Determine positions in all conditions PHASE 1 COSTAL NAVIGATION

 

Determine positions in all conditions

Determine position and the accuracy of resultant position fix by any means

Transferring of position lines

Running Fix is a process whereby your ship’s position can be obtained by transferring an LOP to intersect another LOP with a steaming interval between them.

Caution when executing a running fix

In all cases of transferring an LOP, whether in chartwork or in practical navigation, you assume that you know course made good during the interval between the two LOPs. The accuracy of the fix obtained rests on this assumption. Hence you must NOT rely implicitly on a fix obtained through a running fix. You must verify your position by other means as soon as practicable. 

Procedure

Your attention is invited to the accompanying diagram. While on a passage, you observe a bearing of charted object A at say 0800. At 1030, you observe the bearing of object B. Obviously, B was not visible while observing A and vice versa. Otherwise, you would have taken simultaneous bearings of both and obtained fixes. If only one charted object is available, then two bearings of the same object can be taken with appropriate time interval, for this purpose. If leeway and wind direction are given, then it should be applied to the given CS to obtain the corrected CS, on which further plotting should be carried out. Draw both the LOPs. To perform a running fix, take any point C on the first LOP, and from it draw the course steered at the engine speed for the interval of 2½ hours to a point you call D. From D, apply the set and drift of current for the interval of 2½ hours and call the point reached, E. Transfer the first LOP through E and where this intersects the second LOP is the fix at 1030. CE is the CMG and SMG from 0800 to 1030.

If needed, draw a parallel to EC, from F and where it intersects the first LOP was the fix at 0800.

Example of running fix

In the following diagram, at 1500, point P bore 310º(T) while steering a course of 247º(T) at 15 knots. At 1900, point Q bore 030º(T). If a current was setting 020º(T) at 2 knots, find the position of your ship at 1900 and also at 1500. State the course and speed made good.

Note: Points C, D and E are used here for easy identification during explanation only. They are not to be marked on the chart.

Step 1: Draw the two LOPs

Draw the given bearings of 310º(T) and 030º(T) through P and Q.

Step 2: Plot position to transfer 1st LOP

From any point C on first LOP, draw the course of 247º(T) at 15 knots for 4 hours to point D and thence the set and drift of current for 4 hours -  020º(T) at 2 knots to point E.

Step 3: Obtain fix

Through point E, transfer the first LOP. The point of intersection of the transferred LOP and the 2nd LOP is the 1900 fix. Read off its latitude and longitude.

In this case lat 35º 35.0’N long 60º 15.5’E.

Step 4: Obtain CMG and SMG

The line CE represents the course and distance made good - 253º(T) 54.6 NM, that is 13.6 knots in this case.

Step 5: Obtain the earlier fix

From the 1900 fix, draw the CMG backwards to the first LOP. The point of intersection is the fix at 1500. In this case lat 35º 51.0’N long 61º 09.0’E. 

Step 6: State your answers

1900 position: lat 35º 35.0’N long 60º 15.5’E; 

1500 position: lat 35º 51.0’N long 61º 09.0’E; 

CMG: 253º(T); SMG: 13.6 knots.

Position Fixing with Leeway and Current

Drifting, with or without position fixing, when your ship is moving. You are given your ship’s movement through the water and are expected to find your movement over the ground, being affected by current and wind. 

Example of drifting due to current and leeway

Refer to the accompanying diagram. Your ship is in position A at 1500. Course of 050º(G) error 1º(L) was set. Your ship’s engine speed is 16 knots. An estimated tidal stream was setting the ship 125º(T) at 2.5 knots. Leeway 3º for a NW’ly wind. Find the course and speed made good. At what time would lighthouse B come abeam?

Step 1: Choose the plotting interval

Seeing the scale of the chart, one hour seems appropriate.

Step 2: Find the true course steered

Gyro course steered	050° (G)
Gyro error	1° (L)
True course steered	051°(T)

Step 3: Plot the starting position on the chart.

In this case, the starting position is A at 1500 hours.

Step 4: Compute the leeway course

True CS --- 051° T

Leeway ---         + 3°

Corrected CS --- 054° T

Step 5: Lay off the 1600 DR position at the end of the interval

In this case, plotting interval is 1 hour. Hence lay off 16 NM on the leeway course of 054º(T).

Step 6: Plot EP

From the 1600 DR position draw the set for 1 hour @2.5 knots = 2.5 NM in the direction 125º(T).

Step 7: Obtain course and speed MG.

Join the starting fix and the EP. That is the estimated course and distance made good in 1 hour. In this case, CMG = 062º(T), distance = 17.0 NM.

Speed made good = distance MG ÷ plotting interval = 17.0 ÷ 1 = 17.0 knots.

