDIS APPROACH

In the course of navigation programmes for ECDIS

purposes it became apparent that the standard text

books of navigation were perpetuating a flawed

method of calculating rhumb lines on the Earth con-

sidered as an oblate spheroid. On further investiga-

tion it became apparent that these incorrect methods

were being used in programming a number of calcu-

lator/computers and satellite navigation receivers.

Although the discrepancies were not large, it was

disquieting to compare the results of the same rhumb

line calculations from a number of such devices and

find variations of a few miles when the output was

given, and therefore purported to be accurate, to a

tenth of a mile in distance and/or a tenth of a minute

of arc in position. The problem was highlighted in

the past and the references at the end of this paper

show that a number of methods have been proposed

for the amelioration of this problem.

This paper presents and recommends the guide-

lines that should be used for the accurate solutions.

Most of these may be found in standard geodetic text

books, such as, but also provided are new formulae

and schemes of solution which are suitable for use

with computers or tables. The paper also takes into

account situations when a near-indeterminate solu-

tion may arise. Some examples are provided which

demonstrate the methods. The data for these prob-

lems do not refer to actual terrestrial situations but

have been selected for illustrative purposes. Prac-

tising ships' navigators will find the methods de-

scribed in detail in this paper to be directly applica-

ble to their work and they also should find ready

acceptance because they are similar to current prac-

tice. In almost none of the references cited at the end

of this paper has been addressed the practical task of

calculating, using either a computer or tabular tech-

niques.

The paper presents the review of different ap-

proaches to contact formulae for the computation of

the position, the distance, and the azimuth along a

great ellipse. The proposed alternative formulae are

to be primarily used for accurate sailing calculations

on the ellipsoid in a GIS environment as in ECDIS

and other ECS. Among the ECDIS requirements is

the need for a continuous system with a level of ac-

curacy consistent with the requirements of safe nav-

igation. At present, this requirement is best fulfilled

by the Global Positioning System (GPS). The GPS

system is referenced to World Geodetic System

1984 Datum (WGS 84). Using the ellipsoid model

instead of the spherical model attains more accurate

calculation of sailing on the Earth. Therefore, we

aim to construct a computational procedure for solv-

ing the length of the arc of a great ellipse, the way-

points and azimuths along a great ellipse. We an-

nounce our aspiration to provide the straightforward

formulae involving the great elliptic sailing based on

two scenarios. The first is that the departure point

and the destination point are known. The second is

that the departure point and the initial azimuth are

given (direct and inverse geodetic problems on ref-

erence ellipsoids).

5.1 ECDIS Calculations

As a minimum, an ECDIS system must be able to

perform the following calculations and conversions

[Weintrit, 2009]:

− geographical coordinates to display coordinates,

and display coordinates to geographical coordi-

nates;

− transformation from local datum to WGS-84;

− true distance and azimuth between two geograph-

ical positions;

− geographic position from a known position given

distance and azimuth (course);

514

− projection calculations such as great circle and

rhumb line courses and distances;

− “RL-GC” difference between the rhumb line and

great circle in sailing along the great circle (or

great ellipse?).

5.2 Route planning calculations

The ECDIS allows the navigator to create waypoints

and routes including setting limits of approach and

other cautionary limits. Both rhumb line and great

circle routes can be defined. Routes can be freely

exchanged between the ECDIS and GPS or ARPA.

Route checking facility allows the intended route to

be automatically checked for safety against limits of

depth and distance as defined by the navigator.

The mariner can calculate and display both a

rhumb line and a great circle line and verify that no

visible distortion exists between these lines and the

chart data.

Authors predict the early end of the era of the

rhumb line. This line in the natural way will go out

of use. Nobody after all will be putting the naviga-

tional triangle to the screen of the ECDIS. Our

planned route is not having to be a straight line on

the screen. So, why hold this line still in the use?

Each ship’s position plotted on the chart can be the

starting point of new updated great circle GC, or

saying more closely, great ellipse GE.

