Editor's Note: This article was ori~i-nalhy published in the Mard -1 1979 issueof Surface Warfare magazine, andreprinted 8 !tears later. The airthor, LTblur V. Hall of the LISS Power (DD839) retired front the naval service in1979. We receive numerous requests forcopies, so in honor of :APT Hall's arti-cle, we have added information to flueoriginal article using contributions fromMarine Safetil International . As moreSailors lrtta111 qualifrcatioll t1S 00D, orstand watch as Junior Officer of the Deckor Junior Officer of the Watch, we hopeto receive marnf more innovativeapproaches to shiphandling for publish-ing!

A s the techniques andmachines designed to aid inall phases of operations

become more numerous and sophis-ticated, it seems a pleasure to realizethat in one vital area no substitutehas been found, or is likely to be forthe exercise of good human judg-ment and thought, aided by rathersimple equipment of long standing .

The conning officer does of coursehave numerous aids to help performeven the most elementary maneuvers .CIC, radar, and the maneuveringboard are close at hand . Informationconcerning contacts is clearly dis-played . Everything on the bridgeseems to be calm, in order, ready .Probably it is . But somehow, some-thing can come tip fast enough andunexpectedly enough so that an almostinstantaneous decision will have to bemade without much outside help .

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Even if help and recommendationsare quickly fortlhcomning, there is stillthe question, is this correct? Is it thesafest, quickest and best way? Theresponsibility for making this deter-mination does not leave the bridge; itis there to stay.

The conning officer then has anobligation to use every means avail-able to quickly and accurately snakethis determination . The 001) isrequired to study pertinent tacticaldoctrine, the maneuvering cliaracter-

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sistics of his own ship and other types,and maneuvering board . He isrequired to give and receive instruc-tion so that his decision will alwaysbe acceptable . There is however, afaculty which an OOD can developand use effectively with just a littleanalysis and practice, but which few

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A One radian is approximately 60 degrees, We use the radian rule for small angles (30 degrees or less)to approximate the length of the short side of a triangle by assuming the ratio is 1 to 60 compared to the

long side .

a

seem to consciously use . It is nomore than mental aritlhnietic, basedon a variety of simple rifles .

When the OOD uses "thumbrules" in maneuvering, lie is merelyapplying a little pre-cooked math toa specific situation . When lie exer-cises „seaman's eye", he is also mak-ing mathematical calculations, buton a level below (or above) the levelof conscious figuring. Let's look fora minute at one simple device :radian measure .

Most OOD's have run across theradian at one time or another in ele-mentary math courses. Few appreciateit properly.

What is a Radian?A radian is defined as the angle

which subtends an arc of a circle

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til for 0By LT John V. Hall, USN and Marine Safety International

the arc subtended by an angle of 1degree is going to be approxi-mately 1 /60 of the radius, 2degrees, 2/60 of the radius, 10degrees, 1/6 of the radius, and soforth . For small angles (30 degreesor less), the arc (a) and chord (c)are approximately equal .

Simply put: for every angle of 1degree, the ratio f the long side of atriangle to the short side is 1 to 60 .So for the triangle shown, if thelong side is 600 yards, then the shortside is 10 yards .

For instance :You see a fishing boat about two

miles ahead,, 3 degrees on your star-,A Applying the radian rule here, 1 degree to 60 degrees is the some ratio as the side opposite of one

board bow. He seems to be DIW.degree to the adjacent side - 10 yards to 600 yards .

How close do you come if you holdyour course?

where the length of the arc equals the

If we round 57.4 to 60, we can

True this is an instant's work on aradius. In the drawing, arc a, radius

say that the length of the arc cre-

maneuvering board . But why notr and chord c are shown . When a = r

ated an angle of 60 degrees is going convert that 3 degrees to 1/20 of athe angle shown is one "radian" or

to be approximately equal to the

radian (3 degrees / 60 degrees), and57.4 degrees .

radius. Consequently the length of

then take 1/20 of the range, thus

4,000 YO ,

A While bearing drift, the constant change of an object's bearing from the ship, will help determine if a danger of collision exists, the actual range of the closest

point of approach ((PA) can be found using the radian rule .

