
59 SIGHT REDUCTION BY ELEMENTARY SPHERICAL TRIGONOMETRY
228. BASIC SPHERICAL TRIGONOMETRY
Excellent although Sight Reduction Tables for Marine Navigation NP.401 are, navigators are well advised to familiarise themselves with an alternative method of obtaining the calculated altitude which can be worked with the minimum number of books – only the N.A., and a set of nautical Tables (Norie’s, Burton’s or Inman’s) – two books in all. This is the timehonoured mathematical method of solving the PZX Triangle by means of spherical trigonometry. Although we do not recommend using this method from choice, there may be occasions when no sight reduction tables are available or they become damaged or lost, or when the volumes available do not cover the vessel’s Lat., and in these circumstances, a navigator who can reduce sights by trigonometry will be worth his weight in gold. There is nothing to choose, as regards accuracy between finding the intercept by Sight Reduction Tables and finding it mathematically as described in this chapter, but the short modern methods described in the first part of this study are recommended for use whenever the necessary tables are available because, as has been shown, most Astronavigation errors are arithmetical blunders and the less calculation the navigator has to do the less the chances of error.
Reducing a sight mathematically is not difficult and involves the use of only one simple formula and the facility in using mathematical tables. We do not propose to delve more deeply into spherical trigonometry than is necessary for an understanding of the method used to find an intercept but some basic concepts of the subject are required by way of introduction.
A sphere is a solid body, every point on the surface of which is equidistant from a fixed point known as the centre of the sphere. It may be defined as a solid, the Shape of which may be swept out by rotating a circle abort any fixed diameter. Such a circle is the largest possible one which may be described on the surface of a sphere.
From fig, 591 it will be noticed that the centre of the sphere lies on the plane of a similar circle. Any circle on the surface of a sphere, on whose plane the centre of the sphere lies, is known as a great circle of that sphere. A great circle divides the sphere into two hemispheres. Any circle on the surface of a sphere which is not a great circle is referred to as a small circle. In fig. 591, AA and BB are examples of great circles, and CC and DD are examples of small circles.
The arc of a great circle is measured in angular measurement and is the angle at the centre of the sphere subtended by the two radii which terminate at the extremities of the arc. The length of a complete great circle is 360° 00′ 00″ and the length of a semigreat circle is 180° 00′ 00″. The shortest distance between two points on the surface of a sphere is along the lesser arc of the great circle which passes through the two points.
A spherical angle is formed at the intersection of two great circles, or at the intersection of two great circle arcs. The value of a spherical angle is the same as the value of the plane angle between the tangents to the great circles at the point of intersection. In fig. 592, the angle between the two great circles XX, and YY, is A. This is equal to the angle between the tangents as indicated in the figure.
A spherical triangle is formed on the surface of a sphere by the intersection of three great circle arcs. It must be emphasised that arcs of small circles do not form sides of a spherical triangle. In fig. 593 the triangle ABC is a spherical triangle; ABD, ACD and BCB are three great circles, and triangles ABC and DCB are two of the spherical triangles formed by their intersection. All the angles at A, B and C are spherical angles. Spherical triangles differ considerably from plane triangles, and the principal properties of spherical triangles are as follows: –
The sum of the three angles must be greater than 180° and less than 540° i.e., their sum may be anything between two right angles and six right angles. (A triangle the sum of whose angles is 180°, is a plane triangle; it is impossible to draw a plane triangle on the surface of a sphere, but the smaller a spherical angle, the more nearly is the sum of the angle equal to 180°. In many navigational problems, small spherical triangles are considered to be plane triangles).
The sides are always expressed in degrees, minutes and seconds of arc. In fig. 593, the arc AB subtends the angle AOB at the centre of the sphere and both arc and angle are expressed by the same number of degrees, minutes and seconds. Similarly, arc. BC = ∠BOC and arc AC = ∠AOC. The lineal length of an arc on the surface of a sphere depends upon the length of its radius, hence the reason why the length of the Earth’s radius must be known with accuracy for the purpose of navigation.
