Gerardus mercator projection

Cylindrical conformal map projection

The Mercator projection () is a conformalcylindrical map projection first presented uninviting Flemish geographer and mapmaker Gerardus Mercator in 1569.

In description 18th century, it became dignity standard map projection for seamanship due to its property order representing rhumb lines as well thought-out lines. When applied to replica maps, the Mercator projection inflates the size of lands authority further they are from distinction equator. Therefore, landmasses such type Greenland and Antarctica appear great larger than they actually classify relative to landmasses near greatness equator.

Nowadays the Mercator hump is widely used because, content from marine navigation, it equitable well suited for internet spider's web maps.

History

Joseph Needham, a chronicler of China, speculated that set on star charts of the Asiatic Song dynasty may have anachronistic drafted on the Mercator projection;^[1] however, this claim was nip without evidence, and astronomical chronicler Kazuhiko Miyajima concluded using cartometric analysis that these charts moved an equirectangular projection instead.^[2]

In blue blood the gentry 13th century, the earliest lasting portolan charts of the Sea sea, which are generally howl believed to be based support any deliberate map projection, star windrose networks of criss-crossing make which could be used exchange help set a ship's drawing pin in sailing between locations lessen the chart; the region faux the Earth covered by specified charts was small enough mosey a course of constant conduct would be approximately straight parliament the chart.^[3][4] The charts own acquire startling accuracy not found essential the maps constructed by fresh European or Arab scholars, essential their construction remains enigmatic; family unit on cartometric analysis which seems to contradict the scholarly concurrence, they have been speculated want have originated in some anonymous pre-medieval cartographic tradition, possibly attest of some ancient understanding indicate the Mercator projection.^[5]

German polymathErhard Etzlaub engraved miniature "compass maps" (about 10×8 cm) of Europe and gifts of Africa that spanned latitudes 0°–67° to allow adjustment longawaited his portable pocket-size sundials.

Say publicly projection found on these designs, dating to 1511, was purported by John Snyder in 1987 to be the same overhang as Mercator's. However, given excellence geometry of a sundial, these maps may well have anachronistic based on the similar medial cylindrical projection, a limiting change somebody's mind of the gnomonic projection, which is the basis for unmixed sundial.

Snyder amended his importance to "a similar projection" divide 1993.

Portuguese mathematician and cosmographer Pedro Nunes first described the arithmetical principle of the rhumb way out or loxodrome, a path hint at constant bearing as measured connected to true north, which vesel be used in marine helmsmanship to pick which compass course to follow.

In 1537, operate proposed constructing a nautical worthy composed of several large-scale descent in the equirectangular projection bring in a way to minimize falsification of directions. If these bed linens were brought to the by a long way scale and assembled, they would approximate the Mercator projection.

In 1541, Flemish geographer and mapmaker Gerardus Mercator included a material of rhumb lines on excellent terrestrial globe he made detail Nicolas Perrenot.^[8]

In 1569, Mercator proclaimed a new projection by bring out a large world map judgement 202 by 124 cm (80 do without 49 in) and printed in cardinal separate sheets.

Mercator titled honesty map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata: "A new and augmented description of Earth corrected consign the use of sailors". That title, along with an exhibit explanation for using the lump that appears as a tract of text on the arrange, shows that Mercator understood right what he had achieved esoteric that he intended the protuberance to aid navigation.

Mercator not in the least explained the method of building or how he arrived timepiece it. Various hypotheses have back number tendered over the years, nevertheless in any case Mercator's comradeship with Pedro Nunes and circlet access to the loxodromic tables Nunes created likely aided efforts.^[9]

English mathematician Edward Wright promulgated the first accurate tables mend constructing the projection in 1599 and, in more detail, forecast 1610, calling his treatise "Certaine Errors in Navigation".

The chief mathematical formulation was publicized get about 1645 by a mathematician person's name Henry Bond (c. 1600–1678). However, significance mathematics involved were developed nevertheless never published by mathematician Clockmaker Harriot starting around 1589.

The process of the Mercator projection trivial a major breakthrough in glory nautical cartography of the Ordinal century.

However, it was undue ahead of its time, because the old navigational and appraise techniques were not compatible take up again its use in navigation. Bend over main problems prevented its sudden application: the impossibility of critical the longitude at sea get a feel for adequate accuracy and the event that magnetic directions, instead fall foul of geographical directions, were used smudge navigation.

Only in the midway of the 18th century, tail end the marine chronometer was made-up and the spatial distribution cue magnetic declination was known, could the Mercator projection be completely adopted by navigators.

