Lateral marks


Navigation marks indicate the port and starboard sides of waterways. In IALA A regions (Europe, Australia, Africa, and certain Asian countries), the port side is marked with red during the day and night, while the starboard side is marked with green. In IALA B regions (North and South America, Japan, North and South Korea, and the Philippines), it’s the opposite. The marking line is determined by the boat’s direction from the open sea or as per authorities’ provisions in the agreement with neighboring countries.

green redred green

Modified lateral marks signify waterway bifurcation, highlighting the main channel. They feature distinct shapes like cylinders, pillars, or spars, often accompanied by corresponding light indicators.

Cardinal marks


Cardinal marks indicate a channel’s deepest waters, safe passages, and warnings. The system encompasses four quadrants, marked by right directions NW-SE, NE-SE, SE-SW, and SW-NW from the center. These marks, featuring pillar or spar shapes, are accompanied by corresponding flashing lights at intervals.

Isolated danger marks


They are set or anchored on an isolated danger or above the isolated danger of the smaller proportion with navigable water all around it. The mark can be bypassed on all sides.

Safe water marks


This mark indicates that safe water is situated within 360 degrees, usually applicable in the labeling of the central line or of the center of the channel.

Special marks


Special marks denote some areas or objects: marks of systems for collecting data on the open sea, marks of the places of discharge of materials, marks of military exercises, and marks of cables and pipelines.

New dangers marks


New danger marks indicate recently discovered hazards not documented in nautical publications. They are placed near the danger or within its vicinity until a permanent mark is established. These hazards encompass both natural and artificial obstacles.


Rhumb line sailing definition

Loxodrome is a navigational curve cutting all meridians at a constant angle, often used for its simplicity in maintaining a steady course. Despite its advantages, like a constant compass heading, it covers a longer distance than the great circle route. Navigational calculations involve two types of loxodromic problems: one determines course and distance from known latitude and longitude differences. At the same time, the other calculates latitude difference and spacing given course and distance. These problems are solved through equations derived from three loxodromic triangles.

loks trokut1

First loxodromic triangle or triangle of course and small distances

loks trokut2

Second loxodromic triangle or a triangle of medium widths

loks trokut3

Third loxodromic triangle or Mercator’s Triangle


Rhumb line sailing calculation

First loxodromic problem:
Departure position P1 (φ1, λ1) to the arrival position of P2 (φ2, λ2). The navigation course (Kl) and loxodromic distance (Dl) are unknown.

loks 1

Δλ = (λ2 – λ1) – relative coordinate, the difference of longitudes
ΔφM = (φM2 – φM1) – relative coordinate, the difference of Mercator’s widths
K – loxodromic angle in the triangle loxodromic

Calculation of Mercator’s widths:


Determination of loxodromic course:
Sailing in Ist navigation quadrant:
(Δφ> 0; Δλ> 0) Kl = K

Sailing in IInd navigation quadrant:
(Δφ <0; Δλ> 0) Cl = 180° K

Sailing in IIIrd navigation quadrant:
(Δφ <0; Δλ <0) Cl = 180° K

Sailing in the IVth navigation quadrant:
(Δφ> 0; Δλ <0) Cl = 360° K

Second loxodromic problem:
Departure position P1 (φ1, λ1) and specific loxodromic course (Kl), and crossed distance (Dl). It is necessary to calculate the relative coordinates (Δφ, Δλ).

loks 2

Calculation of latitude of the point of arrival:
φ2 = φ1 Δφ

Calculation of longitude of the point of arrival:
λ2 = λ1 Δλ

Special cases of loxodromic sailing:
1. Sailing on the equator:
Δφ = 0°, DI = Δλ ‘, K = 090° or 270°

2. Sailing on the parallel:
Δφ = 0°, DI = Δλ ∙ cos’, K = 090° or 270°

3. Sailing on the meridian:
Δλ = 0°, DI = Δφ ‘, K = 0° or 180°

Great circle sailing definition

A great circle is a shorter arc of a large circle whose plane passes through the Earth’s departure and arrival positions. Contrasted with the loxodrome, it offers a shorter path, requires frequent course adjustments, and leads to higher latitudes. Great circles include the equator and meridians. Spherical triangles arise when all sides are large circle arcs, allowing spherical trigonometry to solve elements of great circles, enhancing navigational calculations on Earth’s curved surface.

Great circle sailing calculation

In solving the problem of the great circle sailing, the coordinates of departure and arrival positions are given, so sides 90 – A and 90 – B and the angle between them are known in the great circle spherical triangle. Other initial elements of the great circle (Do and KPC) can be calculated using the cosine theorem of spherical trigonometry.

The great circle distance:

orto 1

Great circle initial and final course:

orto 2

orto 3

Sailing eastward: Kop = α; Kok = 180° – β
Sailing westward: Kop = 360° – α; Kok = 180° β

Great circle vertex:

orto 4

orto 5

Waypoints of Great Circles:

orto 6

orto 6 – 1

Composite sailing

Composite sailing is employed when a great circle’s summit enters high latitudes, involving selecting a border parallel that the vessel won’t exceed. Key elements include border points where the great circle intersects the loxodrome, distances of two great circles and one loxodrome, initial and final great circle courses, general loxodromic course, and waypoints.

Border points:

komb 1

komb 2

Total distance:

komb 3 

komb 4 

komb 5

komb 6

komb 7

Estimated Time of Arrival

The Earth’s surface is divided into 24 time zones, each determined by its central meridian. The zero zone ranges from -7.5° to 7.5°, with the Greenwich meridian as the central meridian. Time adjustments are made during maritime travel, shifting one hour forward when heading east and one hour back when heading west for every 15° of longitude. Crossing the date line deducts or adds a day. ETA is calculated by converting departure time to UT, adding travel time, and adjusting for local time.


