Tipos de inventario

Rhumb lines were first discussed in the sixteenth century byPedro Nunes as curves of constant azimuth which spiralled from pole to pole. The more academic word loxodrome (Greek loxos: oblique + dromos: running) appeared early in the seventeenth century. At that time both of these terms excluded simple parallel or meridian sailing but modern usage includes these cases. This is unfortunate. We shall permit rhumb lines to include all possible directions but we shall restrict the use of loxodrome to azimuths which are neither paral-lels nor meridians. This is consistent with the definition of the loxodrome in mathematics as a spherical helix: seeWeisstein(2012). The distinction is important for there are two topologically distinct classes of rhumbs: (a) closed parallels; (b) open lines, loxodromes, running from pole to pole with meridians as degenerate cases. The importance of rhumb lines follows from the conformality property: α= β implies that a rhumb line with constant α is projected to a straight line on the Mercator projection.

-75 -60 -30 0 30 60 75

Figure 2.14: Rhumb line on the sphere and the Mercator projection

Figure2.14shows a loxodrome crossing the equator at 30^◦W and maintaining a constant azimuth of 83^◦: it spirals round the sphere covering a finite distance from pole to pole even though it makes an infinite number of turns about the axis. (These statements are proved below). On the projection, it is a repeated straight line of infinite total length as|y| → ∞.

The intercepts with the Greenwich meridian are calculated later using Equation2.43.

Figure 2.15: Infinitesimal elements of a rhumb on the sphere and the projection

Consider the rhumb distance, r12, on the sphere between P(φ1, λ1) and Q(φ2, λ2). If the rhumb is a parallel the distance is the radius of the parallel circle times the change of longitude. On a meridian the distance is simply the radius of the sphere times the change of latitude. For an infinitesimal element of a loxodrome on the sphere, PQ in Figure2.15a we have cos α= R dφ /ds. Since α is a constant this integrates trivially. In summary:

r₁₂= R cos φ (λ₂− λ1), parallel, (2.37)

r₁₂= R(φ₂− φ1), meridian, (2.38)

r₁₂= R sec α (φ2− φ1), loxodrome. (2.39) Therefore, to calculate the distance along a loxodrome, we need know only the constant azimuth and the change of latitude. This is an important result. Note (a) the meridian result can be deduced as the α→ 0 limit of the loxodrome result (although the figure is inappro-priate); (b) the parallel result is not related to the loxodrome result by any limiting process.

The latter is a reflection of the different topological nature of parallels and loxodromes.

The above equations show that the length of a loxodrome is finite. Setting φ1=−π/2 and φ₂ = π/2 in Equation 2.39 we obtain the total length from one pole to another as π R sec α . This reduces to π R on a meridian.

Equations of the loxodrome

To find the equation, on the sphere, of the loxodrome which starts at the point(φ1, λ1) at an azimuth α note that on the projection it is the straight line

y− y1= (x− x1) cot α . (2.40)

where y1= Rψ(φ₁) and x₁= Rλ₁. Using Equations2.33and2.34gives

ψ(φ ) = ψ(φ1) + (λ− λ1) cot α , (2.41)

λ(φ ) = λ1+ tan αh

tanh⁻¹sin φ− tanh⁻¹sin φ1

i

, (2.42)

φ(λ ) = sin⁻¹tanhtanh⁻¹sin φ1+ (λ− λ1) cot α . (2.43) As an example take λ1= φ1=−10^◦ and λ = λ2= φ2= 40^◦. Transforming λ to radian measure, Equation2.41gives

tan α= π 180

50

[ψ(40)− ψ(−10)]= 0.929

and therefore α = 42.9^◦. Note that this is just the shortest rhumb line through the two points. If the rhumb makes one complete revolution before getting to the second point, then replacing ∆λ= 50^◦by ∆λ = 410^◦we find α = 82.5^◦and so on.

Once α has been determined further points on the rhumb are found from Equation2.43.

For example, for the loxodrome with α = 83^◦, φ1 = 0 and λ1 =−30^◦ (Figure 2.14), we calculate the intercepts on the Greenwich meridian (λ= 0, 360, 720, . . .) as

3.7^◦, 43.1^◦, 67.3^◦, 79.4^◦, 85.0^◦, 87.7^◦, 88.9^◦, 89.5^◦, 89.8^◦ . . ..