Step 8: Compute the beam bearing

Ship’s head	051º(T)
Stbd beam so add 90º	+90º
True beam bearing	141º(T)

Step 9: Lay off abeam position

The point where the beam bearing cuts the course MG is the position when B would come abeam.

Step 10: Obtain the interval to steam

Measure the distance from A to the beam position.

Interval to steam = distance to beam bearing ÷ speed MG = 35.6 ÷ 17.0 = 2.09 hours = 02h 06m

Step 11: Compute ETA

Staring time	1500
Steaming interval	0206
ETA beam bearing	1706

Step 12: State the answers

Course and speed MG = 062º(T) 17.0 knots. ETA beam bearing of B = 1706 hours.

Running fix by transferring position circle, with current and leeway

Data given --- Course steered (CS), ship's speed, current direction and rate, wind direction and leeway, and distances of two objects A and B from the ship with time interval in-between.

• With A as centre and given distance, draw the first position circle and note the time.
• After the given time interval draw a second position circle with B as centre and the given distance. (If only one object is available then two distances shall be taken from the same object with time interval, and two position circles shall be plotted with same object as centre.)
• At A draw the CS and apply leeway appropriately to obtain corrected CS₁.
• From the ship's speed calculate the distance run in the given time interval, and measure it on the CS₁ to obtain point C.
• At C draw the current direction.
• From the rate of current calculate the drift for the given time interval, and measure it on the current direction to obtain estimated position D.
• Then AD is the course made good (CMG).
• With D as centre transfer the first position circle to cut the second position circle at point E.
• At E draw the CMG in reverse direction to cut the first position circle at point F.
• F and E are the first and second observed positions of the ship respectively.
• FE is the distance made good (DMG) in the given time interval, from which calculate speed made good (SMG).

Three bearings of same object, with time interval, and with current and leeway

Data given --- Three bearings of object (O), time intervals between bearings, ship's course and speed, current direction, wind direction and leeway.

• Draw the three given bearings OX, OY and OZ from the object O, and note the times of each.
• Calculate time interval T₁ between the first and second bearings and interval T₂ between the second and third bearings.
• Calculate ratio of T₁ : T₂ or ratio of distances run at ship's speed between OX and OY, and between OY and OZ.
• Draw a line through O, on one side and clear of the three bearings. (This line need not be parallel or perpendicular to any other line.)
• Measure convenient lengths on this line, on either sides of O, in the above ratio, to obtain point A on the side of OX, and point B on the side of OZ.
• From A and B draw lines parallel to OY, to cut OX and OZ at C and D respectively.
• Line CD is the CMG but points C and D are NOT positions of the ship.
• From C draw the given CS and apply leeway appropriately to obtain corrected CS₁.
• From the given ship's speed calculate the distance run in the total time interval and measure it on CS₁ to obtain point E.
• At E draw the given current direction to meet CD at F.
• At F transfer the first bearing OX to cut the third bearing OZ at R.
• At R draw reverse CMG to cut the first bearing OX at P and the second bearing OY at Q.
• P, Q and R are the three positions of the ship at the three bearings.
• Measure the distance EF and calculate the rate of current for the given total time interval.
• Measure PR, which is DMG, and using the total time interval calculate SMG.

Out of the four possible data i.e. CS, ship's speed, current direction and rate, any three data will be given in the example. Using these 3 data and the CMG obtained above, construct the course and current vector triangle as explained above, and calculate the remaining data.

Course to steer to sight an object right ahead at certain distance, allowing for current but without leeway

Data given --- Starting position (A), ship's speed, current direction and rate, and distance (d) at which object (O) is to be sighted right ahead.

• Plot the given starting position of the ship at A.
• Using ship's speed calculate time period to cover distance d.
• Using rate of current calculate drift for this time period.
• From O draw the given current direction and calculate drift, to obtain point B.
• Draw an arc of radius d around O.
• Draw line AB to cut the arc at C, which is the position from where O will be sighted right ahead.
• AB is CMG and CB is DMG in the calculated time period. Calculate SMG.
• Draw line CO, which is the CTS.

Course to steer to sight an object right ahead at certain distance, allowing for current but without leeway

Data given --- Starting position (A), ship's speed, current direction and rate, and distance (d) at which object (O) is to be sighted right ahead.

• Plot the given starting position of the ship at A.
• Using ship's speed calculate time period to cover distance d.
• Using rate of current calculate drift for this time period.
• From O draw the given current direction and calculate drift, to obtain point B.
• Draw an arc of radius d around O.
• Draw line AB to cut the arc at C, which is the position from where O will be sighted right ahead.
• AB is CMG and CB is DMG in the calculated time period. Calculate SMG.
• Draw line CO, which is the CTS.

