The modern chart shows us positions of
many recognizable navigation aids like churches and lighthouses, which
facilitate the approach to a coastal area. This concept originated from a
chart by Waghenaer and proved a milestone in the development of European
cartography. This work was called "Spieghel der Zeevaerdt" and included
coastal profiles and tidal information much like the modern chart. It
enables us to find the angle between the North and for example a platform,
as seen from our position.
Taking a bearing on this oilrig with a compass provides us with a compass
course. This course first needs correction for both variation and  via
ship's heading  deviation before plotting a
Line of Position (LOP) in the chart. Our
position is somewhere along this line.
Ranges
A precise way to obtain a LOP (and
without a compass) is to locate two navigational aids in line. The image
on the right shows us four examples of ranges,
each consisting of two nav. aids.
Please, note that:
One of the four ranges consists of two
lights that are intentionally placed to provide a LOP. These pairs of
lights are called Range lights or
Leading lights. In this case they indicate the
channel between the shallows along a true course of 50° .
When looking toward the leading lights, the closest one will be lower.
Therefore, in the middle of the channel
both lights will appear above each other.
Even when there are no manmade
structures available, a range can be found by using natural features such
as coastlines and islets. The example on the left shows a yacht that will
avoid the dangerous wreck as long as the islets don't overlap.
The Position Fix
To construct our position fix we need
two of these lines of position to intersect each other.
Fix is the initial element of the ship's
navigational and dead reckoning (see below)
plot. A fix is the ship's position on the earth at some given point in
time. A fix is determined by the simultaneous intersection of LOP's. Often
however, a triangle occurs when a third LOP is added in the 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
bearing in a wider sector of, for instance 10°. Bearings create more
certainty about our position when they are perpendicular to each other.
Yet, bearings on distant objects bring about more uncertainty in our
position fix as the sector widens. If moving fast you should not put any
time between the bearings.
In the next example we will plot our position fix by taking bearings at
two light vessels just off the coast of Willemsen Land. To plot in the
chart we will use a soft pencil and avoid drawing lines through the chart
symbols. This is to prevent damage to the chart when you have to erase the
construction.
Since we use our steering compass for our bearings the same deviation
table can be used. We will assume the variation to be 1° and the ship's
compass heading 190°. Hence, from the deviation table we find a deviation
of 4°.
The construction:

The first compass bearing on 'Will. N'
is 65°.
cc + var. + dev. = tc, therefore tc = 60°.

Plot the LOP in the chart aligned to
this lightship. Mark 'Time' and 'True Course' along with it.

Mutatis mutandis the second LOP on
'Will. W' is 145°.

The intersection of these two LOP's is
our Position Fix. Mark this with an 'Ellipse' and the 'Time'. The
greater the uncertainty, the greater the ellipse (position area).

Fixed Position around 15^{h}00^{m}
= 39° 58'.9 N , 24° 25'.5 E, approximately. So, welcome to my island but
mind the rocks!


Although we didn't have a third LOP
creating the dreaded triangle, we still have to doubt the accuracy of our
position fix. Then, if three or more LOP's were used, and a nice point was
not achieved, we again are left with some ambiguity. This could be caused
by any number of reasons, including instrument errors, erroneous
identification of a navigation aid, sloppy plotting, or error by the
bearer taker, among others. In this case, we will assume that we are at
the worst possible position (i.e. closest to the nearest navigational
hazard).
To minimize the effect of possible errors the
optimum angular spread should be 90° when two objects are shot
or 120° when three objects are shot.
The Estimated Position
It is sometimes impossible to obtain
more than one LOP at a time. To determine the ship's position using only
one navigation aid, we can use a running fix
(see below). However if a running fix is not possible, we can determine an
estimated position.
An estimated position is based upon whatever incomplete navigational
information is available, such as a single LOP, a series of depth
measurements correlated to charted depths, or a visual observation of the
surroundings. An estimated position can be determined using a single LOP
and the ship's Dead Reckoning Position (DR).
This is done by 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 used for a fix.
Do not rely on an EP as much as a fix. The scale
of reliability, from best to worst:

Fix

Running Fix

Estimated Position

DR position

Dead Reckoning
Dead reckoning is a technique to
determine a ship's approximate position 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 earlier fix, we plot
from that position our course and 'distance travelled since then' and
deduce our current position.
0930: We
start off with a Fix and plot
a DR position for 15 minutes later.
0945: Our
estimation about our speed
and course was correct, so we don't have to
charge the DR position.
1000:
and so on...
S = Speed through water
C = Course (T = true, M = magnetic, C = compass) through water.
Mark with an arrow as indicated.