Equation2.42shows that λ becomes infinite at the poles so that the loxodrome must encircle the pole an infinite number of times as it approaches, even though it is of finite length. This is a geometrical version ofZeno’s paradox.

Mercator sailing

The above results solve the two basic problem of Mercator sailing, by which we mean loxodromic sailing; i.e. the trivial cases of sailing along parallels or meridians are excluded from the discussion. The two problems are

1. Given a starting point P(φ₁, λ₁) and a destination Q(φ₂, λ₂) find the azimuth α of the loxodrome line and the sailing distance, d. (The inverse problem).

2. Given an initial point P(φ₁, λ₁), a loxodrome of azimuth α and a sailing distance d, find the destination Q(φ₂, λ₂). (The direct problem).

We now outline the solution of these problems

For the inverse problem we are given φ₁, λ₁, φ₂, λ₂. First calculate ψ₁, ψ₂ from2.33 (or simply use a table of meridian parts as described below): α then follows from2.41and the distance from2.39(in nautical miles if the latitudes are given in minutes of arc). On a chart the angle is measured from the slope of PQ and and the distance, if moderate, can be approximated without calculation by using dividers to transfer the interval PQ to the latitude scale on the chart, taking care to place the dividers symmetrically with respect to the mid-latitude point so that the latitude scale is approximately uniform for that interval.

The latitude difference in minutes gives the distance in nautical miles.

For the direct problem we are given φ₁, λ₁, r₁₂, α. Equation 2.39gives δ φ ; hence φ₂. Calculate (or use tables for) ψ₁, ψ₂ and then δ λ is found from 2.41; hence λ₂. Again, without calculation, open the dividers to a distance d (given in nautical miles measured as minutes on the latitude scale and transfer the dividers to the line through P with azimuth α, fixing Q; φ₂, λ₂can then be read from the chart.

Meridian parts

At the end of the sixteenth century Edward Wright published Certaine Errors in Naviga-tion(Wright,1599) in which he criticised the Plane Chart, i.e. the equirectangular chart. He stressed that over large regions the plane chart was unreliable in respect of both distance and direction. (See Section2.3). He advocated the use of Mercator’s chart and constructed his own version, stressing that he had no assistance in devising his method. He did exploit Mer-cator’s statement that conformality, not that he used the word, is achieved by compensating the stretching of parallels to the same width as the equator by an equal amount of stretching in the meridian direction. He implemented this by dividing the plane chart he divides it into one minute (of latitude) strips and stretches each by a factor of the sec φ evaluated at the upper edge of the strip; hence giving a slight overestimate. The amount any point moves up is the sum of the increments of all the strips beneath. His results are summarised in a table of cumulative secants at intervals of one second of arc and beginning

sec 1⁰ 1.000000042

sec 1⁰+ sec 2⁰ 2.000000211 sec 1⁰+ sec 2⁰+ sec 3⁰ 3.000000592 sec 1⁰+ sec 2⁰+ sec 3⁰+ sec 4⁰ 4.000001269

In the first edition of the book he listed only rounded values at intervals of 10⁰ but the full table appears in later editions (Monmonier, 2004, Chapter5). We consider only the first table. The cumulative secants, multiplied by a factor of 10, later became called meridional parts and the number of such parts in the interval from zero up to an angle φ is denoted by

MP(φ ) = 10

φ

∑

0

sec φi at intervals of 1⁰. (2.44) N.B. Modern tables andweb calculatorsusually omit this factor of 10.

Some values selected from the published table are shown in the following table: Wright truncated his calculated values for simplicity of use. He also chose the unit of the meridional part (MP) to be one tenth of the length of an equatorial minute of longitude. Therefore if the width of a given chart is W cm. the unit of the meridional part is W/216000 cm. An extra line has been interpolated in the table at 85^◦3⁰ corresponding to the latitude at which Wright’s figure in Certaine Errors was truncated at 108000MP so that the aspect ratio (of the northern hemisphere only) was exactly 2. Note that the first entries, up to 2^◦30⁰, show the increased spacing was undetectable up to that point in the rounded figures of his table.