Plot courses between two positions, allowing for current and leeway, and passing at a safe distance from an object

Data given --- Two positions X and Y, ship's speed, safe distance (d) to be maintained from object (O) located in-between X and Y, current direction and rate, and wind direction and leeway.

• Plot the two positions X and Y.
• Draw a circle around the object O for the given safe distance d.
• From X and Y draw tangents to the circle, to meet each other at A.
• Tangents are the CMGs from X to A, and from A to Y. Measure the CMGs and DMGs.
• Using the given data and calculated CMGs, construct separate "one hour" course and current vector triangles at X and A to calculate both CTSs and SMGs.
• Calculate time taken on each CMG and hence total time taken between the two positions.
• Counteract leeway appropriately to obtain corrected CTSs.
• Using the first SMG and DMG calculate the time for altering course at A.

Plot courses between two positions, passing at a safe distance form an object, without current or leeway

Data given --- Two positions X and Y, ship's speed, safe distance (d) to be maintained from object (O) located in-between X and Y.

• Plot the two positions X and  Y.
• Draw a circle around the object O for the given safe distance d.
• From X and Y draw tangents to the circle, meeting the circle at A and B respectively.
• Tangents are the CMGs from X to A, and from B to Y. Measure the CMGs, which will also be CTSs because there is no current or leeway.
• At A object O will be abeam, after which the ship will sail along an arc of the circle drawn, by maintaining the object abeam all the time, till B.
• At B the ship will be heading on the second CMG / CTS.
• Measure , which is the difference between the two CMGs.
• Calculate distance (miles) along the arc .
• Calculate total distance and time taken between the two positions, assuming that the ship's speed along the arc remains constant.

Safe courses to steer to reach a given position using single position line, with current and leeway

Data given --- Final position, terrestrial position line or celestial position line with position to draw it, ship's speed current direction and rate, and wind direction and leeway.

• Draw the PL in the given position.
• Plot the final position C.
• Take any point A on the PL and draw the first CMG ---

• Perpendicular to the PL, or
• In the direction given in the example.

• Draw a line from C, parallel to the PL, to meet the first CMG at B.
• AB is the first CMG and DMG, but A and B are NOT positions of the ship.
• BC is the second CMG but NOT second DMG. This CMG is same as the PL
• Using the given data and calculated CMGs, construct separate "one-hour" course and current vector triangles at A and B to calculate both CTSs and SMGs.
• For both CTSs counteract leeway appropriately to calculate corrected CTSs. (It is assumed that leeway remains same for both CTSs).
• Using the first DMG and SMG, calculate time for altering course at B.
• Time to reach C cannot be calculated because second DMG is not known.

Safe courses to steer to reach a given position using single position line, with current and leeway

Data given --- Final position, terrestrial position line or celestial position line with position to draw it, ship's speed current direction and rate, and wind direction and leeway.

• Draw the PL in the given position.
• Plot the final position C.
• Take any point A on the PL and draw the first CMG ---

• Perpendicular to the PL, or
• In the direction given in the example.

• Draw a line from C, parallel to the PL, to meet the first CMG at B.
• AB is the first CMG and DMG, but A and B are NOT positions of the ship.
• BC is the second CMG but NOT second DMG. This CMG is same as the PL
• Using the given data and calculated CMGs, construct separate "one-hour" course and current vector triangles at A and B to calculate both CTSs and SMGs.
• For both CTSs counteract leeway appropriately to calculate corrected CTSs. (It is assumed that leeway remains same for both CTSs).
• Using the first DMG and SMG, calculate time for altering course at B.
• Time to reach C cannot be calculated because second DMG is not known.

Obtain ship's positions when two relative bearings of same object on bow are taken, without current or leeway

Data given --- Starting position or CS, two relative bearings of one object, time interval between the two bearings, and ship's speed.

•  is first relative bearing and  is second relative bearing.
• Calculate D₁ = Distance run between the two bearings in given time interval.
• D₂ is distance of the object at subsequent beam bearing.
• Calculate .

• If starting position is given ---

• Plot position circle for distance D₂ around the object, and draw a tangent to it from the starting position.
• This gives CS and the position when the object will be abeam.
• Apply  and  to the CS and draw the two true bearings to intersect the CS at first and second positions of the ship.

• If CS is given ---

• Calculate true bearings of the object when it is  and  on the bow, and when it is subsequently abeam.
• Knowing D₂ and beam bearing obtain beam position.
• From beam position plot reverse CS to obtain positions when object is  and  on the bow.

• Special cases

• If  and , then D₁ = D₂
• If , then D₁ = Distance of object at second relative bearing.

Rendezvous with a ship at the earliest, without current

Data given --- Positions of both ships, speed (S₁) of own ship, CS and speed (S₂) of other ship, wind direction and leeway as applicable to own ship.