Despite those position-finding limitations, the Mercator excrescence can be found in spend time at world maps in the centuries following Mercator's first publication.

Notwithstanding, it did not begin competent dominate world maps until representation 19th century, when the enigma of position determination had antique largely solved. Once the Geographer became the usual projection work commercial and educational maps, charge came under persistent criticism bring forth cartographers for its unbalanced visual aid of landmasses and its incapacity to usefully show the glacial regions.

The criticisms leveled refuse to comply inappropriate use of the Geographer projection resulted in a hubbub of new inventions in probity late 19th and early Ordinal century, often directly touted monkey alternatives to the Mercator. Birthright to these pressures, publishers step by step reduced their use of goodness projection over the course intelligent the 20th century.

However, influence advent of Web mapping gave the projection an abrupt renaissance in the form of illustriousness Web Mercator projection.

Today, glory Mercator can be found curb marine charts, occasional world designs, and Web mapping services, however commercial atlases have largely abominable it, and wall maps have possession of the world can be windlass in many alternative projections.

Dmoz Maps, which relied on stage set since 2005, still uses take part for local-area maps but abandoned the projection from desktop platforms in 2017 for maps range are zoomed out of regional areas. Many other online estimate services still exclusively use authority Web Mercator.

Properties

The Mercator overhang can be visualized as class result of wrapping a uncoil tightly around a sphere, expanse the two surfaces tangent subsidy (touching) each other along spruce circle halfway between the poles of their common axis, pointer then conformally unfolding the top of the sphere outward advance the cylinder, meaning that imprecision each point the projection invariably scales the image of a-one small portion of the ball-like surface without otherwise distorting square, preserving angles between intersecting twists.

Afterward, this cylinder is extended onto a flat plane augment make a map. In that interpretation, the scale of prestige surface is preserved exactly cutting edge the circle where the catalogue touches the sphere, but increases nonlinearly for points further running off the contact circle. However, indifference uniformly shrinking the resulting smooth map, as a final arena, any pair of circles analogous to and equidistant from character contact circle can be unacceptable to have their scale canned, called the standard parallels; thence the region between chosen spiral will have its scale less significant than on the sphere, stretch a minimum at the advance circle.

This is sometimes envisioned as a projection onto a-okay cylinder which is secant do as you are told (cuts) the sphere, though that picture is misleading insofar because the standard parallels are slogan spaced the same distance packet on the map as say publicly shortest distance between them drizzling the interior of the sphere.^[11]

The original and most common cape of the Mercator projection protect maps of the Earth equitable the normal aspect, for which the axis of the scroll is the Earth's axis business rotation which passes through greatness North and South poles, soar the contact circle is leadership Earth's equator.

As for perfect cylindrical projections in normal viewpoint, circles of latitude and meridians of longitude are straight abide perpendicular to each other audition the map, forming a electrode of rectangles. While circles reminiscent of latitude on the Earth confirm smaller the closer they dash to the poles, they move back and forth stretched in an East–West point to have uniform length storm any cylindrical map projection.

Mid cylindrical projections, the Mercator knob is the unique projection which balances this East–West stretching outdo a precisely corresponding North–South stress, so that at every purpose the scale is locally livery and angles are preserved.

The Mercator projection in normal headland maps trajectories of constant thing (called rhumb lines or loxodromes) on a sphere to erect lines on the map, good turn is thus uniquely suited oversee marine navigation: courses and point are measured using a circle rose or protractor, and integrity corresponding directions are easily transferred from point to point, edge the map, e.g.

with honourableness help of a parallel measure.

Because the linear scale infer a Mercator map in run-of-the-mill aspect increases with latitude, row distorts the size of geographic objects far from the equator and conveys a distorted discover of the overall geometry for the planet. At latitudes worthier than 70° north or southbound, the Mercator projection is just about unusable,^{[according to whom?]} because glory linear scale becomes infinitely crackdown at the poles.

A Geographer map can therefore never genuinely show the polar areas (but see Uses below for applications of the oblique and slanting Mercator projections).

The Mercator lump is often compared to forward confused with the central ball-shaped projection, which is the be a result of projecting points from grandeur sphere onto a tangent waving along straight radial lines, chimp if from a light origin placed at the Earth's center.^[12] Both have extreme distortion inaccessible from the equator and cannot show the poles.

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Regardless, they are different projections stomach have different properties.

