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Doubled angle fix

The Doubled angle on the bow fix resembles a running fix though only one navigation aid is used.
double angle on the bow fix
α = 30° , β = 60°
δ = 120° , γ = 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.

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°).
four point fix
α = 45° , β = 90°
δ = 90° , γ = 45°
Isosceles    d1 = d2


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.
special angle fix
α = 21° , β = 32°
d1 = d2
In the example on the right α = 21° and β = 32° are used. Now, the log distance equals the shortest distance between wreck and course line (6 nm).
A few practical pairs:
16 : 22      21 : 32
25 : 41      32 : 59
37 : 72      40 : 79

Remember: the greater the angular spread the better. Hence, of these three fixes the four point fix is the most precise one.
Enter α (1-45°): β:
Mathematics: isosceles triangle fixes

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: Horizon with atmospherical refraction and 
the curvature of the earth's surface.

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:

Distance of vis. horizon (nm) 
with height of eye in metres. 

This is a simplified function. 


Mathematics: horizon distances
 

 

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. 365 kb 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.
Dipping distance
The formula contains the two distances from the visible horizon and can be simplified by the equation: 2.08 x (√Elevation + √Eye height). Many nautical publications contain a table called "distances of the horizon" which can be used instead of the equation.
Use the dipping range to plot a Distance LOP in the chart: a circle equal in radius to the measured distance, which is plotted about the navigation aid. Finally, take a bearing on the object to get a second LOP and a position fix.
Enter Eye height (metres):
Enter Elevation (metres):
Distance is (nm):

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.
Vertical sextant angle...        Looking through sextant.

Range in nm. 
Elevation in metres 
Water Height in metres 
Angle in minutes total.

Though tables can be used for quick reference, this function is valid for objects higher than usually tabulated. An example with a lighthouse of 80 metres: The range can be used as a danger bearing.
Together with a compass bearing one object with known elevation results in a position fix. If more than one vertical sextant angle is combined the optimum angular spread should be maintained.
Enter Angle (minutes total):
Enter Elevation (metres):
Distance is (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.
Angle over horizon...        Looking through sextant.

Range in nm. 
Elevation in metres 
Angle in minutes total
Eye Height in metres.

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.

Enter Eye Height (metres):
Enter Angle (minutes total):
Enter Elevation (metres):
Distance is (nm):


Mathematics: vertical sextant angles
 


Estimation of distance

The most obvious way to estimate distances is of course by using the distance between our eyes. There is NO part of the author in this image !!!!!!! 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 width 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.
The height of the rock equals your height 
since the top aligns with the horizon.
This picture of the 'Pigeon Rocks' near Beirut harbour was taken from a crow's nest at a height of 34 metres.
The distance of the visible horizon (12 nm) is far larger than 34 metres. Therefore, we can - without any other information - estimate that these rocks have a height of 34 metres 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 metres.
Finally, plot both range and bearing in the chart to construct an EP, et Voilà!

Fix by depth soundings

   © sailingissues.com   
A series of depth soundings - in this example every 10 minutes - can greatly improve your position fix:
Fix by depth soundings.
Due to leeway, currents or other factors the two course lines need not be parallel to or of same length as each other.

Yacht charters and learning how to sail in Greece with instruction.
 

 

Overview

Mathematics: isosceles triangle fixes
Mathematics: horizon distances
Mathematics: sextant angles

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23 Sept 2016
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