Lat. MP Lat. MP Lat. MP

10⁰ 100 10^◦ 6030 70^◦ 59667

20⁰ 200 20^◦ 12251 80^◦ 83773 30⁰ 300 30^◦ 18884 85^◦3⁰ 108000

··· ··· 40^◦ 26228 89^◦ 163176

2^◦30⁰ 1500 50^◦ 34746 89^◦50⁰ 226223 2^◦40⁰ 1601 60^◦ 45277 90^◦ ∞

Table 2.3

Wright asserted that the MP values gave the correct spacing of the Mercator parallels.

This is obvious since his construction is just a numerical integration of sec φ replacing Equa-tion2.27. Of course the calculus hadn’t been invented in his day and the ‘log-tan’ formula was derived only one hundred years later, (although Wright’s contemporary and compatriot, Thomas Harriot, seems to have arrived at the formula by his own original method, Lohne (1965)). The numerical integration may be written as

ψ(φ ) = Z _φ

0 sec φ dφ ≈

φ

∑

0

sec φiδ φi δ φ in radians. (2.45) An interval of 1⁰corresponds to δ φ= 0.000291rad = 1/3437.75rad. Therefore

MP(φ ) = 10

φ

∑

0

sec φi= 34377.5 ψ(φ ). (2.46) A table of meridional parts may be used to solve the Mercator sailing problems by calcu-lation. For example, in the inverse problem the direction of the azimuth may be evaluated from Equation2.40:

cot α=∆y

∆x =∆ψ

∆λ =3437.75∆ψ

∆λ⁰ =MP(φ₂)− MP(φ1)

10(λ₂⁰− λ₁⁰) (2.47) and the distance (in nautical miles) from Equation2.39: d = R sec α ∆φ⁰ follows . In the direct problem ∆φ= (d/R) cos α is known and hence we have φ₂. Then λ₂ follows from the last equation.

Wright’s tables of rhumbs

In addition to the table of Meridian parts Wright published tables of rhumb line coordi-nates, meaning by ‘rhumb’ the seven loxodromes of the first quadrant making angles of 11.25^◦, 22.5^◦, 33.75^◦, 45^◦, 56.25^◦, 67.50^◦, 78.75^◦with the equator at the point of cross-ing: these angles are the complements of their azimuths. With these values he was able to plot rhumb lines on globes and also map projections other than mercator. (He mentions stereographic projectionsparticularly.)

Without loss of generality we assume the rhumbs cross the equator at the zero of longi-tude. He observes that if we take a step of 1^◦of longitude (600 MP) along the equator the ordinate of the next point on the first rhumb is 600 tan(11.25^◦) = 117.8097 MP. Therefore, since the rhumb is a straight line on the projection, the ordinates of successive points are simple multiples: 0, 117.81, 235.62, 353.43,··· . He then used the (full) table of meridian parts to invert these MP values to give latitudes at steps of one degree of longitude along the rhumb. Some selected values for the first rhumb are given in the following table:

Long. Lat. Long. Lat. Long. Lat.

0^◦ 0^◦ 60^◦ 11^◦050 360^◦ 58^◦1⁰ 10^◦ 1^◦59⁰ 70^◦ 13^◦47⁰ 2×360^◦ 80^◦36⁰ 20^◦ 3^◦58⁰ 80^◦ 15^◦42⁰ 3×360^◦ 87^◦18⁰ 30^◦ 5^◦57⁰ 90^◦ 17^◦37⁰ 4×360^◦ 89^◦13⁰ 40^◦ 7^◦55⁰ 180^◦ 33^◦40⁰ 5×360^◦ 89^◦46⁰ 50^◦ 9^◦53⁰ 270^◦ 47^◦13⁰ 6×360^◦ 89^◦59⁰

Table 2.4

He gives similar tables for each of the seven rhumbs, all in steps of one degree of longitude from zero up to a value at which the latitude is equal to 89^◦59⁰. For example the table for the fourth rhumb terminates at 540^◦and that for the seventh rhumb at 114^◦.