The American Practical Navigator/Chapter 22

= CHAPTER 22 CALCULATIONS AND CONVERSIONS =

2200. Purpose and Scope
This chapter discusses the use of calculators and computers in navigation and summarizes the formulas the navigator depends on during voyage planning, piloting, celestial 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 be familiar with the use of a basic scientific calculator. The navigator should choose a calculator based on personal needs, which may vary greatly from person to person according to individual abilities and responsibilities.

2201. Use of Calculators in Navigation
Any common calculator can be used in navigation, even one providing only the four basic arithmetic functions of addition, subtraction, multiplication, and division. Any good scientific calculator can be used for sight reduction, sailings, and other tasks. However, the use of a computer program or handheld calculator specifically designed for navigation will greatly reduce the workload of the navigator, reduce the possibility of errors, and increase the accuracy of results over those obtained by hand calculation.

Calculations of position based on celestial observations are becoming increasingly obsolete as GPS takes its place as a dependable position reference for all modes of navigation. This is especially true since handheld, batterypowered GPS units have become less expensive, and can provide a worldwide backup position reference to more sophisticated systems with far better accuracy and reliability than celestial.

However, for those who still use celestial techniques, a celestial navigation calculator or computer program can improve celestial positions by easily solving numerous sights, and by reducing mathematical and tabular errors inherent in the manual sight reduction process. They can also provide weighted plots of the LOP’s from any number of celestial bodies, based on the navigator’s subjective analysis of each sight, and calculate the best fix with lat./long. readout.

On a vessel with a laptop or desktop computer convenient to the bridge, a good choice would be a comprehensive computer program to handle all navigational functions such as sight reduction, sailings, tides, and other tasks, backed up by a handheld navigational calculator for basic calculations should the computer fail. Handheld calculators are dependable enough that the navigator can expect to never have to solve celestial sights, sailings, and other problems by tables or calculations.

In using a calculator for any navigational task, it important 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 accuracy 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.

Some basic calculators require the conversion of degrees, 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, and should do conversions at will. Though many non-navigational computer programs have an onscreen calculator, these are generally very simple versions with only the four basic arithmetical functions. They are thus too simple for many navigational problems. Conversely, a good navigational computer program requires no calculator per se, since the desired answer is calculated automatically from the entered data.

The following articles discuss calculations involved in various aspects of navigation.

2202. Calculations of Piloting
 Hull speed in knots is found by: S = 1.34$$\sqrt{\text{waterline length}}$$ (in feet). This is an approximate value which varies with hull shape.

Nautical and U.S. survey miles can be interconverted by the relationships: 1 nautical mile = 1.15077945 U.S. survey miles. 1 U.S. survey mile = 0.86897624 nautical miles. 

The speed of a vessel over a measured mile can be calculated by the formula: S = $$\frac{\text{3600}}{\text{T}}$$ 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: D = $$\frac{\text{S · T}}{\text{60}}$$ 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 $$\sqrt{\mathrm{h}_\mathrm{f}}$$&emsp;or,&emsp;D = 2.07 $$\sqrt{\mathrm{h}_\mathrm{m}}$$ depending upon whether the height of eye of the observer above sea level is in feet (h$f$) or in meters (h$m$).</li>

<li>Dip of the visible horizon in minutes of arc can be calculated using the formula: <p >D = 0.97' $$\sqrt{\mathrm{h}_\mathrm{f}}$$&emsp;or,&emsp;D = 1.76' $$\sqrt{\mathrm{h}_\mathrm{m}}$$ depending upon whether the height of eye of the observer above sea level is in feet (h$f$) or in meters (h$m$)</li>

<li>Distance to the radar horizon in nautical miles can be calculated using the formula: <p >D = 1.22 $$\sqrt{\mathrm{h}_\mathrm{f}}$$&emsp;or,&emsp;D = 2.21 $$\sqrt{\mathrm{h}_\mathrm{m}}$$ depending upon whether the height of the antenna above sea level is in feet (h$f$) or in meters (h$m$).</li>

