Navigation

(1)

Positions

Longitude and latitude

The earth can be regarded as a spherical object, and The earth can be regarded as a spherical object, and sin

since ce we'we're re deadealinling g witwith h a a 3-d3-dimeimensinsionaonal l shashape pe wewe need coordinates of a different form than the usual need coordinates of a different form than the usual x-an

and d y-ay-axesxes. . ThoThougugh h adaddinding g an an extextra ra z-az-axes xes wouwouldld make sense for submarines, we will most likely be make sense for submarines, we will most likely be fo

founund d on on ththe e susurfrfacace e of of ththis is spsphherere e whwhilile e ususiningg another system of coordinates, that covers our planet another system of coordinates, that covers our planet with imaginary lines called

with imaginary lines called meridiansmeridians andand parallelsparallels,,

see figure 1. All these lines together provide the grid see figure 1. All these lines together provide the grid w

whhicich h eennaabbllees s uus s tto o ddeessccrriibbe e aanny y ppoossiittiioon n iinn longitudes

longitudes andand latitudeslatitudes.. The obvious place to divide

The obvious place to divide the Northern and Southernthe Northern and Southern Hem

Hemispispherheres es was the ewas the equaquatortor. But t. But the dihe divisvision of tion of the Eahe Eastestern anrn and Wesd Westerternn hem

hemispispherheres es wawas s the the sosourcurce e of of mumuch ch polpolitiitical cal tuturmormoil. il. GreGreenwenwich ich (Gr(Greateat Britain) won, placing for example The

Britain) won, placing for example The Netherlands in the Eastern and IrelandNetherlands in the Eastern and Ireland in the Western

in the Western Hemisphere.Hemisphere.

It takes the earth 24 hours for a full rotation of It takes the earth 24 hours for a full rotation of 360°. Thus, every hour we

360°. Thus, every hour we rotate 15° longitude,rotate 15° longitude, see figure 2.

see figure 2.

When it is 12:00 UTC (international standard When it is 12:00 UTC (international standard time) - anywhere in the world - it

time) - anywhere in the world - it is 12:00 Localis 12:00 Local Time in Greenwich and 24:00 Local Time at the Time in Greenwich and 24:00 Local Time at the other side of the planet: 180° E or 180° W: the other side of the planet: 180° E or 180° W: the da

datte e llininee. . CCrorosssisinng g ththis is ssppececiaial l mmererididiaiann changes not only the hour but also the

changes not only the hour but also the date.date. The North Pole has a latitude of 90° N and the The North Pole has a latitude of 90° N and the South Pole 90° S. The meridians cover twice South Pole 90° S. The meridians cover twice

(2)

this angle up to 180° W or E. this angle up to 180° W or E.

Meridians converge at the poles, whereas parallels run parallel to each other Meridians converge at the poles, whereas parallels run parallel to each other and never meet. All meridians and the equator - the biggest parallel - form and never meet. All meridians and the equator - the biggest parallel - form great circles

great circles, and the remaining parallels form so-called, and the remaining parallels form so-called small circlessmall circles. . AA great circle divides the earth in

great circle divides the earth in two exact halves.two exact halves. In figure 3 the position of

In figure 3 the position of Boston in the United States is shown using latitudeBoston in the United States is shown using latitude and longitude in degrees, minutes and

and longitude in degrees, minutes and seconds:seconds: 42° 21' 30" N , 71° 03' 37" W

42° 21' 30" N , 71° 03' 37" W

Most sailors will actually notate seconds in metric fractions of minutes: Most sailors will actually notate seconds in metric fractions of minutes: 42° 21,5' N , 71° 03,6' W or

42° 21,5' N , 71° 03,6' W or 42

42° 2° 21.1.5' 5' N , N , 7171° 0° 03.3.6' 6' WW , s, see ee ththee notation style guidenotation style guide

On small scaled charts we want to be accurate within one minute or one On small scaled charts we want to be accurate within one minute or one nautical mile. On larger scaled charts the accuracy is

nautical mile. On larger scaled charts the accuracy is more likely to be more likely to be withinwithin a tenth of a mile (a cable).

a tenth of a mile (a cable).

IIf f tthhe e eeaarrtth h wweerre e a a ppeerrffeecct t sspphherere e wwitith h aa cir

circucumfmferenerence ce of of rouroughlghly y 40040000 00 kilkilomeometrtres es allall great circles - meridians plus the

great circles - meridians plus the equator - wouldequator - would have the same length and could be used as a have the same length and could be used as a dist

distance unit when ance unit when dividdivided ed into 360 into 360 degredegrees, ores, or 36

360° 0° x x 6060' ' = = 212160600' 0' miminunutetes. s. In In 19192929, , ththee international community agreed on the definition international community agreed on the definition of 1 international nautical mile as 1852 metres, of 1 international nautical mile as 1852 metres, wh

whicich h is is rorougughlhly y ththee averageaverage lelengngth th of of ononee minute of latitude i.e. one minute of arc along a minute of latitude i.e. one minute of arc along a line of longitude (a meridian).

line of longitude (a meridian). Or to put it shortly:

Or to put it shortly: 1 nm = 1'1 nm = 1'

We

We arare e nonow w abable le to to dedescscribribe e anany y poposisitition on in in lalatititutudedes s anand d lolongngititududeses.. Moreover, we can state the distance between two of those positions using Moreover, we can state the distance between two of those positions using nautical miles or minutes. All we need now is a proper way to define speed. nautical miles or minutes. All we need now is a proper way to define speed. For that, sailors use

For that, sailors use knotsknots, the number of nautical miles an hour., the number of nautical miles an hour.

RYA

& &

ASA sailing schools

To put this navigation course into practice a

To put this navigation course into practice a Royal Yachting AssociationRoyal Yachting Association oror American Sailing Association

American Sailing Association approved sailing course is recommended.approved sailing course is recommended. Sailing schools in Greece and Turkey for:

Sailing schools in Greece and Turkey for: RYA competent crew

RYA competent crew

RYA day skipper (non-tidal) RYA day skipper (non-tidal) RYA coastal skipper (non-tidal) RYA coastal skipper (non-tidal) ASA basic coastal cruising (103) ASA basic coastal cruising (103) ASA bareboat chartering (104) ASA bareboat chartering (104) ASA coastal navigation (105) ASA coastal navigation (105)

And the most enjoyable way to learn how to sail is by combining such And the most enjoyable way to learn how to sail is by combining such courses with a yachting vacation in Greece or Turkey. Ideal areas for sailing courses with a yachting vacation in Greece or Turkey. Ideal areas for sailing courses are the

courses are the Saronic Gulf near AthensSaronic Gulf near Athens and theand the Ionian IslandsIonian Islands to theto the west of Greece, which provide reliable

(3)

Greek monuments and temples. Greek monuments and temples.

A little History

Mariners during the 15th century relied on charts called "portolans" to assist Mariners during the 15th century relied on charts called "portolans" to assist them on their voyages.

them on their voyages. PortolanPortolan comes from the Italian wordcomes from the Italian word portolani portolani ,, which were medieval pilot books.

which were medieval pilot books. Th

The e poportrtololaans ns cocontntaiainened d mmapaps s of of cocoasastltlinineses, , lolocacatitionons s of of harbours, river mouths, and man-made features visible from the harbours, river mouths, and man-made features visible from the sseeaa. . TThheey y wweerre e a a ccoommppiillaattiioon n oof f cceennttuurriiees s oof f sseeaaffaarreerr ob

obseservrvatatioionsns. . As As sasaililorors' s' skskilills ls imimprprovoved ed anand d ththe e ususe e of of ththee

compass

compass was more widespread, portolans improved in was more widespread, portolans improved in accuracy.accuracy.