Radio Detection and ranging (RADAR)


RADAR (Radio Detection And Ranging) uses radio waves to detect and measure the distance of objects. It emits short pulses of modulated electromagnetic waves in a wide beam. The waves reflect off obstacles and return to the radar, displaying reflections on the screen. Distance is measured based on the time taken for pulses to return. Azimuths are determined by rotating the transmitter aerial and cursor at the same speed as the ship’s heading, ensuring accurate readings on the screen.

radar antena

We distinguish the X-band radars operating at a frequency of 9000 MHz and the wavelength of 3 cm (λ = 3 cm), which show us more detailed images and are used in small ranges. Next, the S-band radars operate at a frequency of 3000 MHz and a wavelength of 10 cm (λ = 10 cm), mostly used for more extensive ranges and better work in poor weather conditions.

Automatic Radar Plotting Aids (ARPA)

ARPA (Automatic Radar Plotting Aids) is a radar device with an integrated computer that processes data for navigation, facilitating collision avoidance at sea. It monitors selected vessels, assesses the situation, and provides warnings through visual and acoustic signals if collision danger is detected. ARPA allows for test maneuver simulations, determining parameters like avoidance course, speed, and start time delay to enhance navigational safety.

ECDIS and electronic chart


ECDIS (Electronic Chart Display and Information System) is a navigational tool that displays selective information from the systemic electronic map (SENC) and positional data from navigational sensors. It facilitates route planning and trip surveillance and displays relevant navigational parameters, utilizing a standardized database designed for ECDIS systems.

ecdis display

Electronic Charts come in raster and vector formats. Raster maps resemble traditional navigation charts, displaying the same information. In contrast, vector maps use mathematical analysis to represent coastlines and features on the screen graphically. ECDIS systems utilize electronic charts, showing the vessel’s position and various navigational data. These systems offer features like easy map updates, vessel maneuvering characteristics, radar overlay, route planning, weather reports, and direct navigation plan control on the screen, enhancing navigation planning and control.

Global Positioning System (GPS)


GPS (Global Positioning System) is a worldwide navigational system facilitating position determination and navigation in various weather conditions. It relies on distance calculations from GPS satellites broadcasting their pre-calculated positions. The distances of four visible satellites are measured independently through time-delay analysis of transmitted signals. GPS signals are broadcasted on frequencies 1572.42MHz and 1227.6MHz, so GPS signals require at least four satellites for navigation. With an orbit inclination of 55%, these satellites form a constellation of 24 to 32, positioned at around 10980m in height. The system demands precise satellite clock synchronization with GPS time and requires user equipment with an accurate clock for distance measurements.


Astronomical navigation methods involve mathematical principles that use celestial body positions to aid orientation at sea. Nautical miles, equivalent to one minute of a large circle arc, are the basic unit for distance measurement. Astronomical navigation employs spherical trigonometry to solve spherical triangles defined by points on the sphere’s main circle. Spherical angles form at the midpoint of two major circles, creating a spherical angle, while spherical length measures the smaller arc between two points on the main circle.

Celestial sphere


The celestial sphere is a spherical surface projecting all celestial bodies, simplifying celestial observation and movement study. Two spherical coordinates define celestial bodies’ positions. Key points include the zenith (above the observer), nadir (center below), north and south celestial poles (above Earth’s poles). The celestial equator is the Earth’s equator projection, and the celestial horizon stems from the earthly horizon. The celestial meridian is the local observer’s meridian projection, while the ecliptic plane, Earth’s orbit path, intersects the celestial sphere, forming the ecliptic circle. This framework aids in understanding celestial bodies’ positions and movements during navigation.

The horizon system of coordinates


In astronomical navigation, two types of coordinate systems exist those dependent on the observer’s position and those that are not. The horizon coordinate system centers around the zenith and nadir, established by extending a perpendicular from the observer to the celestial sphere. Key circles include the celestial horizon, celestial meridian, and vertical circles. Altitude (H) measures the arc from the celestial horizon to a celestial body. At the same time, azimuth (W) is the arc from the northern (or southern) point to the vertical circle intersecting the celestial body, or the angle between the north (south) point and the intersection on the horizon.

The equator system of coordinates


The local equatorial system centers around the celestial equator’s plane. The north and south celestial poles are established by extending the Earth’s axis to the celestial sphere. Key circles include the celestial equator and meridians. Declination (δ) measures the arc from the celestial equator to a celestial body or the angle to its center. Positive δ indicates a northward position, while negative δ signifies a southward position. Hourly angle (s) measures the arc from the upper meridian to a celestial body’s hourly circle or the corresponding pole angle. It ranges from 0° to 360° westward or 0° to 180° eastward and westward.

The celestial equator system of coordinates


The celestial-equatorial system revolves around the celestial equator and the celestial meridian of a celestial body. Key coordinates include declination (δ) and sidereal hour angle (360° – α). Right ascension (RA) measures the arc from the vernal point to the celestial meridian through a celestial body or the angle in the pole between these. RA ranges from 0° to 360° counter-clockwise. Sidereal hour angle (SHA) measures the arc from the vernal point to a meridian passing through the celestial body, counting from 0° to 360°.

The celestial ecliptic system of coordinates


In the ecliptic coordinate system, the poles are the north and south poles of the ecliptic, obtained when the axis is perpendicular to the ecliptic plane. Key circles include the ecliptic and ecliptic meridians. Coordinates are given as latitude or ecliptic width (β) and longitude or ecliptic length (λ). Latitude is an arc from the ecliptic plane to the celestial body or the corresponding angle, ranging from 0° to 90°, positive in the northern hemisphere (N) and negative in the southern (S). Longitude is the arc from the vernal point to the celestial body on the ecliptic, counting from 0° to 360° progressively.



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