5.3 Most important questions

It is an important question whether in the ECDIS

time still Mercator projection is essential for marine

navigation. We really need it? And what about loxo-

drome? Also not? So, let start navigation based on

geodesics. It is high time to forget the rhumb line

navigation and great circle navigation, too. But the

first we need clear established methods, algorithms

and formulas for sailing calculations. But it is al-

ready indicating the real revolution in navigation -

total revolution. We will be forced to make the revi-

sion of such fundamental notions as the course, the

heading and the bearing.

And another very important question: do you re-

ally know what kind of algorithms and formulae are

used in your GPS receiver and your ECS/ECDIS

systems for calculations mentioned in chapter 5.1?

We are sure, your answer is negative. So, we have a

problem – a serious problem.

6 REVIEW OF RECENTLY PUBLISHED

PAPERS

From 1950 till 2010 the following professional mag-

azines and journals published some papers about

navigation on the great ellipse and on the spheoridi-

cal Earth: The Journal of Navigation [Bennett, 1996;

Bourbon, 1990; Carlton-Wippern, 1992; Chen, Hsu,

& Chang, 2004; Earle, 2000, 2005, 2006, 2008;

Hickley, 1987; Hiraiwa, 1987; Nastro & Tancredi,

2010; Pallikaris & Latsas, 2009; Prince & Williams,

1995; Sadler, 1956; Tseng & Lee, 2007; Tyrrell,

1955; Walwyn, 1999; Williams, 1950; Williams,

1996; Williams & Phythian, 1989, 1992; Zukas,

1994], International Hydrographic Review [Pallika-

ris, Tsoulos & Paradissis, 2009a], Coordinates

[Pallikaris, Tsoulos & Paradissis, 2010], Navigation

- The Journal of The Institute of Navigation [Kaplan,

1995; Miller, Moskowitz & Simmen, 1991], Bulletin

Geodesique [Bowring, 1983, 1984; Rainsford, 1953,

1955; Sodano, 1965], Journal of Marine Science and

Technology [Tseng & Lee, 2010], The Journal of the

Washingtin Academy of Sciences [Lambert, 1942],

The Canadian Surveyor [Bowring, 1985], Survey

Review [Vincenty, 1975, 1976], Surveying and

Mapping [Meade, 1981], The Professional Geogra-

pher [Tobler, 1964], College Mathematics Journal

[Nord, Muller, 1996; Schechter, 2007].

The following particular problems were discussed

among the others:

− practical rhumb line calculations on the spheroid

[Bennet, 1996],

− geodesic inverse problem [Bowring, 1983],

− direct and inverse solutions for the great elliptic

and line on the reference ellipsoid [Bowring,

1984],

− loxodromic navigation [Carlton-Wippern, 1992],

− formulas for the solution of direct and inverse

problems on reference ellipsoids using pocket

calculators [Meade, 1981],

− geometry of loxodrome on the ellipsoid

[Bowring, 1985],

− geometry of geodesics [Busemann, 1955],

− geodesic line on the surface of a spheroid [Bour-

bon, 1990],

− great circle equation [Chen, Hsu & Chang, 2004],

− novel approach to great circle sailing [Chen, Hsu

& Chang, 2004],

− vector function of traveling distance for great cir-

cle navigation [Tseng & Lee, 2007],

− great circle navigation with vectorial methods

[Nastro & Tancredi, 2010],

− vector solution for great circle navigation [Earle,

2005],

− vector solution for navigation on a great ellipse

[Earle, 2000],

− navigation on a great ellipse [Tseng & Lee,

2010],

515

− great ellipse solution for distances and headings

to steer between waypoints [Walwyn, 1999],

− great ellipse on the surface of the spheroid [Wil-

liams, 1996],

− vector solutions for azimuth [Earle, 2008],

− sphere to spheroid comparisons [Earle, 2006],

− great circle versus rhumb line cross-track distance

at mid-longitude [Hickley, 1987],

− modification of sailing calculations [Hiraiwa,

1987],

− practical sailing formulas for rhumb line tracks on

an oblate Earth [Kaplan, 1995],

− distance between two widely separated points on

the surface of the Earth [Lambert, 1942],

− traveling on the curve Earth [Miller, Moskowitz

& Simmen, 1991],

− new meridian arc formulas for sailing calcula-

tions in GIS [Pallikaris, Tsoulos & Paradissis,

2009a],

− new calculations algorithms for GIS navigational

systmes and receivers [Pallikaris, Tsoulos & Par-

adissis, 2009b],

− improved algorithms for sailing calculations

[Pallikaris, Tsoulos & Paradissis, 2010],

− new algorithm for great elliptic sailing (GES)