Al

Also Colso Columumbubus s usused theed these porse portotolalans on his jouns on his journrneyeys. Pors. Portutuguguesese e chcharartt makers added the meridian line, a point useful for latitude sailing as well as makers added the meridian line, a point useful for latitude sailing as well as for navigating solely by compass. A geographic feature could now be located for navigating solely by compass. A geographic feature could now be located through the use of its distance in degrees of latitude from a ship's point of through the use of its distance in degrees of latitude from a ship's point of departure. Note that the use of latitude and longitude was understood since departure. Note that the use of latitude and longitude was understood since th

the te time ime of of PtoPtolemlemyy , t, the he secsecond ond cencentutury ry CE.CE. Du

Duriring ng ththe e fififtfteeeentnth h cecentnturury y PoPortrtugugal al leled d ththe e EuEuroropepean an woworlrld d in in seseaa exploration. The golden age of discovery for Portugal lasted almost a century exploration. The golden age of discovery for Portugal lasted almost a century until the Dutch eventually seized their trade routes from them.

until the Dutch eventually seized their trade routes from them. As we move to the next chapter of this course we

As we move to the next chapter of this course we enter the sixteenth centuryenter the sixteenth century when the Mercator chart was invented.

when the Mercator chart was invented.

Glossary

1.

1. ParallelsParallels: Circles parallel to the equator, ranging from 0° to 90° N or: Circles parallel to the equator, ranging from 0° to 90° N or

S. Only the equator is a great circle. S. Only the equator is a great circle.

2.

2. MeridiansMeridians: half-circles converging at the poles, ranging from 0° to: half-circles converging at the poles, ranging from 0° to 180° E or W.

180° E or W. Each pair of opposing meridians forms a great circle.Each pair of opposing meridians forms a great circle.

3.

3. Prime meridianPrime meridian: 0° or the Greenwich meridian which - together with: 0° or the Greenwich meridian which - together with the date line meridian - divides the

the date line meridian - divides the Western and Eastern hemispheres.Western and Eastern hemispheres.

4.

4. Great circleGreat circle: The intersection of a sphere and a plane that passes: The intersection of a sphere and a plane that passes

through the sphere's centre. through the sphere's centre.

5.

5. Small circleSmall circle: The intersection of a sphere and a plane that doesn't: The intersection of a sphere and a plane that doesn't pass though the sphere's centre.

pass though the sphere's centre.

6.

6. TimTime e zonzoneses: : By By coconvnvenentition on 24 24 zozonenes, s, eaeach ch 1515° ° lolongngititudude e wiwidede.. Hence, noon at Greenwich gives midnight at 180° E.

Hence, noon at Greenwich gives midnight at 180° E.

7.

7. GMGMT, T, UTUTC, C, ZuZululu: The : The outoutdatdated ed accaccronronym ym GMT GMT (Gr(Greeneenwiwich ch MeaMeann

Time) is roughly the same as UTC or Zulu, and is also the local time at Time) is roughly the same as UTC or Zulu, and is also the local time at Greenwich when daylight saving isn't used. Note that UTC is an atomic time Greenwich when daylight saving isn't used. Note that UTC is an atomic time

(4)

scale which only approximates GMT, so best to use the modern term “UTC”. scale which only approximates GMT, so best to use the modern term “UTC”. Antonym: Local time elsewhere. For example, local time in Athens = UTC + Antonym: Local time elsewhere. For example, local time in Athens = UTC + 2.

2.

8.

8. Date lineDate line: The 180° meridian which extends from or : The 180° meridian which extends from or is opposite to theis opposite to the

prime meridian. Here, not only the hour

prime meridian. Here, not only the hour changes when crossing the meridian,changes when crossing the meridian, but also the date.

but also the date.

9.

9. LatitudeLatitude: Position property defined by the number of : Position property defined by the number of degrees north ordegrees north or

south of the equator, varies from 0°

south of the equator, varies from 0° to 90°.to 90°.

10.

10. LongitudeLongitude: Position property defined by the number of degrees east: Position property defined by the number of degrees east

or west of the prime meridian, varies from 0° to 180°. or west of the prime meridian, varies from 0° to 180°.

11.

11. PositionPosition: Latitude first and longitude second. For example: Athens in: Latitude first and longitude second. For example: Athens in Greece 37° 58' N , 23° 43' E.

Greece 37° 58' N , 23° 43' E.

12.

12. Nautical mileNautical mile: One nm is one minute (') on the vertical scale on the: One nm is one minute (') on the vertical scale on the chart. 1' equals 1852 metres. Nautical miles are divided into 10

chart. 1' equals 1852 metres. Nautical miles are divided into 10 cables.cables.

13.

13. KnotsKnots: Nautical miles per hour.: Nautical miles per hour.

Chapter-2

Chapter-2 NauticalNautical

char

(5)

Skip over navigation

Compass

navigation

Skip over navigation Skip over navigation

Marine compass

In

In ChinaChina compasses have been in use since the Han dynasty (2nd centurycompasses have been in use since the Han dynasty (2nd century BCE

BCE to 2nd century CE) when they were referred to as “south-pointers”.to 2nd century CE) when they were referred to as “south-pointers”.

However at first these magnets were only used for geomancy much like in However at first these magnets were only used for geomancy much like in the art of Feng Shui.

the art of Feng Shui.

Eventually, during the Sung dynasty (1000 CE) many trading

Eventually, during the Sung dynasty (1000 CE) many trading ships were thenships were then able to sail as far as Saudi Arabia using compasses for marine navigation. able to sail as far as Saudi Arabia using compasses for marine navigation. Between 1405 and 1433, Emperor Chu Ti's Treasure Fleet of the Dragon Between 1405 and 1433, Emperor Chu Ti's Treasure Fleet of the Dragon Throne ruled the entire South Pacific and the Indian Ocean, a territory that Throne ruled the entire South Pacific and the Indian Ocean, a territory that ranges from Korea and Japan to the

ranges from Korea and Japan to the Eastern coast of Africa.Eastern coast of Africa.

At this time Western mariners were still rather ignorant of the navigational At this time Western mariners were still rather ignorant of the navigational us

use e of thof the e mamagngnetet. . PePetrtrus Perus Perigigririnunus s vavan Marin Maricocoururt t wrwrotote e a firsa first t trtreaeatitisese on the magnet itself: “De Magnete” (1269). And though its nautical use was on the magnet itself: “De Magnete” (1269). And though its nautical use was already mentioned in 1187 by the

already mentioned in 1187 by the English monk Alexander Neckham, the useEnglish monk Alexander Neckham, the use on

onboboarard d ononly ly cacame me ababouout t ararouound nd ththe e 1313th th anand d 1414th th cecentnturury y in in ththee Mediterranean Sea.

Mediterranean Sea.

Much later, in 1545, Pedro de Medina (Sevilla 1493-1567) wrote the Spanish Much later, in 1545, Pedro de Medina (Sevilla 1493-1567) wrote the Spanish st

stanandadard rd wowork rk “A“Artrte e de de NaNavevegagar” r” on on mamaririne ne cocompmpasass s nanavivigagatitionon. . ThThisis masterpiece was first translated in Dutch (1580) and was -O Irony- used by masterpiece was first translated in Dutch (1580) and was -O Irony- used by Ja

Jacocob van Heeb van Heemsmskekerk rk whwhen then the Dute Dutch dech deststroroyeyed the Spad the Spaninish flsh fleeeet neart near Gibraltar in 1607. The drawback was of course Van Heemskerk's own death Gibraltar in 1607. The drawback was of course Van Heemskerk's own death during this victory.

during this victory.