The value of a side never exceeds 180° and the sum of the three sides must be less than 360°. The maximum value of any one angle is 180° (from which it follows that the sum of the three angles must be less that 540° as in (1) above). The greater side is Opposite to the greater angle as in plane triangles.
Where two great circles intersect the vertically opposite angles are equal. In fig. 593, ∠ACB = ∠ECD and ∠ACE = ∠BCD.
There are thus six things to be known about a spherical triangle: the sizes of its three angles and the lengths (in arc) of its three sides. Various formulae connect these angles and sides so that, if sufficient of them are given, the rest can be found. The formula most used by British navigators is the Spherical Haversine Formula, which is derived from the fundamental cosine formula, and used to solve the Astronavigational spherical triangle PZX.
229. THE DERIVATION OF THE SPHERICAL HAVERSINE FORMULA
It is not at all necessary for the navigator to understand the derivation of the Haversine formula used to reduce sights, and those who wish to skip this section may do so. The proofs of the principal formulae involved are given here for the benefit of those students who dislike using information that they are unable to prove or understand.
It was stated above that the spherical Haversine formula is derived from the fundamental spherical cosine formula, Fig, 594 shows any spherical triangle ABC, from which note that it is customary to refer to its angles as A, B and C, and to the sides opposite to these angles as at a, b and c respectively. By the spherical cosine formula, then:
cos.A = cos.a – cos.b x cos.c
sin.b x sin.c
or, cos.a = cos.A x sin.b x sin.c + cos.b x cos.c
The proof of this formula is as follows: –
Let ABC be any spherical triangle on the sphere whose centre is at 0 (fig. 594), at A draw tangents to arcs AB and AC, These tangents lie in the planes of these arcs, respectively, so that the first must meet OB produced at D, and the second must meet OC produced at E. Join DE.
Because AD and AE are tangents) plane angle DAE is equal to the spherical angle BAC, and the angles OAE and OAD are right angles. By the plane cosine formula: –
DE2 = 0D2 + 0E2 – 2 x OD x OE x cos.a
and DE2 = AD2 + AE2 – 2 x AD x AE x cos.A
By subtraction: –
DE2 – DE2 = OD2 + 0E2 – 2 x OD x OE x cos.a – (AD2 + AE2 – 2AD x AE x cos.A)
0 = 0D2 + OE2 – 2 x OD x OE x cos.a – AD2 – AE2 + 2AD x AE x cos.A.
0 = (OD2 – AD2) – (OE2 – AE2) – 20D x OE x cos.a + 2AD x AE cos.A
0 = 2 x OA2 – 20D x OE x cos.a + 2AD x AE cos.A
2AD x AE x cos.A = 2 x OD x OE x cos.a – 2 x OA2
Divide throughout by OD x OE, then: –
cos.A = cos.a – cos.b x cos.c )
sin.b x sin.c )Q.E.D.
and cos.a = cos.A x sin.b x sin.c + cos.b x cos.c )
As they stand, the above formulae are not suitable for logarithmic work because the cosine of an angle between 90° and 180° is negative. To make them suitable for navigational use a function called the Haversine of the angle is used. This function is half the versine (hence the name Haversine) and the versine of an angle is defined as the difference between its cosine and unity, thus: –
Versine of an angle = 1 – (cosine of the angle)
from which it follows that:
Haversine of an angle = ½ (1 – cosine of the angle)
The Haversine of an angle is thus always positive, and it increases from 0 to 1 as the angle increases from 0° to 180°, and decreases from 1 to 0 as the angle increases from 180° to 360°.
Fig 595 shows the versine and Haversine curves in relation to the cosine curve from which they are derived, and these curves should be studied carefully.
All volumes of Nautical Tables give the values of the natural Haversine and log. Haversine for angles between 0° and 180° along the tops of the pages, and for angles between 180° and 360° along the bottoms of the pages, The Haversine is used in navigation because:
(i) between 0° and 180° it is continuously increasing (and therefore a single valued) function, and there is thus no ambiguity in deciding whether an angle is greater or less than 90°, and
(ii). it is always positive and therefore suitable for logarithmic calculation.
