TAB 5 Navigational Mathematics - Chapter 22 - Navigational Calculations.pdf - N. Bowditch - was72

CHAPTER 22

NAVIGATIONAL CALCULATIONS

INTRODUCTION

2200. Purpose And Scope

these skills do not fail when the calculator or computer

does.

In using a calculator for any navigational task, it im-

portant to remember that the accuracy of the result, even if

carried to many decimal places, is only as good as the least

accurate entry. If a sextant observation is taken to an accu-

racy of only a minute, that is the best accuracy of the final

solution, regardless of a calculator’s ability to solve to 12

decimal places. See Chapter 23, Navigational Errors, for a

discussion of the sources of error in navigation.

In addition to the 4 arithmetic functions, a basic navi-

gational calculator should be able to perform reciprocals,

roots, logarithms, trigonometry, and have at least one mem-

ory. At the other end of the scale are special navigational

computer programs with almanac data and tide tables, inte-

grated with programs for sight reduction, great circle

navigation, DR, route planning, and other functions.

This chapter discusses the use of calculators and com-

puters in navigation and summarizes the formulas the

navigator depends on during voyage planning, piloting, ce-

lestial navigation, and various related tasks. To fully utilize

this chapter, the navigator should be competent in basic

mathematics including algebra and trigonometry (See

Chapter 21, Navigational Mathematics), and possess a good

working familiarity with a basic scientific calculator. The

brand of calculator is not important, and no effort is made

here to recommend or specify any one type. The navigator

should choose a calculator based on personal needs, which

may vary greatly from person to person according to indi-

vidual abilities and responsibilities.

2201. Use Of Calculators In Navigation

Any common calculator can be used in navigation, even

one providing only the 4 basic arithmetic functions of addi-

tion, subtraction, multiplication, and division. In general,

however, the more sophisticated the calculator and the more

mathematical functions it can perform, the greater will be its

use in navigation. Any modern hand-held calculator labeled

as a “scientific” model will have all the functions necessary

for the navigator. Programmable calculators can be preset

with formulas to simplify solutions even more, and special

navigational calculators and computer programs reduce the

navigator’s task to merely collecting the data and entering it

into the proper places in the program. Ephemeral (Almanac)

data is included in the more sophisticated navigational cal-

culators and computer programs.

Calculators or computers can improve celestial naviga-

tion by easily solving numerous sights to refine one’s

position and by reducing mathematical and tabular errors

inherent in the manual sight reduction process. In other

navigational tasks, they can improve accuracy in calcula-

tions and reduce the possibility of errors in computation.

Errors in data entry are the most common problem in calcu-

lator and computer navigation.

While this is extremely helpful, the navigator must

never forget how to do these problems by using the tables

and other non-automated means. Sooner or later the calcu-

lator or computer will fail, and solutions will have to be

worked out by hand and brain power. The professional nav-

igator will regularly practice traditional methods to ensure

2202. Calculator Keys

It is not within the scope of this text to describe the

steps which must be taken to solve navigational problems

using any given calculator. There are far too many calcula-

tors available and far too many ways to enter the data. The

purpose of this chapter is to summarize the formulas used

in the solution of common navigational problems.

Despite the wide variety of calculators available from

numerous manufacturers, a few basic keystrokes are com-

mon to nearly all calculators. Most scientific calculators

have two or more registers , or active lines, known as the x-

• The +,–,

, and

keys perform the basic arithmet-

radians = 400 grads.

• The +/– keys changes the sign of the number in the

x-register.

• The C or CL key clears the problem from the

calculator.

• The CE key clears the number just entered, but not

the problem.

• The CM key clears the memory, but not the

problem.

• The F, 2nd F, Arc, or Inv key activates the second

339

ical functions.

• The Deg Rad Grad key selects degrees, radians, or

grads as the method of expression of values.

360

340

NAVIGATIONAL CALCULATIONS

function of another key. (Some keys have more

than one function.)

• The x 2 key squares the number in the x-register.

• The y x key raises the number in the y-register to

the x power.

• sin, cos, and tan keys determine the trigonometric

function of angles, which must be expressed in de-

grees and tenths.

• R—>P and P—>R keys convert from rectangular

to polar coordinates and vice versa.

• The p key enters the value for Pi, the circumfer-

ence of a circle divided by its diameter.

• The M, STO, RCL keys enter a number into mem-

ory and recall it.

• M+, M– keys add or subtract the number in mem-

ory without displaying it.

• The ln key calculates the natural logarithm (log e)

of a number.

• log calculates the common logarithm of a number.

• The 1/x key calculates the reciprocal of a number.

(Very useful for finding the reciprocal of trigono-

metric functions).

3600

where S is the speed in knots and T is the time in

seconds.

• The distance traveled at a given speed is computed

by the formula:

where D is the distance in nautical miles, S is the speed

in knots, and T is the time in minutes.