Magnetic Variation

In

In ththe e finfin-d-de-se-sièciècle le of of ththe e sixsixteeteentnth h cencenturtury y marmarineiners rs belbelievieved ed ththat at ththee m

(6)

ssuuggggeessttiioon n ootthheerrwwiisse e hhaad d bbeeeen n ddeenniieed d bby y PPeeddrro o dde e MMeeddiinnaa..

Magn

Magnetic observatetic observations ions madmade e by by exploexplorers rers in in subssubsequenequent t decadecades des showshoweded ho

howewevever r ththat at ththesese e susuggggesestitionons s wewere re trtrueue. . BuBut t it it totook ok ununtitil l ththe e eaearlrlyy nineteenth century, to pinpoint the magnetic north pole somewhere in Arctic nineteenth century, to pinpoint the magnetic north pole somewhere in Arctic Canada (78° N , 104° W). From then on the angle between the true North Canada (78° N , 104° W). From then on the angle between the true North and the

and the Magnetic NorthMagnetic North could be precisely corrected for. This correction anglecould be precisely corrected for. This correction angle

is called

is called magnetic variationmagnetic variation or declination.or declination. It is believed that

It is believed that the Earth's magnetic field is produced by electrical currentsthe Earth's magnetic field is produced by electrical currents that originate in the hot, liquid, outer core of the rotating Earth. The flow of that originate in the hot, liquid, outer core of the rotating Earth. The flow of electric currents in this core is continually changing, so the magnetic field electric currents in this core is continually changing, so the magnetic field produced by those currents also changes. This means that at the surface of produced by those currents also changes. This means that at the surface of the Earth, both the strength and direction of the magnetic field will vary over the Earth, both the strength and direction of the magnetic field will vary over th

the e yeyearars. s. ThThis is grgradaduaual l chchanange ge is is cacalllled ed ththee seculsecular ar variatvariationion of of ththee magnetic field. Therefore, variation changes not only with the location of a magnetic field. Therefore, variation changes not only with the location of a vessel on the earth but also

vessel on the earth but also varies in time.varies in time. Th

The e corcorrecrectiotion n for for mamagngnetietic c varvariatiation ion for for youyour r loclocatation ion is is shshown own on on thethe nearest!

nearest! nautical chart's compass rosenautical chart's compass rose. In this example we find a variation of 4°. In this example we find a variation of 4°

15' W in 2009, with an indicated annual correction of 0° 08' E. Hence, in 15' W in 2009, with an indicated annual correction of 0° 08' E. Hence, in 2011 this variat

2011 this variation is ion is estiestimatemated d to be to be 3° 59', 3° 59', almoalmost 4° st 4° WestWest. . This meanThis meanss that if we sail 90° on the chart (the true course), the compass would read that if we sail 90° on the chart (the true course), the compass would read 94°.

94°.

Another example: let's say the compass rose gives a variation of 2° 50' E in Another example: let's say the compass rose gives a variation of 2° 50' E in 2007, with a correction of 0° 04' E per year. In 2009 this variation is 2007, with a correction of 0° 04' E per year. In 2009 this variation is estimated to be 2° 58', almost 3° East. Now, if we sail 90° on the chart, the estimated to be 2° 58', almost 3° East. Now, if we sail 90° on the chart, the compass would read 87°.

compass would read 87°.

Correcting for variation

(7)

These overlayed compass roses show These overlayed compass roses show the difference between true north and magnetic north when the magnetic the difference between true north and magnetic north when the magnetic variation is 10° West.

variation is 10° West. From the image we find:

From the image we find: tc = cc + vartc = cc + var in

in whwhicich h ““cccc” and “” and “tctc” ” ststanand d fofor r “c“comompapass ss cocourursese” ” anand d “t“trurue e cocourursese”,”, respectively.

respectively. To convert a

To convert a true course into a compass coursetrue course into a compass course we need first assign a “-” we need first assign a “-”

to a

to a WestWestern and a ern and a “+” to “+” to a a EasEastern variattern variation. Note that this makes sense!ion. Note that this makes sense! because of the clockwise direction of the compass rose. Here,

because of the clockwise direction of the compass rose. Here, the inner circlethe inner circle is turned 10°

is turned 10° anticlockwisanticlockwise, hencee, hence _--1100°° .. Now, use the

Now, use the same but re-written equation:same but re-written equation: cc = tc - var

cc = tc - var

235° = 225° - (-10°) 235° = 225° - (-10°)

So, to sail a true course of 225°, the helmsman has to steer a compass So, to sail a true course of 225°, the helmsman has to steer a compass course of 235°.

course of 235°. To convert a

To convert a compass course into a true coursecompass course into a true course we can use the originalwe can use the original

equation. If we have steered a compass course of 200°, we have to plot a equation. If we have steered a compass course of 200°, we have to plot a tru

true coue course orse of 20f 203° in 3° in ththe chae chart if trt if the vahe variariatiotion is 3n is 3° Eas° East t or a tror a true cue couroursese of

of 19190° 0° if if ththe ve varariaiatition on is is 1010° W° Wesestt ..

Magnetic deviation

Magnetic deviation is the second correctable error. The deviation error is Magnetic deviation is the second correctable error. The deviation error is caused by magnetic forces within your particular boat. Pieces of metal, such caused by magnetic forces within your particular boat. Pieces of metal, such as an engine or an anchor, can cause magnetic forces. And also stereo and as an engine or an anchor, can cause magnetic forces. And also stereo and oth

other er eleelectrctric ic equequipmipment ent or or wirwiringing, , if if too too cloclose se to to the the comcompaspass, s, intintrodroduceuce errors in

errors in compass heading.compass heading.

Furthermore, the deviation changes with the ship's heading, resulting in a Furthermore, the deviation changes with the ship's heading, resulting in a deviation table as shown below. The vertical axis states the correction in deviation table as shown below. The vertical axis states the correction in degrees West or East, where East is

(8)

The horizontal axis states the ship's heading in degrees divided by ten. Thus, The horizontal axis states the ship's heading in degrees divided by ten. Thus, when you sail a compass course of 220°, the deviation is 4° W. (Note, that when you sail a compass course of 220°, the deviation is 4° W. (Note, that on most modern sailing yachts the deviation is usually not larger than 3°). on most modern sailing yachts the deviation is usually not larger than 3°). When a compass is newly installed it often shows larger deviations than this When a compass is newly installed it often shows larger deviations than this and

and neeneeds ds comcompenpensatsation ion by by carcarefuefully lly plaplacincing g smsmall all mamagngnets ets aroaround und ththee compass. It is the remaining error that is

compass. It is the remaining error that is shown in your deviation table.shown in your deviation table.

You can check your table every now and then by placing your boat in the line You can check your table every now and then by placing your boat in the line of a pair of

of a pair of leading lightsleading lights and turning her 360 degrees.and turning her 360 degrees.

Correcting for both deviation and variation

Converting a

Converting a compass course into a true coursecompass course into a true course, we can still use our, we can still use our equation but we need to add the

equation but we need to add the correction for deviation:correction for deviation: cc + var + dev = tc

cc + var + dev = tc Example 1

Example 1: The compass course is 330°, the deviation is +3° (table) and: The compass course is 330°, the deviation is +3° (table) and

the variation is +3° (chart); the variation is +3° (chart);

330° cc + 3° var + 3° dev = ?° tc 330° cc + 3° var + 3° dev = ?° tc

giving a true course of 336° which we can plot in our chart giving a true course of 336° which we can plot in our chart Example 2

Example 2: The compass course is 220°, the deviation is -4° (table) and the: The compass course is 220°, the deviation is -4° (table) and the

variation is still

variation is still +3° (chart).+3° (chart).