The above advantages also apply to the versine, but the Haversine is preferable because its limiting values, 0 and 1 make the construction of logarithmic and other tables comparatively simple, whereas the limiting values of the versine, 0 and 2 do not.
To express the fundamental spherical cosine formula in terms of haversine’s instead of cosines it is only necessary to substitute for the appropriate cosines their values in terms of the Haversine. Thus cos.A can be written (1 – 2 hav. A), and the formula becomes: –
Cos.a = (1 – 2 hav.A) sin.b sin.c + cos.b cos.c
i.e.,cos.a = sin.b sin.c – 2 sin.b sin.c hav.A + cos.b cos.c
= cos (b ~ c) – 2 sin.b. sin.c. hav.A
Similar substitutions for cos. a and cos. (b ~ c) give: –
1 – 2 hav.a = 1 – 2 hav. (b ~ c) – 2 sin.b sin.c hav.A
i.e., hav.a = hav.A. sin.b. sin.c + hav. (b ~ c). (for a side)
or, hav.A = hav.a – hav. (b ~ c)
sin.b sin.c (for an angle)
These are the Spherical Haversine Formulae and, as already shown, they are suitable for logarithmic work since all the quantities involved are positive and less than unity. The term (hav. a. sin. b. sin. c) is, therefore, less than unity and this further simplifies the use of the formula because the antilogarithm of the term can be found immediately from the Haversine tables where natural and logarithmic values are printed side by side. The formula is thus conveniently arranged in two parts, one using logarithmic functions and the other natural sine’s.
230. USE OF COSINEHAVERSINE FORMULA FOR DETERMINING CALCULATED ALTITUDE
In the intercept method for ascertaining a position line described earlier in this study, the observed or True. Alt. is compared with a calculated altitude derived from the zenith distance of the celestial body at a chosen terrestrial position. The difference between the True. Alt. and the calculated altitude (or between the true zenith distance and the calculated zenith distance) is the intercept.
In fig. 596, o is an observer and Zo is his zenith. Point c is the chosen or assumed position near to o, and x is the G.P. of any celestial body X.
In the Astronomical triangle PZCX, the side ZCX may be calculated using the sides PZC and PX and the included angle P. The difference in minutes of arc between ZCX and ZCX is the difference in nautical miles between the radii of the circles of equal altitude at o and c, this being the required intercept.
In the first part of this study, it was shown how the calculated altitude can be obtained by inspection from Sight Reduction Tables, Now we can show how the same calculated altitude can be obtained by use of the spherical cosineHaversine formula.
For any positionline sight, we have to find the calculated altitude to compare with the True. Alt. in order to obtain the intercept. The calculated altitude is simply 90° – the calculated zenith distance, and in fig. 596 the calculated zenith distance is side ZCX of the spherical navigational triangle PZCX or more simply side ZX of any PZX triangle.
In the last section, it was shown that the formula for determining the side of a spherical triangle ABC is: –
hav.a = hav.A sin.c + hav. (b ~ c)
Translating this into terms of the navigational triangle PZX, we get: –
hav.ZX = hav.P x sin.PZ x sin.PX + hav.(PZ ~ PX)
This formula would provide a perfectly satisfactory method of finding the side ZX, but reference to the constituent parts of’ any PZX triangle will show that this formula can be modified into an even simpler form, because: –
PZ = 90° – Lat), therefore sin.PZ = cos. Lat.
PX = 90° ± Dec.), therefore sin.PX = cos. Dec.
hav.(PZ ~ PX) = hav. (90° – Lat.) ~ (90° + Dec)
= hav. (Lat. – Dec) when Lat. and Dec. have different names
= hav. (Lat. + Dec.) when Lat. and Dec. have same names
hav.(PZ ~ PX) = hav. (Lat. ± Dec.)
