Marine

NAVIGATION

Piloting

Doubled angle fix

The Doubled angle on the bow fix resembles a running fix though only one navigation aid is used. a = 30° , b = 60°
d = 120° , g = 30°
Isosceles    d₁ = d₂ In the example on the right the initial angle (30°) on the bow is doubled (60°) yielding an isosceles triangle. The distance travelled between the bearings is the same as the distance from the visible wreck.

• Start with a bearing with a small angle (smaller than 45°) on the bow (a), note the log.
• Proceed along the course till the angle on the bow is doubled (b), read the log: d₁ is 10 nm.
• Use the log distance to find the position on the second LOP. Isosceles, so d₂ is also 10 nm.
• Label it with an ellipse and "RFix" but realize it is less precise than a running fix that involves two navigation aids.

Four point fix

If the first angle on the bow is 45°, a special situation occurs: The Four point fix, so called since 45 degrees equals 4 points on the compass (1 point = 11,25°). a = 45° , b = 90°
d = 90° , g = 45°
Isosceles    d₁ = d₂

• Start with a bearing with 45° on the bow (a), note the log.
• Proceed along the course till the angle on the bow is 90° (b), read the log: d₁ is 4 nm
• Use the log distance to find the position on the second LOP. Isosceles, so d₂ is also 4 nm.
• Label it with an ellipse and "RFix".

Special angle fix

The Special angle fix requires the mariner to know some special pairs of angles (a : b) that give the distance travelled between bearings as equal to the distance abeam.
a = 21° , b = 32°
d₁ = d₂ In the example on the right a = 21° and b = 32° are used. Now, the log distance equals the shortest distance between wreck and course line (6 nm).
A few practical pairs:

Distance of the horizon

On a flat world there would be no difference between the visible and sensible horizon. However, on Earth the visible horizon appears several arc minutes below the sensible horizon due to two opposing effects:

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

Atmospheric refraction bends light rays passing along the earth's surface toward the earth. Therefore, the geometrical horizon appears elevated, forming the visible horizon.
The distance of the visible horizon is a (semi-empirical) function of Eye Height:

 

Dipping range

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 simple formula. The object's elevation (the height of a light above chart datum) can be found in the chart or other nautical publication such as the 'List of Lights'. Note that in some charts elevation is referred to a different datum than soundings. Click on the image on the right to view a magnificent lighthouse.

Vertical sextant angle

Similarly, a distance LOP can be obtained by using a sextant to measure the angle (arc) between for instance the light and chart datum of a lighthouse or any other structure of known elevation. Once the angle is corrected for index error the distance can be found in a table called: "Distances by Vertical Sextant Angle", which is based on the following equation.

• The angle in minutes total, thus 1° 12' = 72' total, and corrected for index error.
• Elevation in meters.
• Water height in meters above or below chart datum of object.
• Distance or Range in nautical miles.
• Ascertain whether the base of the object is beyond the horizon
• Corrected angle should be greater than 20'.

• Measured angle is 1° 19', index error is +6': angle = 73'.
• Let's assume water height at 3 meters above Mean Level datum.
• Range = 1.854*(80-3/73) = 1.96 nm.

Often, the correction for water height can be left out. Though, realizing that the horizon is closer than one might think, another correction is sometimes needed. In the Mediterranean Sea for example we can see mountain tops with bases lying well beyond the horizon. Mutatis mutandis, the structures, which they bear have bases beyond the horizon as well.

This is the equation for finding the distance of an object of known elevation located beyond the horizon. In the denominator of this equation a compensating factor is included by which the measured angle should be reduced.

Estimation of distance

The most obvious way to estimate distances is of course by using the distance between our eyes. If we sight over our thumb first with one eye then with the other, the thumb moves across the background, perhaps first crossing a tower second crossing a bridge.

The chart might tell that these structures are 300 m apart.
Use the ratio of: distance between eye and outstretched arm/distance between pupils: usually 10.
The objects are 3 kilometres away.
Other physical relationships are useful for quick reference. For example, one finger held at arm's length covers about 2° arc, measured horizontally or vertically.
Two fingers cover 4°. Three fingers cover 6° and give rise to the three finger rule:
"An object that is three fingers high is about 10 times as far away as it is high."
 

Estimation with horizon

The image on the right shows us that it is possible to estimate the height of any object that crosses the horizon as seen from our own point of view. This picture of the 'Pigeon Rocks' near Beirut harbour was taken from a crow's nest at a height of 34 meters.
The distance of the visible horizon (12 nm) is far larger than 34 meters. Therefore, we can - without any other information - estimate that these rocks have a height of 34 meters as well.

Factum: All tops crossing the horizon and with bases at sea level are on eye level.

Furthermore, if we see these rocks over a vertical angle of for example 7° = 0.1225 rad., then the range is 34/0.1225 = 277 meters.
Finally, plot both range and bearing in the chart to construct an EP, et Voilà!
 

Fix by depth soundings

A series of depth soundings - in this example every 10 minutes - can greatly improve your position fix:

• correct your soundings for tide, etc.;
• copy the DR course line on a transparent sheet;
• write the depths adjacent according to the times of the soundings;
• move the sheet over the chart to find its best location.

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.
Range or Distance LOP: Obtained by using a stadimeter, sextant or radar. A circle equal in radius to the measured distance is plotted about the navigation aid; the ship must be somewhere on this circle.
Running fix: A position determined by crossing lines of position obtained at different times and advanced or retired to a common time.
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 drift of a current. DR positions are drawn in advance to prevent sailing into danger. A DR position will be plotted:

• every hour on the hour;
• at the time of every course change or speed change;
• for the time at which a (running) fix is obtained, also a new course line will be plotted;
• for the time at which a single LOP is obtained;
• and never draw a new course line from an EP position!

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