220° cc + 3° var + -4° dev = ?° tc 220° cc + 3° var + -4° dev = ?° tc giving a true course of

giving a true course of 219°.219°. Example 3

Example 3: The compass course is still 220°, therefore the deviation is still: The compass course is still 220°, therefore the deviation is still -4° (table) but let's use a variation of

-4° (table) but let's use a variation of -10° this time.-10° this time. 220° cc + -10° var + -4°

220° cc + -10° var + -4° dev = ?° tcdev = ?° tc giving a true course of

giving a true course of 206°.206°. Converting a

Converting a true course into a compass coursetrue course into a compass course is a little less straightis a little less straight forward, but it is still

forward, but it is still done with the same equation.done with the same equation. Example 4

Example 4: The true course from the chart is 305° and the variation is +3°: The true course from the chart is 305° and the variation is +3°

(chart), yet we don't know the deviation; (chart), yet we don't know the deviation; ?° cc + 3° var +

?° cc + 3° var + ?° dev = 305° tc?° dev = 305° tc Luckily, we can rewrite this so

Luckily, we can rewrite this so this reads:this reads: cc + dev = 305° tc - + 3° var = 302° cc + dev = 305° tc - + 3° var = 302°

In plain English: the difference between the true course and the variation In plain English: the difference between the true course and the variation (305 - + 3) = 302 should also be the summation of the compass course and (305 - + 3) = 302 should also be the summation of the compass course and the deviation. So, we can tell our helms person to

the deviation. So, we can tell our helms person to steer 300°, since with a ccsteer 300°, since with a cc of 300° we have a deviation of +2° (As can be deduced from the deviation of 300° we have a deviation of +2° (As can be deduced from the deviation table above).

(9)

Example 5

Example 5: The true course from the chart is 150° and we have a Western: The true course from the chart is 150° and we have a Western variation of 7 degrees (-7°). We

variation of 7 degrees (-7°). We will use the rewritten equation to get:will use the rewritten equation to get: 150° tc - - 7° var

150° tc - - 7° var = cc + dev = 157°= cc + dev = 157°

From the deviation table we find a compass course of 160° with a deviation From the deviation table we find a compass course of 160° with a deviation of -3°. of -3°. Voilà! Voilà!

Magnetic course

The

The magnetic coursemagnetic course (mc) is the heading after magnetic variation has been(mc) is the heading after magnetic variation has been

considered, but without compensation for magnetic deviation. This means considered, but without compensation for magnetic deviation. This means that we are dealing with the rewritten equation from above:

that we are dealing with the rewritten equation from above: tc - var = cc + dev =

tc - var = cc + dev = mcmc..

Magnetic courses are used for three reasons: Magnetic courses are used for three reasons:

1.

1. To convTo convert a true coert a true course inturse into a compao a compass courss course like we sase like we saw in the lasw in the lastt paragraph.

paragraph. 2.

2. On vesseOn vessels with more thals with more than one steerinn one steering compag compass, also morss, also more deviatie deviationon tables are in use; hence only a magnetic or true course is plotted in tables are in use; hence only a magnetic or true course is plotted in the chart.

the chart. 3.

3. BeBeararinings gs tatakeken n wiwith th a a hahanndhdheleld d cocommpapass ss ofofteten n ddonon't 't rereququirire e aa correction for deviation, and are therefore useful to

correction for deviation, and are therefore useful to plot in the chart asplot in the chart as magnetic courses.

magnetic courses.

Note, that the actual course lines the navigator draws in

Note, that the actual course lines the navigator draws in the chart are alwaysthe chart are always true courses! These can subsequently be labeled with the true course or the true courses! These can subsequently be labeled with the true course or the corresponding magnetic or compass course if appropriate. In the

corresponding magnetic or compass course if appropriate. In the next chapternext chapter

we will be

we will be plotting courses in the chart.plotting courses in the chart.

To summarise, we have three types of “north” (true, magnetic and compass To summarise, we have three types of “north” (true, magnetic and compass nor

northth) ) liklike e we we hahave ve ththree typeree types s of of coucourserses: s: tc, mc tc, mc and cc. and cc. All thesAll these e areare related by deviation and

related by deviation and variation.variation.

Glossary

1. 1. MaMaps ps wiwith th isisogogononiic c llinineess :: World - overview 2000 World - overview 2000 World - detailed 2000 World - detailed 2000 World - detailed 2005 World - detailed 2005

(10)

Wor

World ld - - animanimateated d in in timtimeeVariationVariation: The angle between the magnetic north pole: The angle between the magnetic north pole

and

and the the geoggeographraphic nic north orth pole. pole. Also Also callcalled ted the mhe magneagnetic dtic declineclinatioationn ..

2.

2. Secular variationSecular variation: The change of magnetic declination in time with: The change of magnetic declination in time with respect to both strength and direction of its magnetic field.

respect to both strength and direction of its magnetic field.

3.

3. WestWest (-)(-) , East, East (+)(+): Western variations or deviations are designated: Western variations or deviations are designated

with a negative sign by convention due to the compass card's clockwise with a negative sign by convention due to the compass card's clockwise direction.

direction.

4.

4. DeviationDeviation: The error in compass heading caused by electric magnetic: The error in compass heading caused by electric magnetic currents and or metal objects.

currents and or metal objects.

5.

5. Deviation tableDeviation table: A table containing deviations in degrees versus the: A table containing deviations in degrees versus the ship's heading (compass course) in degrees. Usually

ship's heading (compass course) in degrees. Usually plotted in a plotted in a graph.graph.

6.

6. True courseTrue course: Course plotted in the chart i.e. course over the ground: Course plotted in the chart i.e. course over the ground

or “course made good”. The

or “course made good”. The course corrected for compass errors.course corrected for compass errors.

7.

7. Compass courseCompass course: The course (ship's heading) without the correction: The course (ship's heading) without the correction for compass errors.

for compass errors.

8.

8. cc + var + dev = tccc + var + dev = tc: This equation shows the connection between the: This equation shows the connection between the

compass course, its errors and the true course. It can also be read as: tc compass course, its errors and the true course. It can also be read as: tc -var = cc + dev.

(11)

Plotting and

piloting

Lines of position

Th

The e momodedern rn chcharart t shshowows s us us poposisititionons s of of mamany ny recrecogogninizazablblee aiaids ds toto

navigation

navigation like churches and lighthouses, which facilitate the approach to alike churches and lighthouses, which facilitate the approach to a co

coasastatal l arareaea. . ThThis conis concecept oript origiginanateted d frfrom a om a chcharart t by Wagby Waghehenanaer er anandd proved a milestone in the development of European cartography. This work proved a milestone in the development of European cartography. This work was c

was callalled “Sped “Spiegieghehel der Zeel der Zeevaevaerdrdt” t” and iand incncludluded coaed coaststal proal profilfiles and tes and tidaidall information much like the modern chart. It enables us to find the angle information much like the modern chart. It enables us to find the angle between the North and for example an offshore platform, as seen from our between the North and for example an offshore platform, as seen from our position.

(12)

Compass courses

Compass courses True coursesTrue courses

Taking a bearing on this oil rig with a compass provides us with a compass Taking a bearing on this oil rig with a compass provides us with a compass course. This course first needs correction for both variation and - via ship's course. This course first needs correction for both variation and - via ship's heading

-heading - deviationdeviation before plotting abefore plotting a Line of PositionLine of Position (LOP) in the chart(LOP) in the chart

as a true course. as a true course.