<li>Dip of the sea short of the horizon can be calculated using the formula: <p >Ds = 60 tan$-1$ $$\left(\frac{\mathrm{h}_\mathrm{f}}{6076.1 \mathrm{d}_\mathrm{s}} + \frac{\mathrm{d}_\mathrm{s}}{8628}\right)$$ where Ds is the dip short of the horizon in minutes of arc; h$f$ is the height of eye of the observer above sea level, in feet and d$s$ is the distance to the waterline of the object in nautical miles.</li>

<li>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 significance, these assumptions produce no large errors. <p >D = $$\sqrt{\frac{\tan^2\mathrm{a}}{0.0002419^2} + \frac{\mathrm{H-h}}{0.7349} - \frac{\tan \mathrm{a}}{0.0002419}}$$ where D is the distance in nautical miles, a is the corrected 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 (0.0002419 and 0.7349) account for refraction.</li> </ul>

2203. Tide Calculations
<ul > <li>The rise and fall of a diurnal tide can be roughly calculated from the following table, which shows the fraction of the total range the tide rises or falls during flood or ebb.</li>

</ul>

2204. Calculations of Celestial Navigation
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 representation of the circle of equal altitude, particularly at high observed altitudes.

<ul > <li>The dip correction is computed in the Nautical Almanac using the formula:

<p >D = 0.97$$\sqrt{\text{h}}$$

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.</li>

<li>The computed altitude (Hc) is calculated using the basic formula for solution of the undivided navigational triangle:

<p >sin h = sin L sin d + cos L cos d cos LHA,

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:

<p >Hc = sin$-1$[(sin L sin d) + (cos L cos d cos LHA)].

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.</li>

<li>The azimuth angle (Z) can be calculated using the altitude azimuth formula if the altitude is known. The formula stated in terms of the inverse trigonometric function is:

<p >Z = cos$-1$$$\left(\frac{\text{sin d - (sin L sin Hc)}}{\text{(cos L cos Hc)}}\right)$$

If the altitude is unknown or a solution independent of altitude is required, the azimuth angle can be calculated using the time azimuth formula:

<p >Z = tan$-1$$$\left(\frac{\text{sin LHA}}{\text{(cos L tan d) - (sin L cos LHA)}}\right)$$

The sign conventions used in the calculations of both azimuth formulas are as follows: <ol > <li>if latitude and declination are of contrary name, declination is treated as a negative quantity; <li>if the local hour angle is greater than 180°, it is treated as a negative quantity.</li> </ol>

If the azimuth angle as calculated is negative, add 180° to obtain the desired value.</li>

<li>Amplitudes can be computed using the formula:

<p >A = sin$-1$(sin d sec L)

this can be stated as

<p >A = sin$-1$$$\left(\frac{\text{sin d}}{\text{cos L}}\right)$$

where A is the arc of the horizon between the prime vertical and the body, L is the latitude at the point of observation, and d is the declination of the celestial body.</li> </ul>

2205. Calculations of the Sailings
<ul > <li>Plane sailing is based on the assumption that the meridian through the point of departure, the parallel through the destination, and the course line form a plane right triangle, as shown in Figure 2205.

<p >Figure 2205. The plane sailing triangle.

From this:&emsp;cos C = $$\frac{\text{l}}{\text{D}}$$,&emsp;sin C = $$\frac{\text{p}}{\text{D}}$$,&emsp;and&emsp;tan C = $$\frac{\text{p}}{\text{l}}$$.

From this:&emsp;l = D cos C,&emsp;D = l sec C,&emsp;and&emsp;p = D sin C.

From this, given course and distance (C and D), the difference of latitude (l) and departure (p) can be found, and given the latter, the former can be found, using simple trigonometry. See Chapter 24.</li>

<li>Traverse sailing combines plane sailings with two or more courses, computing course and distance along a series of rhumb lines. See Chapter 24.</li>

<li>Parallel sailing consists of interconverting departure and difference of longitude. Refer to Figure 2205. <p > DLo = p sec L,&emsp;and&emsp;p = DLo cos L </li>

<li>Mid-latitude sailing combines plane and parallel sailing, 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.