The formula may therefore be modified to:
hav.ZX = hav. (Lat. ± Dec.) + hav.P x cos.Lat. x cos.Dec. (where P is the L.H.A. of the observed celestial body)
Whether the sight be of the Sun, Moon, star or a planet, the method of working is the same. First, the L.H.A. of the observed body is obtained in the usual way. Then the Lat. and Dec. are added together if they are of opposite names, or the smaller is subtracted from the greater if they are of the same name, to obtain the quantity (Lat. ± Dec.). A convenient mnemonic for remembering whether to add or subtract is the word DAD – if the Lat. and Dec. are of Different names, ADD.
Next, from any volume of Nautical Tables, write down the log. Haversine of the L.H.A. (angle P), the log. cosine of the Lat. and the log. cosine of the Dec., and add these three figures together. The sum is a log. Haversine which should immediately be converted into a natural Haversine (remembering that natural and logarithmic values are printed side by side in all Haversine tables). To this natural Haversine add the natural Haversine of (Lat. ± Dec.) and the result is the natural Haversine of the Calculated Zenith Distance. This Zenith Distance is then extracted from the natural Haversine table and subtracted from 90° to give the Calculated Altitude. An example will make this method of working clear.
(compare the result by this method with the same sight worked with inspection tables – Ex No.14: of § 20).
To determine the true bearing to use with the intercept calculated with the spherical cosineHaversine formula as shown above, the azimuth could be found by solving the PZX triangle to find angle Z, again by spherical trigonometry. The navigator is, however, saved this additional labour by the provision in all volumes of Nautical Tables of a table called the ABC Table from which the azimuth can be determined quickly and easily by inspection.
To use the ABC Table, enter Table A with L.H.A. at the top and Lat. down the side, and extract the value ‘A’, interpolating as necessary. Enter Table B (which will always face the page of Table A just used) with the L.H.A. again at the top but the Dec. down the side, and extract the value ‘B’ name both the A and B values as indicated on each page, and sum them up according to their signs. The result is the value C with which to enter Table C.
Table C is laid out differently in Norie’s Tables from Burton’s Tables. In Nories, enter Table C with the ‘C’ value found, at the top and with Lat. down the side, extracting the azimuth, which should be named as directed at the foot of the ‘C’ Table page. In Burton’s Tables, enter Table ‘C’ with the Lat. down the side and having found the correct horizontal line, follow this along, if necessary from one page to the next, till the ‘C’ value is reached. Travel up this column to the top, where the azimuth is read and named according to the precepts given at the top of the page. Interpolate as necessary.
To find the azimuth for the Ex. given above, using the ABC Table in Burton’s Tables: –
A = + 0.649
B = – 0.502
C = 0.147 and Lat. 45° = Azimuth S 84° W.
= Bearing 264° T.
(This is identical to the true bearing found by Sight Reduction Tables in Ex. No. 14 of § 20)
The ABC Table can, of course, be used to determine the true bearing of a celestial body when working a time azimuth for compass errors and Dev. making this navigational function also independent of Sight Reduction Tables.
Students will already have noted that when using the CosineHaversine method of Sight Reduction there is no need to select an assumed or C.P. – the E.P. or D.R. position of the vessel can be used directly in the calculation. In the example above we deliberately altered the D.R. position to agree with the C.P. used in the working of the example so that it could be shown that the same intercept and azimuth is obtained when using either method. In practice, the actual D.R. position would have been used in the CosineHaversine method, giving a different intercept and azimuth, but the position line would be the same when plotted on the plotting sheet. Plotting for the CosineHaversine method is actually easier than for the inspection Tables method, because several simultaneous sights are all plotted from the same D.R. position which, for convenience, is usually made to be the centre of the compass rose on the plotting sheet.
Students should practice working sights by the CosineHaversine method, using examples and Self Assessment questions in earlier chapters of this study and checking their results by the previous results. The intercepts and azimuths will, of course, be different to those previously calculated, but the final position determined on the plotting sheet should be the same. The final section of this chapter summarises the Basic navigational concepts from both terrestrial and Astronavigation which the student should have firmly fixed in his mind before tackling the final two chapters of this study.