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getting your 200 yard CPA? Tooclose? Five hundred yards would bebetter. Well, 500 is 1/8 of 4000, and1/8 of a radian is what? Between 7and 8 degrees . To clear him by 500yards, starting now, you should steerabout 8 degrees off his bearing soyou come left 5 degrees .

Another raplil1Crati0ra 0f the radian rule :While alongside during an unrep,

one of our primary concerns is keep-ing a safe. distance from the deliveryship. Since we routinely steer coursesto the left and right of replenishmentcourse while adjusting position, it

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Applying the radian rule to speed vectors, we see a 1 degree course change at 13 knots during calm

weather conditions causes the ship to open at 20 feet per minute .

A The Amphibious (ommand Ship (right) has an overall length of approximately 600ft . A 1 degree star-board course change swings the stern toward the guide ship by at least 10 feet, the decrease in pressuredue to water moving faster through a narrower opening could make steering more difficult .

would be useful to know how thisaffects the position of our stern .There are a lot of variables at workduring an unrcp but figuring out thelateral movement of the stern (at leasttheoretically) for a heading change isrelatively easy to do . For simplicity,assume our ship is 600 feet long . Itwe change heading 1 degree, and. ifwe assume the ship pivots at the bow(a "worst case" scenario), we applythe radian rule ratio of 1/60 and con-clude that the stern moves left 10feet. Since we know that the shipwill pivot somewhere aft of the bow,

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tile 10 fool figure represents the mostthe stern will move toward the deliv-ery ship for every one degree ofheading change .

Y'L t raiiother applicat iora . . .

Most unreps are conducted at aspeed of 13 knots . When the receiv-ing ship steers a heading to the rightor left of Romeo Corpen she shouldeither open or close depending onwhat side of the delivery ship she ison. It is useful to know what the rateof lateral opening or closing is forevery degree of heading difference .In the diagram to the right, the twoship's vectors are shown with a head-ing difference of one degree . If wetake 1/60 of 13 knots (which happensto be 1300 feet per minute) we get21 .6 feet per minute (fpm), which weround off to 20 fpm . Thus, in theabsence of other forces, for every onedegree of heading difference fromRomeo Corpen we open (or close) atthe rate of about 20 feet per minute .

Getting out of closequarters -- the "fivedegree" maneuver :

It, by some circumstance, you findyourself dangerously close to theoiler, say 60 or 80 feet, you have toopen the distance quickly becauseinteractive forces are increasing expo-nentially and you are running out oftime. You initiate emergency break-away; but you can't floor the acceler-ator until the rigs are clear, and theincrease in speed of water passingriver the hull may cause the waterpressure between ships decrease fur-ther, increasing chances for collision .You are so close that k you're con-cerned that if you come right, yourstern may hit the delivery ship .Nevertheless, that's what you have todo and what's more, you have tomake a decisive course change .

Assume you are to starboard ofthe oiler. As we learned above, for

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A The blind area created by a particularly rakish bow or low superstructure can be a dangerous region . Small boats should never drive under the bow .

every degree of heading change tothe right the stern move left and willclose the oiler by as much as 10 feet(for a 60() foot long ship) . At the sametime, however, each degree of head-ing change to the right adds about 20feet per minute to your opening vec-tor. Thus, if you make a coursechange to the right of five degrees,your stern will close by at most 50feet. At the same time you willachieve an opening rate of 100 feetper minute or more . In fact, as theship comes right even though thestern rotates toward the deliveryship, it closes by much less than 50feet because the opening vector takeseffect and the ship tracks to the right .

It is recommended that this "fivedegree" maneuver be executed bygiving a course order (i .e. "Helm,come right smartly to XXX ." ) vice arudder order and that the helmsmanbe a qualified master helmsman whohas been briefed to anticipate such anorder in the event of an emergencybreakaway. You may choose to use

rudder orders, but the helmsmanmay be carrying significant amountsof rudder already to hold the orderedcourse and will know what's neededto execute a smart course change.The real challenge in this maneuvermay be returning to base coursebefore you open up too far .

The Three Minute andOne Minute Rules

Experienced OODs are wellversed in the three minute rulewhich states that the distance inyards traveled in 3 minutes is equalto the speed in knots multiplied

A At a signal's execution, a ship must expeditiously maneuver to station while minding

safety of navigation .