Dead reckoning is crucial since it can
provide an approximate position in the future. Each time a fix or running
fix is plotted, a vector representing the ordered course and speed
originate from it. The direction of this course
line represents the ship's true course, and the length represents the
distance one 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 1 meter deep should raise an eyebrow...
Guidelines:

Plot a new course line from each new
fix or running fix.

Never draw a new course line from an
EP.

Plot a DR position every time course
or speed changes.

Also, plot a DR position when a single
LOP is obtained.

Label the DR position with a
semicircle and 'DR'.


Running Fix
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 earlier time may be advanced to
the time of the later LOP. These two LOP's should not be parallel to each
other; remember that the optimal angular spread is 90°. The position
obtained is termed a running fix because the ship has "run" a certain
distance during the time interval between the two LOP's.
0905:
Our tacking sequence starts with a solid
position Fix.
0916:
We obtain a single LOP and construct a
corresponding (same time) dead reckoning position. Our estimated
position is constructed by drawing the shortest line between the DR and
the LOP (perpendicular).
0926:
No LOP's at all.
We tack and construct a DR position.
0934:
We obtain a LOP on Oil Rig 2.
To use the first LOP we advance it over a construction line between the
two corresponding DR positions.
We use both its direction and distance.
To use the LOP obtained at the earlier
time, we must advance it to the time of the second LOP. This is done by
using the dead reckoning plot. First, we measure the distance between the
two DR positions and draw a construction line,
which is parallel to a line connecting the two DR positions.
Note that if there are no intervening course changes between the two DR
positions, it's easiest just to use the course line itself as the
construction line.
Now, using the parallel rulers we advance the first LOP along this
construction line over the distance we measured. Et Voilá, the
intersection is our RFix.
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 with the LOP's.
Advancing a LOP:

The distance: equal to the distance
between the two DR positions.

The direction: equal to the direction
from the first to the second DR position.

Label the Running Fix with an ellipse
and "RFix".


Danger Bearing
Like the dead reckoning positioning the
danger bearing is an important tool to keep the ship out of trouble.
First, the navigator identifies the limits of safe, navigable water and
determines a bearing to a prominent landmark. This bearing is marked as
"No More Than" (NMT) or "No Less Than" (NLT), depending on
which side is safe. Hatching is included on the side that is hazardous,
along with its compass bearing.
When a distance instead of a direction is used a
danger range is plotted much the same way as the danger bearing.
Turn Bearing
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 waters (again like the
danger bearing and dead reckoning). The turn bearing is taken on an
appropriate navaid and is marked 'TB'. As you pass the object its
bearing will slowly change. When it reaches the Turn bearing turn the
vessel on her new course.
This type of bearing is also used for selecting an anchorage position or
diving position.
Snellius Construction
Willebrord Snellius  a 16th century
mathematician from Leiden NED  became famous for inventing the loxodrome
and his method of triangulation.
The Snellius construction was first used to
obtain the length of the meridian by measuring the distance between two
Dutch cities.
He took angles from and to church towers of villages in between to reach
his objective. We now can use the Snellius method to derive our position
from three bearings without the use of LOP's. We can leave out deviation
and variation, which simplifies things. Also, since only relative angles
are needed a sextant can be used to measure navigation aids at greater
distances.
The construction:

Compass bearings are 320° on A, 360°
on B and 050° on object C.

The angle between A and B = 40°.

The angle between B and C = 50°.

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

Add the two perpendicular bisectors
(yellow) of lines AB and BC.

Draw at object A a construction line
(blue) 40° inland of line AB.

Draw at object C a second construction
line (blue) 50° inland of line CB.


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 angles between objects at
greater distances, which with a compass would be less precise.
Notation
Overview

Line Of
Position (LOP): The locus of points along
which a ship's position must lie. A minimum of two LOP's are necessary
to establish a fix. It is standard practice to use at least three LOP's
when obtaining a fix, to guard against the possibility of and, in some
cases, remove ambiguity.

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

Leading lights
or Range lights: A pair of lights or day
marks deliberately placed to mark a narrow channel.

Position fix:
The intersection of various LOP's.

Cross bearing:
The use of LOP's of several navigational aids to obtain a position fix.
Remember to use an optimal angular spread.

Running fix:
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.

Dead reckoning:
Determining a position by plotting courses and 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.

Estimated
position: Combine a corresponding DR position
with a single LOP to get an EP position.

Snellius
construction: Another way to combine three
compass bearings to obtain a position fix. The advantage over a cross
bearing is that both magnetic variation and deviation don't need to be
taken into account.

Course:
(C) The direction in which a vessel is steered or is intended to be
steered (direction through the water).

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

Set:
(SET) The direction in which the current is flowing (see chapters 6,7
and 8).

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

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

Default time
is 24 hour clock ship time else UTC.