Our position is somewhere along this line. Our position is somewhere along this line.

Ranges

A precise way to obtain a LOP, and A precise way to obtain a LOP, and without a compass, is to locate two aids to navigation in line. The map of without a compass, is to locate two aids to navigation in line. The map of Laura Island on the right shows four examples of

Laura Island on the right shows four examples of rangesranges, each consisting of , each consisting of two aids to

two aids to navigation.navigation. Please, note that:

Please, note that:

More distance between the two landmarks enhances accuracy. More distance between the two landmarks enhances accuracy.

And less distance between the vessel and the closest aid to navigation also And less distance between the vessel and the closest aid to navigation also enhances accuracy.

enhances accuracy.

One of these four ranges consists of two lights that are intentionally placed One of these four ranges consists of two lights that are intentionally placed to provide a LOP. These pairs of lights are called

(13)

lights

lights. In this case they indicate the approach towards the marina and mark. In this case they indicate the approach towards the marina and mark th

the e chchanannenel l bebetwtweeeen n ththe e dadangngererouous s rorockcks s aalolong a ng a trtrue couue coursrse e of 50°of 50° .. Wh

When looen lookiking towng towarards any leads any leadiding ligng lighthts, the neas, the nearerest one wilst one will l be lowbe lowerer ..

Therefore, in the middle of the channel both Therefore, in the middle of the channel both lights will appear vertically above each other.

lights will appear vertically above each other.

Even when there are no man-made structures available, a range can be Even when there are no man-made structures available, a range can be found by using natural features such as coastlines and islets. The

found by using natural features such as coastlines and islets. The example onexample on the left shows a yacht that will avoid the dangerous wreck as long as the the left shows a yacht that will avoid the dangerous wreck as long as the islets don't overlap.

islets don't overlap.

Position fix

If two LOPs intersect we can construct a

If two LOPs intersect we can construct a position fixposition fix: the ship's position on: the ship's position on the earth.

the earth. Of

Oftten en hhowoweveverer, , a a ttririaannggle le ococccururs s wwhhen en a a tthhirird d LLOP OP is is aaddddeed d in in tthhee construction. This indicates that there are errors involved in at least one of construction. This indicates that there are errors involved in at least one of the bearings taken. In practice, we should consider each LOP as the average the bearings taken. In practice, we should consider each LOP as the average be

beararining in g in a wa widider ser secectotor of r of fofor inr inststanance 1ce 10°0° ..

The

The optoptimuimum m angangulaular r sprspreadead isis 90°90° (t(two wo obobjecjectsts) ) oror 120°120° (three(three objects). Moreover, bearings on distant objects bring about

objects). Moreover, bearings on distant objects bring about more uncertaintymore uncertainty in our position fix

in our position fix as the sector widens. Finally, if moving fast you as the sector widens. Finally, if moving fast you should notshould not put any time between the

put any time between the bearings.bearings.

The next example features a nocturnal landfall on Willemsen Island - you are The next example features a nocturnal landfall on Willemsen Island - you are welcome to visit, but mind the rocks. The position fix is plotted by taking welcome to visit, but mind the rocks. The position fix is plotted by taking

(14)

bearings at two light-vessels as their lights appear over the

bearings at two light-vessels as their lights appear over the horizonhorizon . The. The

variation is -1° and the ship's compass heading is 190°. Since we use our variation is -1° and the ship's compass heading is 190°. Since we use our st

steeeeriring cong commpapass ss fofor r ouour bear bearirinngsgs, we , we cacan usn use the sae the sameme deviation tabledeviation table..

That means a deviation of -4° with which we can calculate (

That means a deviation of -4° with which we can calculate ( cc + var + dev = tccc + var + dev = tc))

the true courses. the true courses.

ction ction ompass ompass earing on Will. earing on Will. is 72° is 72° ru

rue e cocoururse se isis 7°

7° lo

lot t LOLOP P wiwithth m

me e & & ttrruuee ourse ourse ompass ompass earing on Will. earing on Will. is 173° is 173° ru

rue e cocoururse se isis 68°

68° lo

lot t LOLOP P wiwithth m

me e & & ttrruuee ourse ourse raw an ellipse raw an ellipse h heerre e tthhee OPs intersect OPs intersect o

ottaatte e ttiimmee nd

nd ““Fix Fix ”” longside

longside os

osititioion n is is 3232°° 4

4,,22' ' N N , , 2244°° 6,7' E

6,7' E

Wi

Withthouout t a a ththirird d LOLOP P - - foformrmining g ththe e drdreaeadeded d trtriaiangngle le - - ththerere e is is ththe e fafalslsee suggestion of accuracy. Yet, instrument errors, erroneous identification of an suggestion of accuracy. Yet, instrument errors, erroneous identification of an aid to navigation, sloppy plotting, etc. can and will cause navigation errors. aid to navigation, sloppy plotting, etc. can and will cause navigation errors. Therefore, if close to e.g. rocks, you should assume to be at the worst Therefore, if close to e.g. rocks, you should assume to be at the worst possible position (i.e. closest to

possible position (i.e. closest to the navigational hazard).the navigational hazard). The

The lines plotted in the chart are always true courseslines plotted in the chart are always true courses and these areand these are labeled with true courses by default; the “

labeled with true courses by default; the “T T ” is optional. If labeled with the” is optional. If labeled with the corresponding

corresponding magnmagnetic etic courcoursese or or cocommpapasss s ccouoursrse e aaddd d an an ““M M ” o r “” o r “C C ”,”, respectively.

respectively.

Estimated position

It

It is is sosomemetitimemes s imimpopossssibible le to to obobtatain in momore re ththan an onone e LOLOP P at at a a titimeme. . ToTo de

(15)

running fix

running fix. However if a running fix is not possible, we can determine an. However if a running fix is not possible, we can determine an estimated position

estimated position..

An

An esesttimimatated ed poposisitition on is is bbasased ed upuponon whatever incomplete navigational information is available, such as a single whatever incomplete navigational information is available, such as a single LOP, a series of depth measurements correlated to charted depths, or a LOP, a series of depth measurements correlated to charted depths, or a visual observation of the

visual observation of the surroundings.surroundings. In the example on the right we

In the example on the right we see an estimated position constructsee an estimated position constructed using aed using a sin

single gle LOP and LOP and ththe e shship'ip'ss dead dead reckoreckoning ning positpositionion ((DDRR)) . T. Thhiis is is ds doonne be byy

drawing a line from the DR position at the time of the LOP perpendicular to drawing a line from the DR position at the time of the LOP perpendicular to the LOP. An EP is denoted by a square instead of an ellipse.

the LOP. An EP is denoted by a square instead of an ellipse. Do not rely on an EP as much as a fix. The

Do not rely on an EP as much as a fix. The scale of reliabilityscale of reliability, from best to, from best to worst: worst: Fix Fix Running fix Running fix Estimated position Estimated position DR position DR position

Dead reckoning

Dea

Dead reckd reckonioning ng is a teis a techchniqnique to due to deteetermirmine a shne a ship'ip's apps approxroximaimate pote positsitionion by applying to the last established charted position a vector or series of by applying to the last established charted position a vector or series of vectors representing true courses and speed. This means that if we have an vectors representing true courses and speed. This means that if we have an earlier fix, we plot from

earlier fix, we plot from that position our course and “distance travelled sincethat position our course and “distance travelled since then” and deduce our

then” and deduce our current position.current position. e

e start off with a Fix and plot a DR positionstart off with a Fix and plot a DR position or 15 minutes later.

or 15 minutes later.

ur estimation about our speed and course ur estimation about our speed and course as correct, so we don't have to charge the as correct, so we don't have to charge the R position.