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A In the 2 .5 minutes it takes to turn at 20 knots, the guide will travel 1500 yards. The ratio of 1500yards to the range at station is slightly less than half a radian, approximately 25 degrees . Turn whenthe guide bears 095, 25 degrees greater than guide's bearing when at station .

100 . Thus a ship going 10 knotstravels 1000 yards in 3 minutes . Forslow speeds it may sometimes beuseful to use the one Minute rulewhich is essentially the three minuterule divided by three . The oneminute rule says that the distancetraveled in feet in one minute isequal to the speed in knots multi-plied by 100 . Thus a ship approach-ing a berth at 2 knots will travel 200feet in one minute .

The InvisibleDistance Formula

The Invisible distance formula isnot quick math. It has to be calcu-lated in advance but once calculatedit provides a useful tool for closequarters situations . A good exampleof its use is the situation where youare entering a berth and need to esti-mate the distance ahead to a fixedobject such as the quay wall . Fromthe bridge, your line of sight over thebullnose will intersect the surface ofthe water at some distance ahead ofthe bow which, given your height ofeye and the typical draft and trip, ofthe vessel will be a fixed distance .

Typically for CG; and DOGs this dis-tance is about S0 or 90 yards .However, using the drawing and for-mula below you can calculate what itwill be for your particular ship . Youwill find, that differences in height ofeye and small changes in ship's draftwill not affect the result very much .

One of the things an OOD isexpected to know is, of course, themaneuvering characteristics, or "tac-tical curves" of his ship . Usually this

A In the 2 .5 minutes it takes to turn at 20 knots, the guide will travel 1500 yards . The ratio of 1500yards to the range at station is slightly less than half a radian, approximately 25 degrees . Turn when theguide bears 095, 25 degrees greater than guide's bearing when at station .

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will comprise a volume of figuresand graphs, which can scarcely be ;memorized. I lowever, it is impera°tive for the UUD to have a ratheraccurate knowledge of advance,transfer, turning diameter and turn-ing rates for various speeds and rud-der angles . It is not too much toangles .memorize figures for, say, ten, fifteen,twenty, and twenty-five knots, thesame rudder angles, and 180' of turn .In any event, the data should be mostreadily accessible, preferably ill ahandy tabulated form .

Now suppose that. you are a con-siderable distance ahead of yourguide, and you are told to take sta-tion much closer, dead ahead . Youdecide to turn and head back on thereciprocal course, almost "down thethroat" . He's making 20 knots, youare at 25, and suddenly there You arewith a closing range of nearly 45knots and a decision to make . It iswell to come prepared .

Your station is 1000 yards deadahead of the guide, and you havedecided to turn back to base courseat 25 knots with 15 degrees rudder,slowing to 20 a s you begin tosteady tip on base course. Now youabsolutely must know how log;that turn is going to take .

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For safety's sake you are going toassume it takes three minutes . Inthree minutes the guide willapproach you by a matter of 2000yards, due to his own motion .Therefore, when his range is 3000yards you Will turn, and when youhave completed your turn andslowed to 20 knots, you Will be justabout 1000 yards ahead of him,actually a trifle more . However, iffrom necessity or choice you aregoing to do it as described here,there is no other way than to makethese calculations . CIC can do nomore and you may be lucky if theyreact quickly enough to do as much .A maneuvering board is no help . Itwill have to be done in your head .

Here you are coming into whatmight be screen station, 3500 yardsfrom the guide and somewhat abaftof his beam . His course is 000°,

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A Target angle offers useful information to quickly plan or visually confirm the course of a contact calculated by CIC .

speed 18, and when on station hewill bear 070'. You are coming downa more or less reciprocal course alongthe track shown at 20 knots .

You chose your course, by eye orby maneuvering, so as to he dis-placed towards the guide's track bythe diameter of your turning circle .Now as you close him, you are againwondering where to start your turn .

Begin by assuming that your turnwill take place at a point. The pointchosen is the one you reach on com-pletion of your turn. when there, youhope to be on station. This assump-tion will introduce error, but notmuch in a situation similar to thisexample. This turn is going to take(you know or have looked it up) 2minutes at 20 knots, with standardrudder. Again you are going to slowto 15 knots towards the end ofthe turn .