• Distance to the visible horizon in nautical miles can be

calculated using the formula:

D 1.17 h f

, or

2.07 h m

Some basic calculators require the conversion of de-

grees, minutes and seconds (or tenths) to decimal

degrees before solution. A good navigational calculator,

however, should permit entry of degrees, minutes and

tenths of minutes directly.

depending upon whether the height of eye of the ob-serv-

er above sea level is in feet (h f ) or in meters (h m ).

• Dip of the visible horizon in minutes of arc can be cal-

culated using the formula:

Though many non-navigational computer programs

have an on-screen calculator, these are generally very sim-

ple versions with only the four basic arithmetical functions.

They are thus too simple for many navigational problems.

Conversely, a good navigational computer program re-

quires no calculator, since the desired answer is calculated

automatically from the entered data.

D 0.97' h f

, or

1.76' h m

depending upon whether the height of eye of the observer

above sea level is in feet (h f ) or in meters (h m )

2203. Calculations Of Piloting

• Hull speed in knots is found by:

• Distance to the radar horizon in nautical miles can

be calculated using the formula:

1.34 waterline length (in feet).

D 1.22 h f

, or

This is an approximate value which varies accord-

ing to hull shape.

2.21 h m

• Nautical and U.S. survey miles can be interconverted

by the relationships:

depending upon whether the height of the antenna

above sea level is in feet (h f ) or in meters (h m ).

1 nautical mile = 1.15077945 U.S. survey miles.

• Dip of the sea short of the horizon can be calculated

using the formula:

1 U.S. survey mile = 0.86897624 nautical miles.

60 tan 1

–

h f

6076.1 d s

d s

8268

----------------------

------------

• The speed of a vessel over a measured mile can be-

calculated by the formula:

where Ds is the dip short of the horizon in minutes

------------

-------

NAVIGATIONAL CALCULATIONS

341

of arc; h f is the height of eye of the observer above sea

nac using the formula:

0.97 h

level, in feet and d s is the distance to the waterline of

the object in nautical miles.

where dip is in minutes of arc and h is height of eye in

feet. This correction includes a factor for refraction.

The Air Almanac uses a different formula intended for

air navigation. The differences are of no significance in

practical navigation.

• Distance by vertical angle between the waterline

and the top of an object is computed by solving the

right triangle formed between the observer, the top of

the object, and the waterline of the object by simple

trigonometry. This assumes that the observer is at sea

level, the earth is flat between observer and object,

there is no refraction, and the object and its waterline

form a right angle. For most cases of practical signifi-

cance, these assumptions produce no large errors.

• The computed altitude (Hc) is calculated using the ba-

sic formula for solution of the undivided navigational

triangle:

sin

L d

sin

cos

L d LHA,

cos

tan 2 a

0.0002419 2

–

0.7349

–

tan

0.0002419

in which h is the altitude to be computed (Hc), L is the

latitude of the assumed position, d is the declination of

the celestial body, and LHA is the local hour angle of the

body. Meridian angle (t) can be substituted for LHA in

the basic formula.

Restated in terms of the inverse trigonometric function:

where D is the distance in nautical miles, a is the cor-

rected vertical angle, H is the height of the top of the

object above sea level, and h is the observer’s height of

eye in feet. The constants (.0002419 and .7349) ac-

count for refraction.

sin

–

[

(

sin

)

(

cos

LHA

)]

2204. Tide Calculations

When latitude and declination are of contrary name,

declination is treated as a negative quantity. No special

sign convention is required for the local hour angle, as in

the following azimuth angle calculations.

• The rise and fall of a diurnal tide can be roughly cal-

culated from the following table, which shows the

fraction of the total range the tide rises or falls during

flood or ebb.

• The azimuth angle (Z) can be calculated using the al-

titude azimuth formula if the altitude is known. The

formula stated in terms of the inverse trigonometric

function is:

Hour

Amount of flood/ebb

1/12

1– sin d

–

cos L cos Hc

(

sin

L sin Hc

)

2/12

cos

-----------------------------------------------------

(

)

3/12

If the altitude is unknown or a solution independent of

altitude is required, the azimuth angle can be calculated

using the time azimuth formula:

2/12

1/12

2205. Calculations Of Celestial Navigation

tan

–

-----------------------------------------------------------------------------

sin LHA

cos L tan d

(

)

–

(

sin L cos LHA

)

Unlike sight reduction by tables, sight reduction by

calculator permits the use of nonintegral values of latitude

of the observer, and LHA and declination of the celestial

body. Interpolation is not needed, and the sights can be

readily reduced from any assumed position. Simultaneous,

or nearly simultaneous, observations can be reduced using

a single assumed position. Using the observer’s DR or MPP

for the assumed longitude usually provides a better repre-

sentation of the circle of equal altitude, particularly at high

observed altititudes.

The sign conventions used in the calculations of both

azimuth formulas are as follows: (1) if latitude and dec-

lination are of contrary name, declination is treated as a

negative quantity; (2) if the local hour angle is greater

than 180

to obtain the desired value.