Method 1 ---

• Plot own ship at point A and the other ship at point B on the chart and join them.
• Plot CS of the other ship. (Leeway is not applicable to the other ship).
• Measure distance traveled by other ship in 1 hour (or 2 hours if scale of chart is small) on its course, to obtain point D.
• From D draw an arc of radius equal to the distance traveled by own ship in the same time interval of 1 or 2 hours, to cut the line AB at point E.
• ED is CTS of own ship. Transfer it to point A to meet CS of other ship at point C, which is the rendezvous position.
• Counteract leeway appropriately to calculate corrected CTS of own ship.
• Knowing speeds of both ships calculate time of rendezvous, which should be same for both ships.

Method 2 ---

This method is adopted if plotting work on chart is not possible because the positions of both ships and position of rendezvous are either very far apart or not on the same chart.

• Construct approximate  ABC on answer sheet, wherein ---

• A is position of own ship and B is position of other ship.
• C is unknown rendezvous position after unknown T hours.
• AC is unknown CTS and unknown distance run by own ship at speed S₁ in T hours i.e.  miles.
• BC is known CS and unknown distance run by other ship at speed S₂ in T hours i.e.  miles.

• Calculate distance AB by Plain Sailing procedure.
• Calculate bearing of A from B and, knowing CS of other ship, calculate  inside the  ABC.
• Calculate  inside  ABC by formula --- .
• Knowing CS of other ship, and calculated  and , calculate CTS of own ship.
• Knowing  and , calculate .
• Calculate time period T, by formula --- .
• Knowing speeds S₁ or S₂ and time period T, calculate distance AC or BC.
• Knowing distance AC or BC, and positions of both ships, calculate rendezvous position C by Plain Sailing procedure. It should be same for both ships.
• Knowing CTS of own ship, counteract leeway appropriately to calculate corrected CTS.

Rendezvous with a ship at given time

Given data --- Positions of both ships, CS and speed of other ship, time of meeting, current direction and rate, and wind direction and leeway as applicable to own ship.

Method 1 ---

• Plot own ship at point A and the other ship at point B on the chart.
• Plot CS of the other ship. (Leeway is not applicable to the other ship.)
• Using the given data, construct "one hour" course, speed and current vector triangle at point B to calculate CMG and SMG of other ship.
• Knowing CMG and SMG of other ship, and the given rendezvous time, calculate distance run by other ship and plot rendezvous postion C.
• Knowing positions A and C, measure distance AC and CMG of own ship.
• Knowing rendezvous time and distance AC, calculate SMG of own ship.
• Using the given and calculated data, construct "one hour" course, speed and current vector triangle at point A to calculate CTS and speed of own ship.
• Knowing CTS of own ship, counteract leeway appropriately to calculate corrected CTS.

Method 2 ---

This method is adopted if plotting work on chart is not possible because the positions of both ships and position of rendezvous are either very far apart or not on the same chart.

• Select position of other ship at any convenient point B on the chart.
• Using the given data, construct "one hour" course, speed and current vector triangle at B to calculate CMG and SMG of other ship.
• Knowing the above data and time of rendezvous, calculate rendezvous position C.
• Knowing position of own ship A, rendezvous position C, and time of rendezvous, calculate CMG and SMG of own ship.
• Construct "one hour" course, speed and current vector triangle at A, to calculate CTS and speed of own ship.
• Knowing CTS of own ship, counteract leeway appropriately to calculate corrected CTS.

Rendezvous with a stopped ship at the earliest

Given data --- Positions of both ships, speed of own ship, other ship is stopped and drifting, current direction and rate, wind direction and leeway as applicable to own ship.

Method 1 ---

• Plot own ship at point A and the other ship at point B on the chart.
• Measure course from own ship to other ship, which is CS of own ship, and measure distance AB.
• Knowing speed of own ship, calculate time interval T to reach point B.
• Knowing CS of own ship, counteract leeway appropriately to obtain corrected CS.
• Apply direction and rate of current at point B for time T to obtain rendezvous position C.
• ABC is course, speed and current vector triangle for time T.
• AC is CMG and DMG of own ship.
• Knowing time T calculate SMG of own ship.

Method 2 ---

This method is adopted if plotting work on chart is not possible because the positions of both ships and position of rendezvous are either very far apart or not on the same chart.

• Knowing position of both ships calculate course from own ship at point A to other ship at point B, which is CS of own ship.
• Calculate distance AB, and knowing speed of own ship calculate time interval T to reach point B.
• Knowing CS of own ship, counteract leeway appropriately to obtain corrected CS.
• Apply direction and rate of current at point B for time T and calculate rendezvous position C.
• ABC is course, speed and current vector triangle for time T.
• Calculate AC, which is CMG and DMG of own ship.
• Knowing time T calculate SMG of own ship.