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While you spend 2 minutes turn-ing, the guide will go how far alonghis track? About 1500 yards, by thethree minute rule . Those 1500 yardsare less than halt of your 3500 yardsfinal range . The angle A above istherefore a little less than half aradian, say 25 degrees. Thereforewhen the guide bears 005, you willbegin your turn .

Now when you look at it objec-tively, what could be simpler to cal-culate? All you used. was theknowledge you had anyway, plus theradian rule. You make no attempt tobe absolutely precise. You used sev-eral approximations, for simplicity'ssake. Probably you turned a littlelate, and corrected for that immedi-ately on coming out of your turn .What gave your mental calculationstheir value Was the fact that youknew you would come out quiteclose to station .

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At the moment you began yourturn, the situation looked like this :

You turned with the guide at arange of 3000 yards . With your turn-ing diameter of 1150 yards, the angleA was a little over one-third of aradian, say 25 degrees . The distancefrom C to I' was therefore only 2700yards. If your turn had, in fact taken3 minutes, as you assumed, youwould have ended up only 700 yardsahead, which could be cutting itrather fine, to say nothing of puttingyou 300 yards behind station . Thatextra half minute you allowed forthe turn served to compensate for300 yards, inasmuch as half a minuterepresented very close to 300 yardsin the advance of the guide alonghis track .

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A To the observer on the bow, the north face of Pier 1 is "OPEN", A is the "DISTANCE TO" Pier 1, B is the offset of the bow "FROM" the face of Pier 1, and ( is the

offset of the port side "TO" the end of Pier 1 . To the observer on the stern, the south face of Pier 1 is "OPEN", the north face of Pier 1 is "CLOSED", the stern "HAS

SWING CLEARANCE" on buoy T, or the swing clearance on buoy T is distance D .

How do most OOD's figure recip-rocal bearings? A quick glance at themaneuvering hoard is no doubt easi-est and surest . But if you want to doit in your head, do you try to add orsubtract 180 degrees? This canbecome awkward, and can lead toerrors. Why not add 200" and thensubtract 20, or conversely subtract200 and add 20? This is of courseextremely simple, but have all OOD'sthought of it?

Few OOD's realize target angles,reciprocal bearings and angle-on-thebow can be useful tools to determinea contact's course .

The contact bears 168' . The recip-rocal is 348', the contact is 20 degreeson the starboard bow. Contact course348° minus 20 equals 328" . The solu-tion is found in an instant, and if

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your estimate was good, will be bet-ter than CIC's, which may not beavailable anyway for two orthree minutes .

Visualizing relationships is a largepart of using "quick math" to helpyou determine contact course, CPA,and other information, but it can alsohelp you communicate better.

Conning officers often rely onreports from the foc'sle and fantail tojudge the ships position and move-ment. This is only useful if everyoneis speaking the same language. Thedrawing below suggests several termsand definitions that might prove use-ful in communicating between thebridge and various lookouts .

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Doubling the Angle on

the Bow (A ClassicDR tool)

Another application of basictrigonometry can be useful inobtaining ail estimated positionfrom a single landmark . Shoot thebearing to a fixed object and recordthe difference between your head-ing and the bearing (angle on thebow) . Maintain course and con-tinue to shoot bearings to theobject until the angle on the bow istwice what it was at the first bear-ing . At that point, your distancefrom the fixed object is equal to thedistance traveled between the twobearings. The bearing and distancethus obtained can be plotted as anestimated position .

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A Running fixes can be especially useful when visual bearings are only available on one side of the ship .

It would be impossible to exhaustthe situations in which an OODshould be able to find fast and rea-sonably accurate mental solutions tomaneuvering problems . The varietyof techniques, tricks, and thumb rulesis also endless. I believe, however,that it is appropriate to stress thatthis type of thinking, and mentalpreparation, should be just one typi-cal part of an OOD's continuous andanalytical effort to qualify himself tomeet any situation which may arise .

The radian rule and the threeminute rule are excellent aids to anOOD in the execution of his con-ning responsibilities on the bridge .Their use is for assisting in makingrapid and accurate maneuveringdecisions . The following test repre-sents applications of mathematics

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that OOD's should be able to do(sans caluculators) .

(The test is not included here -go online towyv .na~~sea .navy.m1l/swm<rar,ineand download the test and50 variations!)

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