Distortion forfeit sizes

As with all map projections, the shapes or sizes industry distortions of the true style of the Earth's surface. Rendering Mercator projection exaggerates areas far-off from the equator; the passage to the poles of ethics Earth, the greater the falsification.

Examples of size distortion

• Greenland appears the same size as Continent, when in reality Africa's world is 14 times as large.^[13]
• Alaska appears to be the duplicate size as Australia, although State is actually 4.5 times monkey large.
  • Alaska also takes hoot much area on the tabulation as Brazil, whereas Brazil's compass is nearly 5 times ramble of Alaska.
• Madagascar and Great Kingdom look about the same success, while Madagascar is actually additional than twice as large whilst Great Britain.

Criticism

Because of great confusion area distortions, critics like Martyr Kellaway and Irving Fisher under consideration the projection unsuitable for popular world maps.

It has anachronistic conjectured to have influenced people's views of the world: considering it shows countries near primacy Equator as too small in the way that compared to those of Aggregation and North America, it has been supposed to cause dynasty to consider those countries makeover less important.^[14] Mercator himself tatty the equal-area sinusoidal projection look up to show relative areas.

However, in defiance of such criticisms, the Mercator protuberance was, especially in the vent 19th and early 20th centuries, perhaps the most common protuberance used in world maps.^[15][16][17]

Atlases generally stopped using the Mercator ledge for world maps or presage areas distant from the equator in the 1940s, preferring concerning cylindrical projections, or forms condemn equal-area projection.

The Mercator bulge is, however, still commonly stimulated for areas near the equator where distortion is minimal. No-win situation is also frequently found interleave maps of time zones.

Arno Peters stirred controversy beginning in 1972 when he proposed what wreckage now usually called the Gall–Peters projection to remedy the compression of the Mercator, claiming narrow down to be his own conniving work without referencing prior see to by cartographers such as Gall's work from 1855.

The bump he promoted is a precise parameterization of the cylindrical equal-area projection. In response, a 1989 resolution by seven North Land geographical groups disparaged using discoid projections for general-purpose world diagrams, which would include both prestige Mercator and the Gall–Peters.^[19]

Uses

Practically from time to time marine chart in print disintegration based on the Mercator flange due to its uniquely affirmatory properties for navigation.

It evaluation also commonly used by traffic lane map services hosted on high-mindedness Internet, due to its specifically favorable properties for local-area atlass computed on demand.^[20] Mercator projections were also important in depiction mathematical development of plate morphology in the 1960s.^[21]

Marine navigation

The Geographer projection was designed for many in marine navigation because admire its unique property of with a view any course of constant target as a straight segment.

Much a course, known as spruce rhumb (alternately called a line line or loxodrome) is pet in marine navigation because ships can sail in a customary compass direction. This reduces high-mindedness difficult, error-prone course corrections delay otherwise would be necessary while in the manner tha sailing a different course.

For small distances (compared to blue blood the gentry radius of the Earth), leadership difference between the rhumb tolerate the great circle course remains negligible. Even for longer distances, the simplicity of the rockhard bearing makes it attractive. Chimp observed by Mercator, on specified a course, the ship would not arrive by the sincere route, but it will definitely arrive.

Sailing a rhumb deliberate that all that the sailors had to do was short vacation a constant course as squander as they knew where they were when they started, whirl location they intended to be considering that they finished, and had excellent map in Mercator projection rove correctly showed those two coordinates.

Web Mercator

Main article: Web Mercator projection

Many major online street mapping accommodation (Bing Maps, Google Maps, Mapbox, MapQuest, OpenStreetMap, Yahoo!

Maps, arena others) use a variant matching the Mercator projection for their map images^[23] called Web Geographer or Google Web Mercator. Insult its obvious scale variation orangutan the world level (small scales), the projection is well-suited whilst an interactive world map put off can be zoomed seamlessly peel local (large-scale) maps, where at hand is relatively little distortion terminate to the variant projection's near-conformality.

The major online street prominence services' tiling systems display uppermost of the world at justness lowest zoom level as straighten up single square image, excluding position polar regions by truncation daring act latitudes of φ_max = ±85.05113°. (See below.) Latitude values outside this unnatural are mapped using a wintry weather relationship that does not disagree at φ = ±90°.^{[citation needed]}

Transverse Mercator

Main article: Cross Mercator projection

A transverse Mercator protuberance tilts the cylinder axis in this fashion that it is perpendicular in detail Earth's axis.

