Horizontal Sextant Angles (HSA)
By observing the horizontal angle subtended between two charted objects, a geometrical construction can be used to obtain the arc of a position circle on which the observer must be situated. To obtain a fix by HSA it would be necessary to observe two horizontal angles, thus requiring a minimum of 3 objects.
A common method of writing HSA questions is in the form:
'A' x° 'B' y° 'C'
which means that a horizontal angle subtended between objects 'A' and 'B' is x° while at the same time the horizontal angle subtended between objects 'B' and 'C' is y°.
Principle: The HSA method is based on the geometrical principle that:
'The angle subtended at the centre of a circle by a chord or arc of that circle is twice the angle subtended at any point on the circumference by the same chord of arc'.
Case 1: HSA less than 90°.
If angle APB = Θ
The angle AOB = 2Θ
For your own interest prove this relationship, but for practical Chartwork the fact can be accepted.
Adapting this diagram to suit the HSA problem, P represents a position on the circumference of a position circle on which the ship must lie, and 0 is the HSA subtended at the ship between two objects A and B.
The problem on the chart involves starting off with only the two objects observed and finishing with the required position circle plotted. To do this, join AB; it is then necessary to measure from each end of AB two equal construction angles so that when the resulting lines are drawn, they will intersect to indicate the position of the centre (O) of the required circle. With centre O and radius OA = OB, draw in the position circle and make sure it does pass through both A and B.
If the same procedure is then followed with a second HSA, the ship's position will be where the two position circles intersect (but not at the centre object).
In the above diagram the required two equal construction angles are OAB and OBA. Each angle is equal to (90° - 0) as the three angles of triangle OAB must add up to 180°.
Each Construction Angle = 90° - HSA
Case 2: HSA more than 90°
Then reflex angle AOB = 2Θ
Angles OAB, OBA are each equal to (0 - 90°)
Each Construction Angle = HSA - 90°
Note that, in the first diagram (Case 1), O and P lay on the same side of AB, but in this case, for 0 to be more than 90°, O and P must lie on opposite sides of AB.
Thus:
1. If the HSA is less than 90°, the construction angles must be laid off towards the ship.
2. If the HSA is more than 90°, the construction angles must be laid off on the opposite side of AB, i.e. away from the ship
Example Given:
Find the ships position if the following HAS are observed
‘A’ 55o ‘B’ 105o ‘C’
Note :
In the case of AB, angle Θ is less than 90o and in BC it is more than 90o.
Therefore the construction angles will be as follows;
AB angle = 90o – 55o = 35o
(Laid off on the same side of AB as the vessel)
BC angle = 105o – 90o = 15o
(Laid off on the other side of BC from the vessel)
See Diagram below
Case 3: HSA = 90°
This case is based on that geometrical principle that:
'The angle in a semi-circle is a right angle'.
From the diagram it can be seen that O, the centre of the position circle, must be the mid-point of AB.
SUMMARY
For Chartwork it is merely necessary to remember 3 basic rules. The first two are for finding the value of the construction angles to be laid off from each end of the line joining two objects observed, and on which side of the line they should be measured. When these angles are plotted, the lines drawn intersect to indicate the centre of the required position circle.
The third rule gives the centre of the required position circle direct, i.e. it lies at the mid-point of the line joining two objects.
Warning in applying the third rule a very common mistake is made when, having correctly bisected the line geometrically, a candidate Forgets to Draw the Position Circle on which the ship is situated; instead, the ship's position is placed somewhere on the perpendicular bisector.
1. If the HSA is less than 90°, subtract the HSA from 90° and lay off the resultant towards ship.
2. If the HSA is more than 90° from the HSA and lay off the resultant away from ship.
3. If the HSA equals 90°, bisect the line joining the two objects.
Do not use the terms 'to seaward' and 'to landward' for 'towards ship' and 'away from ship' respectively as HSA questions are frequently based on offshore lighthouses or light-vessels, in which cases 'to seaward' would apply to both directions.
In such situations where a ship could, theoretically, be either side of the line joining the two objects observed, many candidates fear that examination questions could leave them in doubt as to which side of the line to assume.
These fears are groundless - in any HSA question where there is possible ambiguity of direction in laying off construction angles, the candidate must be given the basic information that would be available on board ship. This can be done in one or more of the following ways:
1. Stated DR position.
2. Means of obtaining a DR or EP, probably in the form of a given run from a previous position.
3. Questions in the form 'A' x°, 'B' y° 'C' should follow the convention that the three objects are stated in the order in which they would be viewed from the ship, i.e. 'A' would be seen to the left of 'B' , and 'C' would be seen to the right of 'B'.
4. Questions of the type involving Compass and Gyro bearings where the Error is unknown (see later notes) - the bearings give an approximation of the ship's position.
Although a candidate is only expected to construct two position circles when two HSA's are given in a question, in fact a third position circle could be drawn as a check on accuracy. In the example on page 17 it will be seen that a horizontal angle (160°) is subtended between 'A' and 'C', and if the appropriate position circle is now constructed it should pass through the intersection between the two existing position circles.
In Hydrographic Surveying, HSA's are frequently used in coastal waters to give very accurate positions of the survey ship/launch/boat.
The accuracy check used by Surveyors, rather than the method described above, is to measure a third angle between the centre object and a fourth object.
The reason is that Surveyors use very accurate vernier station pointers (see later notes) to fix positions. Having plotted a position the station pointer is held firmly in place while one of the moveable pointers is adjusted to measure the check angle from the central fixed pointer - if the plotted position is correct, the adjusted pointer should pass exactly through the fourth object. This method is used to fix positions accurately and quickly so that the survey can proceed at maximum possible speed.
MEASUREMENT OF HORIZONTAL ANGLES
Although they are referred to as Horizontal Sextant Angles, it is not necessary to use a sextant to measure angles to an accuracy of minutes or seconds of arc if the subsequent plotting, whether by geometrical construction or by station pointer, can only be undertaken to a practical accuracy of the nearest degree or half-degree.
As previously mentioned, Hydrographic Surveyors do measure angles with a sextant as the plotting is undertaken with a station pointer which has a vernier reading to the nearest minute of arc. However, candidates for DoT examinations should expect HSA's to be given in whole degrees only.
HSA's From Compass/Gyro Bearings
If HSA's can only be plotted to the nearest degree, then any device capable of measuring horizontal angles to an accuracy of 1° is acceptable. Hence a common practice is to take Compass or Gyro bearings of three objects and use the difference between successive bearings as HSA's.
This method is used to obtain the ship's position when the bearings cannot be plotted accurately due to the Compass or Gyro Error being unknown, and it is important to remember that the position obtained is accurate and completely independent of the Compass or Gyro Error. Furthermore, from the position plotted on the chart the True bearings of the three objects can be obtained to compare with the respective Compass or Gyro bearings to determine the Error.
Always compare all three bearings as each comparison should, within reason, produce the same Error; if there is a large discrepancy between the three comparisons, there must be a mistake in construction of the position circles. If so, check all the figures used - a common mistake is to lay off construction angles 10° in error or the wrong way.
If the three comparisons gave Errors of, say 6°E, 6°E, 7°E, this would be acceptable and it would be reasonable to call the Compass Error 6°E. If, however, comparisons indicated apparent Errors of, say 4°W, 4°W, 3°E, a definite plotting mistake has been made, so go back and check!
Lesson 13.doc HSA DGR1999
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dariusz.lipinski