• Amplitudes can be computed using the formula:

• The dip correction is computed in the Nautical Alma-

A sin

–

(

sin d sec L

)

----------------------------

----------------

-------------------------

, it is treated as a negative quantity.

If the azimuth angle as calculated is negative, add 180

342

NAVIGATIONAL CALCULATIONS

this can be stated as

• Parallel sailing consists of interconverting departure

and difference of longitude. Refer to Figure 2206.

A sin

– sin d

cos L

-------------

)

DLo

p sec L, and p DLo cos L

where A is the arc of the horizon between the prime ver-

tical and the body, L is the latitude at the point of

observation, and d is the declination of the celestial body.

• Mid-latitude sailing combines plane and parallel sail-

ing, with certain assumptions. The mean latitude (Lm)

is half of the arithmetical sum of the latitudes of two

places on the same side of the equator. For places on

opposite sides of the equator, the N and S portions are

solved separately.

2206. Calculations Of The Sailings

• Plane sailing is based on the assumption that the me-

ridian through the point of departure, the parallel

through the destination, and the course line form a

plane right triangle, as shown in Figure 2206.

In mid-latitude sailing:

From this: cos C= 1

---- , sin C= p

---- , and tan C

--- .

DLo

p sec Lm, and p DLo cos Lm

From this: 1=D cos C, D=1 sec C, and p=D sin C .

• Mercator Sailing problems are solved graphically on

a Mercator chart. For mathematical Mercator solutions

the formulas are:

From this, given course and distance (C and D), the dif-

ference of latitude (l) and departure (p) can be found,

and given the latter, the former can be found, using

simple trigonometry. See Chapter 24.

tan C

----------- or DLo m tan C.

• Traverse sailing combines plane salings with two or

more courses, computing course and distance along a

series of rhumb lines. See Chapter 24.

where m is the meridional part from Table 6.

Following solution of the course angle by Mercator

sailing, the distance is by the plane sailing formula:

L sec C.

• Great-circle solutions for distance and initial course

angle can be calculated from the formulas:

D = cos

–

[

(

sin L

sin

L 2

cos L 1 cos L 2 cos DLo )]

–

sin DLo

tan

-------------------------------------------------------------------------------------------

(

cos L 1

tan L 2

)

–

(

sin L 1

cos DLo

)

where D is the great-circle distance, C is the initial

great-circle course angle, L 1 is the latitude of the point

of departure, L 2 is the latitude of the destination, and

DLo is the difference of longitude of the points of de-

parture and destination. If the name of the latitude of

the destination is contrary to that of the point of depar-

ture, it is treated as a negative quantity.

• The latitude of the vertex ,L v , is always numerically equal

to or greater the L 1 or L 2 . If the initial course angle C is less

than 90

, the vertex is toward L 2 , but if C is greater than

Figure 2206. The plane sailing triangle.

, the nearer vertex is in the opposite direction. The ver-

tex nearer L 1 has the same name as L 1 .

(

DLo

NAVIGATIONAL CALCULATIONS

343

The latitude of the vertex can be calculated from the

formula:

DLo vx

sin

–

sin D vx

cos L x

–

L v

cos

(

cos L 1

sin C

)

A calculator which converts rectangular to polar coor-

dinates provides easy solutions to plane sailings.

However, the user must know whether the difference

of latitude corresponds to the calculator’s X-coordinate

or to the Y-coordinate.

The difference of longitude of the vertex and the point

of departure (DLo v ) can be calculated from the formula:

DLo v

sin

–

----------------

sin

L v

2207. Calculations Of Meteorology And Oceanography

The distance from the point of departure to the vertex

(D v ) can be calculated from the formula:

• Converting thermometer scales between centigrade,

Fahrenheit, and Kelvin scales can be done using the

following formulas:

D v

sin

–

(

cos L 1

sin DLo v

)

----------------------------

- 32

)

• The latitudes of points on the great-circle track can

be determined for equal DLo intervals each side of the

vertex (DLo vx ) using the formula:

--- C

L x

tan

–

(

cos D Lo vx tan L v

)

C 273.15

. However, when L 1 and L 2 are of contrary

name, the other vertex, 180

• Maximum length of sea waves can be found by the

formula:

away, may be the better

one to use in the solution for points on the great-circle

track if it is nearer the mid point of the track.

1.5 fetch in nautical miles .

The method of selecting the longitude (or DLo vx ), and

determining the latitude at which the great-circle cross-

es the selected meridian, provides shorter legs in

higher latitudes and longer legs in lower latitudes.

Points at desired distances or desired equal intervals of

distance on the great-circle from the vertex can be cal-

culated using the formulas:

• Wave height = 0.026 S 2 where S is the wind speed in

knots.

• Wave speed in knots:

1.34 wavelength(in feet), or

–

3.03 wave period (in seconds)

L x

sin

[

sin L v

cos D vx

]

UNIT CONVERSION

Use the conversion tables that appear on the following pages to convert between different systems of units.

Conversions followed by an asterisk are exact relationships.

------------------

cos

(

The DLo v and D v of the nearer vertex are never greater

than 90

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