R position. nd so on… nd so on…

ed through water (not over ground) ed through water (not over ground) rse through water (not over ground) rse through water (not over ground)

True course (default) True course (default)

(16)

=

= MagMagnetnetic ic coucourse rse for for hanhandheldheld d comcompaspass s (no(no correction)

correction)

Compass course for steering compass (deviation Compass course for steering compass (deviation n)

n)

h an arrow, a semi-circle (circular arc) and “ h an arrow, a semi-circle (circular arc) and “DRDR”.”.

Dead reckoning is crucial since it provides an approximate position in the Dead reckoning is crucial since it provides an approximate position in the future. Each time a fix or running fix is plotted, a vector representing the future. Each time a fix or running fix is plotted, a vector representing the ordered course and speed originate from it. The

ordered course and speed originate from it. The directiondirection of thisof this course linecourse line

represents the ship's course, and the

represents the ship's course, and the lengthlength represents the distance onerepresents the distance one would expect the ship to travel in a given time. This extrapolation is used as would expect the ship to travel in a given time. This extrapolation is used as a safety precaution: a predicted DR position that will place the ship in water a safety precaution: a predicted DR position that will place the ship in water 1 metre deep should raise an

1 metre deep should raise an eyebrow…eyebrow…

In the example above the true courses are plotted in the chart, and to assist In the example above the true courses are plotted in the chart, and to assist th

the e hehelmlmsmsman an ththesese e cocoururse se lilinenes s arare e lalabebelllled ed wiwith th ththe e cocorrerrespsponondidingng compass courses.

compass courses.

Guidelines for dead reckoning Guidelines for dead reckoning::

Plot a new course line from each new fix or running fix (single LOP). Plot a new course line from each new fix or running fix (single LOP). Never draw a new course line from an EP.

Never draw a new course line from an EP. Plot a DR position every

Plot a DR position every time course or speed changes.time course or speed changes.

Plot a corrected DR position if the predicted course line proofed wrong, and Plot a corrected DR position if the predicted course line proofed wrong, and continue from there.

continue from there.

Running fix

Under some circumstances, such as low visibility, only one line of position Under some circumstances, such as low visibility, only one line of position can be obtained at a time. In this event, a line of position obtained at an can be obtained at a time. In this event, a line of position obtained at an earlier time may be advanced to the time of the later LOP. These two LOPs earlier time may be advanced to the time of the later LOP. These two LOPs should not be parallel to each other; remember that the optimal angular should not be parallel to each other; remember that the optimal angular spread is 90°. The position obtained is

spread is 90°. The position obtained is termed a running fix because the shiptermed a running fix because the ship has “run” a certain distance during the time interval between the two

(17)

e

e oobbttaaiin n a a ssiinnggllee OP on LANBY 1 OP on LANBY 1 n

nd d pplloot t aa orre

orrespspondonding ing (sa(sameme ime) dead reckoning ime) dead reckoning

o

ossiittiioonn. . TThhee sti

stimamated ted pospositiition on isis o

onnssttrruucctteed d bbyy raw

rawing ing ththe e shshortortestest in

ine e bebetwtweeeen n ththe e DRDR n nd d tthhe e LLOOPP:: erpendicular. erpendicular. o o LLOOPPs s aat t aallll. . WWee a acck k aannd d pplloot t a a DDRR osition. osition. e obtain a LOP on e obtain a LOP on ANBY 2. To use the ANBY 2. To use the irst LOP we advance irst LOP we advance over a construction over a construction ine between the two ine between the two

o

orrrreessppoonnddiinng g DDRR o

ossiittiioonnss. . WWe e uussee ot

oth h itits s ddirirececttioion n & & istance.

istance.

To use the LOP obtained at an earlier time, we must advance it to the

To use the LOP obtained at an earlier time, we must advance it to the time of time of the second LOP. This is done by using the dead reckoning plot. First, we the second LOP. This is done by using the dead reckoning plot. First, we m

meeaassuurre e tthhe e ddiissttaanncce e bbetetwweeeen n tthhe e ttwwo o DDR R ppoossititiioonns s aannd d ddrraaw w aa const

constructioruction n lineline, , wwhihich ch is is papararallllel el to to a a liline ne coconnnnecectiting ng ththe e twtwo o DRDR

positions. positions.

Note that if there are no intervening course changes between the two DR Note that if there are no intervening course changes between the two DR positions, it's easiest just to use the course line

positions, it's easiest just to use the course line itself as the construction line.itself as the construction line. N

Nowow, , uusisinng g tthhe e ppaararallllel el rurullerers s wwe e adadvavancnce e tthhe e ffirirst st LLOP OP alalonong g tthhisis construction line over the distance we measured. Et voilá, the intersection is construction line over the distance we measured. Et voilá, the intersection is our RFix.

our RFix.

If there is an intervening course change, it appears to make our problem If there is an intervening course change, it appears to make our problem harder. Not so! The only DR positions that matter are the two corresponding harder. Not so! The only DR positions that matter are the two corresponding with the LOPs.

with the LOPs.

Guidelines for advancing a LOP Guidelines for advancing a LOP:: Th

The e disdistatancence: : equequal al to to ththe e disdistantance ce betbetweeween n ththe e twtwo o corcorresresponpondinding g DRDR positions.

(18)

Th

The e dirdirectectionion: : equequal al to to the the dirdirectection ion betbetweeween n the the twtwo o corcorresresponpondinding g DRDR positions.

positions.

Draw the advanced LOP with a dotted line and mark with both times. Draw the advanced LOP with a dotted line and mark with both times. Label the Running Fix with an ellipse and "

Label the Running Fix with an ellipse and "RFix RFix " without " without underliningunderlining..

Danger bearing

Like the dead reckoning positioning, the danger bearing is an important tool Like the dead reckoning positioning, the danger bearing is an important tool

tto o kkeeeep p tthhe e sshhiip p oouut t oof f hhaarrmm''s s wwaayy. . FFiirrsstt, , tthhee navigator identifies the limits of safe, navigable water and determines a navigator identifies the limits of safe, navigable water and determines a bearing to for instance a major light. This bearing is marked as “No More bearing to for instance a major light. This bearing is marked as “No More Than” (

Than” (NMT NMT ) or “No Less Than” () or “No Less Than” (NLT NLT ), depending on which side is safe.), depending on which side is safe.

Hatching is included on the side that is hazardous, along with its compass Hatching is included on the side that is hazardous, along with its compass bearing.

bearing. In t

In the exhe examample ople on thn the rige right a tht a true crue courourse of 3se of 325° i25° is pls plottotted (5ed (5° va° variariatiotionn ),), marked with the

marked with the magnetic coursemagnetic course of 320°, practical for a handheld compassof 320°, practical for a handheld compass

that requires no

that requires no deviation correction.deviation correction.

Were we see that light at 350° magnetic which is definitely “More Than” Were we see that light at 350° magnetic which is definitely “More Than” -the rocks and wreck would be between us and -the major light. A possible the rocks and wreck would be between us and the major light. A possible cause could be a (tidal) stream from east

cause could be a (tidal) stream from east to west.to west. When a

When a distancedistance is used instead of ais used instead of a directiondirection, a, a danger rangedanger range is plottedis plotted much the same way as the danger bearing.

much the same way as the danger bearing.