In mid-latitude sailing: <p >DLo = p sec Lm,&emsp;and&emsp;p= DLo cos Lm </li>

<li>Mercator Sailing problems are solved graphically on a Mercator chart. For mathematical Mercator solutions the formulas are: <p >tan C = $$\frac{\text{DLo}}{\text{m}}$$&emsp;or&emsp;DLo = m tan C where m is the meridional part from Table 6 in the Tables Part of this volume. Following solution of the course angle by Mercator sailing, the distance is by the plane sailing formula: <p >D = l sec C </li>

<li>Great-circle solutions for distance and initial course angle can be calculated from the formulas: <p >D = cos$−1$[(sin L$1$ sin L$2$ + cos L$1$ cos L$2$ cos DLo)], and <p >C = tan$−1$ $$\left(\frac{\sin \mathrm{DLo}}{(\cos \mathrm{L}_1 \tan \mathrm{L}_2)-(\sin \mathrm{L}_1 \cos \mathrm{DLo})}\right)$$ where D is the great-circle distance, C is the initial great-circle course angle, L1 is the latitude of the point of departure, L2 is the latitude of the destination, and DLo is the difference of longitude of the points of departure and destination. If the name of the latitude of the destination is contrary to that of the point of departure, it is treated as a negative quantity.</li>

<li>The latitude of the vertex, Lv, is always numerically equal to or greater than L1 or L2. If the initial course angle C is less than 90°, the vertex is toward L2, but if C is greater than 90°, the nearer vertex is in the opposite direction. The vertex nearer L1 has the same name as L1.

The latitude of the vertex can be calculated from the formula: <p >L$v$ = cos$−1$(cos L$1$ sin C)

The difference of longitude of the vertex and the point of departure (DLo$v$) can be calculated from the formula:

<p >DLo$v$ = sin$−1$ $$\left(\frac{\cos \mathrm{C}}{\sin \mathrm{L}_\mathrm{v}}\right)$$.

The distance from the point of departure to the vertex (Dv) can be calculated from the formula: <p >D$v$ = sin$-1$(cos L$1$ sin DLo$v$). </li>

<li>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: <p >L$x$ = tan$-1$(cos D LO$vx$ tan L$v$) The DLo$v$ and D$v$ of the nearer vertex are never greater than 90°. However, when L$1$ and L$2$ are of contrary name, the other vertex, 180° 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.

The method of selecting the longitude (or DLo$vx$), and determining the latitude at which the great-circle crosses 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 (D$vx$) can be calculated using the formulas: <p >L$x$ = sin$-1$[sin L$v$ cos D$vx$], and <p >DLo$vx$ = sin$-1$ $$\left(\frac{\sin \mathrm{D}_\mathrm{vx}}{\cos \mathrm{L}_\mathrm{x}}\right)$$. </li> </ul>

A calculator which converts rectangular to polar coordinates 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.

2206. Calculations Of Meteorology And Oceanography
<ul > <li>Converting thermometer scales between centigrade, Fahrenheit, and Kelvin scales can be done using the following formulas: <p > C° = $$\frac{\text{5 (F° - 32°)}}{\text{9}}$$ , <p > F° = $$\frac{\text{9}}{\text{5}}$$C° + 32°, and <p > K° = C° + 273.15°. </li>

<li>Maximum length of sea waves can be found by the formula:

<p >W = 1.5$$\sqrt{\text{fetch in nautical miles}}$$. </li>

<li>Wave height <p >= 0.026 S$2$ where S is the wind speed in knots. </li>

<li>Wave speed in knots <p >= 1.34 $$\sqrt{\text{wavelength in feet}}$$, or <p >= 3.03 × wave period in seconds. </li> </ul>

UNIT CONVERSION
<p >Use the conversion tables that appear on the following pages to convert between different systems of units.

<p >Conversions followed by an asterisk are exact relationships.

MISCELLANEOUS DATA
<p >Prefixes to Form Decimal Multiples and Sub-Multiples of International System of Units (SI)