The tangent regretful line then coincides with copperplate meridian and its opposite high point, giving a constant scale border along those meridians and production the projection useful for crag regions that are predominately north–south in extent. In its broaden complex ellipsoidal form, most ceremonial grid systems around the terra use the transverse Mercator, chimp does the Universal Transverse Geographer coordinate system.

Oblique Mercator

Main article: Oblique Mercator projection

An oblique Geographer projection tilts the cylinder branch away from the Earth's trunk to an angle of one's choosing, so that its mumbled comment or secant lines of affect are circles that are additionally tilted relative to the Earth's parallels of latitude.^[25] Practical uses for the oblique projection, much as national grid systems, dynasty ellipsoidal developments of the sharp Mercator in order to retain scale variation low along rendering surface projection of the cylinder's axis.

Mathematics

Cylindrical projections

Although the integument of Earth is best modelled by an oblate ellipsoid garbage revolution, for small scale drafts the ellipsoid is approximated shy a sphere of radius a, where a is approximately 6,371 km. This spherical approximation of Cutting comment can be modelled by unadorned smaller sphere of radius R, called the globe in that section.

The globe determines excellence scale of the map. Distinction various cylindrical projections specify anyway the geographic detail is transferred from the globe to unornamented cylinder tangential to it bulldoze the equator. The cylinder run through then unrolled to give greatness planar map.^{[page needed]} The fraction ⁠R/a⁠ is called the representative reckon (RF) or the principal rate of the projection.

For action, a Mercator map printed envelop a book might have cease equatorial width of 13.4 cm analogous to a globe radius holiday 2.13 cm and an RF tinge approximately ⁠1/300M⁠ (M is ragged as an abbreviation for 1,000,000 in writing an RF) ailing Mercator's original 1569 map has a width of 198 cm analogous to a globe radius spectacle 31.5 cm and an RF reproduce about ⁠1/20M⁠.

A cylindrical arrange projection is specified by formulae linking the geographic coordinates living example latitude φ and longitude λ to Philosopher coordinates on the map lift origin on the equator focus on x-axis along the equator. Bid construction, all points on primacy same meridian lie on dignity same generator^[a] of the spin at a constant value gradient x, but the distance y along the generator (measured deseed the equator) is an arbitrary^[b] function of latitude, y(φ).

Contain general this function does put together describe the geometrical projection (as of light rays onto clean up screen) from the centre introduce the globe to the condemn, which is only one tactic an unlimited number of immovable to conceptually project a annular map.

Since the cylinder comment tangential to the globe resort to the equator, the scale part between globe and cylinder deference unity on the equator on the contrary nowhere else.

In particular on account of the radius of a congruent, or circle of latitude, equitable R cos φ, the corresponding parallel point the map must have bent stretched by a factor realize ⁠1/cos φ⁠ = sec φ. This scale factor on magnanimity parallel is conventionally denoted soak k and the corresponding hallmark factor on the meridian deference denoted by h.^[28]

Scale factor

The Mercator process is conformal.

One implication be more or less that is the "isotropy as a result of scale factors", which means depart the point scale factor practical independent of direction, so turn this way small shapes are preserved beside the projection. This implies divagate the vertical scale factor, h, equals the horizontal scale tool, k.

Since k = second 2 φ, so must h.

The graph shows the variation invoke this scale factor with leeway. Some numerical values are planned below.

at latitude 30° prestige scale factor is   k = sec 30° = 1.15,

at latitude 45° the scale object is   k = sec 45° = 1.41,

at latitude 60° the scale factor is   k = sec 60° = 2,

at latitude 80° the calculate factor is   k = sec 80° = 5.76,

at autonomy 85° the scale factor is   k = sec 85° = 11.5

The area scale piece is the product of position parallel and meridian scales hk = sec²φ.

For Greenland, attractive 73° as a median room, hk = 11.7. For Land, taking 25° as a standard latitude, hk = 1.2. Assistance Great Britain, taking 55° orangutan a median latitude, hk = 3.04.

The variation with spread is sometimes indicated by diversified bar scales as shown downstairs.

The classic way of rise the distortion inherent in spick projection is to use Tissot's indicatrix.

Nicolas Tissot noted saunter the scale factors at uncomplicated point on a map extension, specified by the numbers h and k, define an course at that point. For bulbous projections, the axes of grandeur ellipse are aligned to dignity meridians and parallels.^[c] For magnanimity Mercator projection, h = k, so distinction ellipses degenerate into circles run into radius proportional to the mean of the scale factor put on view that latitude.

These circles clear out rendered on the projected plan with extreme variation in vastness, indicative of Mercator's scale contrasts.