Turn bearing

The Turn bearing - like the danger bearing - is constructed in the chart in The Turn bearing - like the danger bearing - is constructed in the chart in advance. It should be used as a means of anticipation for sailing out of safe advance. It should be used as a means of anticipation for sailing out of safe waters (again like the danger bearing and dead reckoning). The turn bearing waters (again like the danger bearing and dead reckoning). The turn bearing is taken on an

is taken on an appropriate aid to navigation and is marked “appropriate aid to navigation and is marked “TBTB”. As you pass”. As you pass

the object its bearing will slowly change. When it reaches the

the object its bearing will slowly change. When it reaches the turn bearingturn bearing

turn the vessel on her new course. turn the vessel on her new course.

(19)

This type of bearing is also used for

This type of bearing is also used for selecting an anchorage position or divingselecting an anchorage position or diving position.

position.

Snellius construction

Wi

Willllebebrorord rd SnSnelellilius us - - a a 1616th centh centutury ry mamaththememataticiciaian n frfrom om LeLeididenen, , ththee Netherlands - became famous for inventing the loxodrome and his method of Netherlands - became famous for inventing the loxodrome and his method of triangulation.

triangulation. The

The Snellius constructionSnellius construction wawas s fifirsrst t usused ed to to obobtatain in ththe e lelenngtgth h of of ththee me

merirididian by meaan by measusuriring the disng the distatancnce e bebetwtweeeen n twtwo o DuDutctch h cicititieses . . He tooHe tookk an

anglgles es frfrom om anand d to to chchururch ch totowewers rs of of vivillallageges s in in bebetwtween een to to rereacach h hihiss objective. Nowadays we use the Snellius method to derive our position from objective. Nowadays we use the Snellius method to derive our position from three bearings without the use of LOPs, and while leaving out deviation and three bearings without the use of LOPs, and while leaving out deviation and variation, which simplifies things. Also, since only relative angles are needed variation, which simplifies things. Also, since only relative angles are needed a sextant can be used to measure navigation aids at greater distances. a sextant can be used to measure navigation aids at greater distances. Closer in a

Closer in a compass can be used.compass can be used. The construction

The construction:: See

See figure figure 1: 1: Compass Compass bearings bearings are are 320° 320° on on A; A; 360° 360° on on B; B; 050° 050° on on objectobject C.

C.

The angle between A and B = 40°. The angle between A and B = 40°. The angle between B and C = 50°. The angle between B and C = 50°.

Draw lines from A to B and from B to C. Draw lines from A to B and from B to C.

Add the two light-blue perpendicular bisectors of lines AB and BC. Add the two light-blue perpendicular bisectors of lines AB and BC. Draw at object A a

Draw at object A a construction line 40° inland of line AB.construction line 40° inland of line AB. Draw at object C a

(20)

See figure 2: At

See figure 2: At object A: draw a lobject A: draw a line perpendicular to the construction line.ine perpendicular to the construction line. At object C: draw another line

At object C: draw another line perpendiculaperpendicular to r to the construction line.the construction line.

The two intersections with the light-blue lines indicate the centres of two The two intersections with the light-blue lines indicate the centres of two circles.

circles.

Finally, draw the first circle using A and B and the second circle using B and Finally, draw the first circle using A and B and the second circle using B and C.

C.

The off shore intersection of the two circle gives us our position fix. The off shore intersection of the two circle gives us our position fix.

(21)

The advantage: deviation and variation can be left out since the angles (here The advantage: deviation and variation can be left out since the angles (here 40° and 50°) are relative ones. Moreover, a sextant can be used to obtain 40° and 50°) are relative ones. Moreover, a sextant can be used to obtain angles between objects at greater distances, that with a compass would be angles between objects at greater distances, that with a compass would be less precise.

less precise.

International notation

International notation conventions for plotting in the chart International notation conventions for plotting in the chart Fix

Fix LOPLOP

R

Ruunnnniinng g FFiixx LOLOP P aaddvvaanncceedd E

(22)

D

Deeaad d RReecckkoonniinngg SSeet t & & DDrriifftt Electronic Fix (GPS)

Electronic Fix (GPS) Electronic Fix (Radar) Electronic Fix (Radar)

Note, that a few countries use an

Note, that a few countries use an alternative symbolalternative symbol

Plotting should be done with a soft pencil. Moreover, avoid drawing lines Plotting should be done with a soft pencil. Moreover, avoid drawing lines through the chart symbols. This is to prevent damage to the chart when you through the chart symbols. This is to prevent damage to the chart when you have to erase the construction.

have to erase the construction.

Learn sailing and navigation via yacht charters with instruction in Greece Learn sailing and navigation via yacht charters with instruction in Greece..

Glossary

1.

1. LinLine e Of Of PosPositiition on (LO(LOP)P): The locus of points along which a ship's: The locus of points along which a ship's

position must lie. A minimum of two LOPs are necessary to establish a fix. It position must lie. A minimum of two LOPs are necessary to establish a fix. It is standard practice to use at least three LOPs when obtaining a fix, to guard is standard practice to use at least three LOPs when obtaining a fix, to guard against the possibility of and,

against the possibility of and, in some cases, in some cases, remove ambiguity.remove ambiguity.

2.

2. Transit fixTransit fix: The method of lining up : The method of lining up charted objects to obtain an LOP.charted objects to obtain an LOP.

3.

3. LeadiLeading ng lightlightss oror RanRange ge liglightshts: : A A papair ir of of lligighhtts s or or ddaay y mmaarkrkss

deliberately placed to mark a narrow channel. deliberately placed to mark a narrow channel.

4.

4. Position fixPosition fix: The intersection of various LOPs.: The intersection of various LOPs.

5.

5. Cross bearingCross bearing: The use of LOPs of several navigational aids to obtain: The use of LOPs of several navigational aids to obtain

a position fix. Remember to use

a position fix. Remember to use an optimal angular spread.an optimal angular spread.

6.

6. Running fixRunning fix: The use of an advanced LOP. Make sure to use only the: The use of an advanced LOP. Make sure to use only the corresponding DR positions. Also don't use the EP for advancing the first LOP. corresponding DR positions. Also don't use the EP for advancing the first LOP.

7.

7. Dead Dead reckonreckoninging: : DetDetermerminiining ng a a pospositition ion by by ploplotttting ing coucourserses s anandd speeds from a known position. It is also used to predict when lights become speeds from a known position. It is also used to predict when lights become visible or to determine the set and rate of a current.

visible or to determine the set and rate of a current.

8.

8. EstiEstimated mated positpositionion: Combine a corresponding DR position with a: Combine a corresponding DR position with a single LOP to get an

single LOP to get an EP position.EP position.

9.

9. Snellius constructionSnellius construction: : AnAnototheher r waway y to to cocombmbinine e ththreree e cocompmpasasss bearings to obtain a position fix. The advantage over a cross bearing is that bearings to obtain a position fix. The advantage over a cross bearing is that both magnetic variation and deviation don't need

both magnetic variation and deviation don't need to be taken to be taken into account.into account.

10.

10. CourseCourse: (C) The direction in which a vessel is steered or : (C) The direction in which a vessel is steered or is intended tois intended to be steered (direction through the

be steered (direction through the water).water).

11.

11. SpeedSpeed: (S) The : (S) The speed of the boat through the water.speed of the boat through the water.

12.

12. SetSet: (SET) The direction in which the current is flowing (see chapters: (SET) The direction in which the current is flowing (see chapters

6,7 and 8). 6,7 and 8).

(23)

13.

13. DriftDrift: (DFT) The speed (in knots) of the current (see chapters 6,7 and: (DFT) The speed (in knots) of the current (see chapters 6,7 and 8).

8).

14.

14. Default headingDefault heading is True course (M is True course (M = magnetic , C = = magnetic , C = compass).compass).