[Pallikaris & Latsas, 2009],

− shortest paths [Lyusternik, 1964],

− sailing in ever-decreasing circles [Prince & Wil-

liams, 1995],

− long geodesics on the ellipsoid [Rainsford, 1953,

1955],

− spheroidal sailing and the middle latitude [Sadler,

1956],

− general non-iterative solution of the inverse and

direct geodetic problems [Sodano, 1965],

− comparison of spherical and ellipsoidal measures

[Tobler, 1964],

− navigating on the spheroid [Tyrrell, 1955; Wil-

liams, 2002],

− direct and inverse solutions of geodesics on the

ellipsoid with application of nested equations

[Vincenty, 1975, 1976],

− loxodromic distances on the terrestrial spheroid

[Williams, 1950],

− Mercator’s rhumb lines: a multivariable applica-

tion of arc length [Nord, Muller, 1996],

− navigating along geodesic paths on the surface of

a spheroid [Williams & Phythian, 1989],

− shortest distance between two nearly antipodean

points on the surface of a spheroid [Williams &

Phythian, 1992],

− shortest spheroidal distance [Zukas, 1994],

− navigating on a spheroid [Schechter, 2007].

7 CONCLUSIONS

This article is written with a variety of readers in

mind, ranging from practising navigators to theoreti-

cal analysts. It was also our goal to present a current

and uniform approaches to sailing calculations high-

lighting recent developments. Much insight may be

gained by considering the examples that have recent-

ly proliferated in the literature reviewed above. We

present our approach to the subject and place special

emphasis on the geometrical base from a general

point of view. Of particular interest are geodesic

lines, in particular great ellipse calculations. The ge-

ometry of modelling structures implies the calculus

essentially, in particular the mathematical formulae

in the algorithms applied in the navigational elec-

tronic device and systems. Thus, is the spherical or

spheroidal model the best fit in the local approxima-

tions of the Earth surface? We show that generally in

navigation the essential calculating procedure refers

to the distance and angle measurement what may be

transferred to more general geometrical structures,

for instance metric spaces, Riemannian manifolds.

The authors point out that the locally modelling

structure has a different “shape” and thus the differ-

ent curvature and the flow of geodesics. That affects

the calculus provided on it. The algorithm applied

for navigational purposes, in particular ECDIS

should inform the user on actually used mathemati-

cal model and its limitations. The question we also

ask affects the range and point in applying the loxo-

drome sailing in case the ECDIS equipped with the

great circle (great ellipse) approximation algorithms

of given accuracy replaces the traditional nautical

charts based on Mercator projection. The shortest

distance (geodesics) depends on the type of metric

we use on the considered surface in general naviga-

tion. The geodesics can look different even on the

same plane if different metrics are taken into consid-

eration. Let us observe for instance the diameter of

the parallel of latitude conical circle does not pass its

centre. That differs from both the plane and spheri-

cal case. Our intuition insists on the way of thinking

to look at the diameter as a part of geodesic of the

researched surface crossing the centre of a circle.

However the diameter depends on the applied met-

ric, thus the shape of the circles researched in the

metric spaces depends on the position of the centre

and the radius. It is also important to know how the

distance between two points on considered structure

is determined, where the centre of the circle is posi-

tioned and how the diameter passes. Changing the

metric causes the differences in the obtained dis-

tances. For example π as a number is constant and

has the same value in each geometry it is used in

calculations. However π as a ratio of the circumfer-

ence to its diameter can achieve different values in

general, in particular π [Kopacz, 2010]. The naviga-

tion based on geodesic lines and connected software

of the ship’s devices (electronic chart, positioning

and steering systems) gives a strong argument to re-

search and use geodesic-based methods for calcula-

tions instead of the loxodromic trajectories in gen-

516

eral. The theory is developing as well what may be

found in the books on geometry and topology. This

motivates us to discuss the subject and research the

components of the algorithm of calculations for nav-

igational purposes.

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