15.

15. Default timeDefault time is 24 hour clock ship time else UTC.is 24 hour clock ship time else UTC.

Piloting and

navigation

Doubled angle fix

The

The Doubled angle on the bow fixDoubled angle on the bow fix resembles aresembles a

running fix though only one navigation aid is used. running fix though only one navigation aid is used. In the example on the right the initial angle (30°) In the example on the right the initial angle (30°) on the bow is doubled (60°) yielding an isosceles on the bow is doubled (60°) yielding an isosceles ttririanangglele . . ThThe e ddisisttanancce e ttraravevelllled ed bbetetwweeeen n tthhee be

beararinings gs is is ththe e sasame me as as ththe e didiststanance ce frfrom om ththee visible wreck.

visible wreck.

Start with the visible wreck having a bearing of less than 45° off the bow ( Start with the visible wreck having a bearing of less than 45° off the bow (α α ),),

note the log distance. note the log distance.

Proceed along the course until the angle on the bow is doubled (

Proceed along the course until the angle on the bow is doubled (ββ), read the), read the

log:

log: dd11 is 10 nm.is 10 nm.

Use the log distance to find the position on the second LOP. It is an isosceles Use the log distance to find the position on the second LOP. It is an isosceles triangle, so

triangle, so dd22 is also 10 nm.is also 10 nm.

Label it with an ellipse and "

Label it with an ellipse and "RFix RFix " but realize it is less precise than a running" but realize it is less precise than a running fix that involves two

fix that involves two navigation aids.navigation aids.

α α = 30° ,= 30° , ββ = 60°= 60° δ δ = 120° ,= 120° , γ γ = 30°= 30° Isosceles d Isosceles d11 = d= d22

(24)

Four point fix

If the first angle on the bow is 45°, a special situation If the first angle on the bow is 45°, a special situation occurs: The

occurs: The Four point fixFour point fix, so called since 45 degrees, so called since 45 degrees equ

equals als 4 po4 pointints on s on the the comcompaspass (1 s (1 poipoint = nt = 11,11,25°25° ).). Start with a bearing with 45° on the bow (

Start with a bearing with 45° on the bow (α α ), note the), note the

log. log.

Proceed along the course till the angle on the bow is Proceed along the course till the angle on the bow is 90° (

90° (ββ), read the log:), read the log: dd11 is 4 nmis 4 nm

Use the log distance to find the position on the second LOP. Isosceles, so Use the log distance to find the position on the second LOP. Isosceles, so dd22

is also 4 nm. is also 4 nm.

Label it with an ellipse and "

Label it with an ellipse and "RFix RFix ".".

Special angle fix

The

The Special angle fixSpecial angle fix requires the mariner to know some special pairs of requires the mariner to know some special pairs of

angles (

angles (aa :: b b) that give the distance travelled) that give the distance travelled

be

betwtweeeen n bebeararinings gs as as eqequaual l to to ththe e didiststanancece a

abbeeaamm ..

In the example on the right

In the example on the right α α = 21° and= 21° and ββ ==

32° are used. Now, the log distance equals the 32° are used. Now, the log distance equals the shortest distance between wreck and course line shortest distance between wreck and course line (6 nm).

(6 nm).

A few practical pairs: A few practical pairs:

16 16 : : 22 22 21 21 : : 3232 25 25 : : 41 41 32 32 : : 5959 37 37 : : 72 72 40 40 : : 7979

Remember: the greater the angular spread the better. Hence, of these three Remember: the greater the angular spread the better. Hence, of these three fixes the four point fix is the most precise one.

fixes the four point fix is the most precise one.

Top of Form Top of Form Enter Enter α α (1-45°):(1-45°): ββ:: Bottom of Form Bottom of Form α α = 45° ,= 45° , ββ = 90°= 90° δ δ = 90° ,= 90° , γ γ = 45°= 45° Isosceles d Isosceles d11 = d= d22 α α = 21° ,= 21° , ββ = 32°= 32° d d11 = d= d22

(25)

Mathematic

Mathematics: s: isosceles triangle fixesisosceles triangle fixes

D

i

s

t

a

n

c

e

o

f

t

h

e

horizon

On a flat world there would be no On a flat world there would be no di

diffffererenence ce bebetwtweeeen n ththe e vivisisiblblee and

and sensensibsible le horhorizoizon. n. HowHoweveever,r, on

on EEarartth h tthhee visvisiblible e horhorizoizonn

a

appppeaears rs ssevevereraal l aarc rc mmininuutteses below the

below the sensible horizonsensible horizon duedue to two opposing effects:

to two opposing effects:

the curvature of the earth's surface; the curvature of the earth's surface; atmospheric refraction.

atmospheric refraction.

Atmospheric refraction bends light rays passing along the earth's surface Atmospheric refraction bends light rays passing along the earth's surface toward the earth. Therefore, the

toward the earth. Therefore, the geometrical horizongeometrical horizon appears elevated,appears elevated, forming the visible horizon.

forming the visible horizon.

The distance of the visible horizon is a (semi-empirical) function of Eye The distance of the visible horizon is a (semi-empirical) function of Eye Height:

Height:

Mathematics: horizon distances

Dipping range

If an object is observed to be just rising above or just dipping below If an object is observed to be just rising above or just dipping below the visible horizon, its distance can be readily calculated using a the visible horizon, its distance can be readily calculated using a simp

simple le formuformula. la. The The objecobject'st's elevationelevation (the height of a light above chart(the height of a light above chart

dat

datumum ) can be foun) can be found in the chd in the chart or otart or otheher nautr nauticaical publ publiclicatiation suon such as thch as thee 'List of Lights'. Note that in some charts elevation is referred to a different 'List of Lights'. Note that in some charts elevation is referred to a different

(26)

d

daattuum m tthhaan n ssoouunnddininggss . . CClilicck k oon n tthhe e iimmaagge e oon n tthhe e rriigghht t tto o vviiew ew aa magnificent lighthouse.

magnificent lighthouse.

The formula contains the two distances from the visible horizon and can be The formula contains the two distances from the visible horizon and can be si

simmplplififieied d by the by the eqequauatitionon: : 2.2.08 x 08 x (√(√ElElevevatatioion n + + √E√Eye ye heheigightht)) . . MaManyny nautical publications contain a table called "distances of the horizon" which nautical publications contain a table called "distances of the horizon" which can be used instead of the

can be used instead of the equation.equation. Use the dipping range to plot a

Use the dipping range to plot a Distance LOPDistance LOP in the chart: a circle equalin the chart: a circle equal

in radius to the measured distance, which is plotted about the navigation aid. in radius to the measured distance, which is plotted about the navigation aid. Finally, take a bearing on the object to

Finally, take a bearing on the object to get a second LOP and a get a second LOP and a position fix.position fix.

Top of Form Top of Form

Enter Eye height (metres): Enter Eye height (metres): Enter Elevation (metres): Enter Elevation (metres):

Distance is (nm): Distance is (nm):

Bottom of Form Bottom of Form

Vertical sextant angle

Similarly, a distance LOP can be obtained by using a sextant to measure the Similarly, a distance LOP can be obtained by using a sextant to measure the ang

angle (ale (arc) brc) betwetween feen for inor instastance tnce the lihe light ght anand chad chart dart datutum of a lim of a lightghthouhousese or any other structure of known elevation. Once the angle is corrected for or any other structure of known elevation. Once the angle is corrected for ind

index ex errerroror the distance can be found in a table called: "Distances bythe distance can be found in a table called: "Distances by

Vertical Sextant Angle", which is based on the following equation. Vertical Sextant Angle", which is based on the following equation.