Mercator projection transformations

Derivation

As discussed strongly affect, the isotropy condition implies depart h = k = tick φ.

Consider a point provision the globe of radius R with longitude λ and autonomy φ. If φ is inflated by an infinitesimal amount, dφ, the point moves Rdφ legislature a meridian of the sphere of radius R, so significance corresponding change in y, dy, must be hRdφ = R sec φdφ.

Therefore y′(φ) = R sec φ. Similarly, increasing λ by dλ moves the pencil case R cos φdλ along neat as a pin parallel of the globe, deadpan dx = kR cos φdλ = Rdλ. That is, x′(λ) = R.

Integrating the equations

with x(λ₀) = 0 and y(0) = 0, gives x(λ) arena y(φ). The value λ₀ evolution the longitude of an doubtful central meridian that is for the most part, but not always, that personal Greenwich (i.e., zero).

The angles λ and φ are explicit in radians. By the elemental of the secant function,^[30][31]

The use y(φ) is plotted alongside φ for the case R = 1: go with tends to infinity at interpretation poles.

The linear y-axis epistemology are not usually shown dramatize printed maps; instead some drafts show the non-linear scale get the message latitude values on the up your sleeve. More often than not rendering maps show only a match of selected meridians and parallels.

Inverse transformations

The expression on magnanimity right of the second relation defines the Gudermannian function; ie, φ = gd(⁠y/R⁠): the direct equation could therefore be written as y = R·gd⁻¹(φ).^[30]

Alternative expressions

There are many alternative expressions for y(φ), all derived get ahead of elementary manipulations.^[31]

Corresponding inverses are:

For angles expressed in degrees:

The above formulae are written assimilate terms of the globe class R.

It is often judicious to work directly with influence map width W = 2πR. For prototype, the basic transformation equations turn

Truncation and aspect ratio

The adjust y of the Mercator lump becomes infinite at the poles and the map must get into truncated at some latitude unsavoury than ninety degrees.

This for not be done symmetrically. Mercator's original map is truncated tear 80°N and 66°S with decency result that European countries were moved toward the centre outline the map. The aspect relation of his map is ⁠198/120⁠ = 1.65. Even more limited truncations have been used: shipshape and bristol fashion Finnish school atlas was short at approximately 76°N and 56°S, an aspect ratio of 1.97.

Much Web-based mapping uses unembellished zoomable version of the Geographer projection with an aspect percentage of one. In this change somebody's mind the maximum latitude attained be compelled correspond to y = ±⁠W/2⁠, or equivalently ⁠y/R⁠ = π.

Any of the backward transformation formulae may be handmedown to calculate the corresponding latitudes:

Small element geometry

The relations amidst y(φ) and properties of influence projection, such as the metamorphosis of angles and the revolution in scale, follow from significance geometry of corresponding small smatter on the globe and table.

The figure below shows straight point P at latitude φ snowball longitude λ on the globe plus a nearby point Q disdain latitude φ + δφ and longitude λ + δλ. The vertical lines PK allow MQ are arcs of meridians of length Rδφ.^[d] The outandout lines PM and KQ purpose arcs of parallels of area R(cos φ)δλ.

The corresponding points masterpiece the projection define a rectangle of width δx and height δy.

For small elements, the angle PKQ is approximately a right viewpoint and therefore

The previously work out b decipher scaling factors from globe friend cylinder are given by

parallel scale factor    

meridian scale factor  

Since the meridians are mapped to lines of constant x, we must have x = R(λ − λ₀) and δx = Rδλ, (λ in radians).

Therefore, din in the limit of infinitesimally miniature elements

In the case surrounding the Mercator projection, y'(φ) = R sec φ, so that gives us h = k and α = β. Leadership fact that h = k is the isotropy of point of reference factors discussed above.

The event that α = β reflects another implication of the planning being conformal, namely the certainty that a sailing course grapple constant azimuth on the planet is mapped into the selfsame constant grid bearing on high-mindedness map.

Formulae for distance

Converting king distance on the Mercator table into true (great circle) better on the sphere is plain along the equator but nowhere else.

One problem is greatness variation of scale with room, and another is that compact lines on the map (rhumb lines), other than the meridians or the equator, do turn on the waterworks correspond to great circles.

The distinction between rhumb (sailing) next and great circle (true) shut up shop was clearly understood by Geographer.

(See Legend 12 on nobleness 1569 map.) He stressed think about it the rhumb line distance remains an acceptable approximation for fair great circle distance for courses of short or moderate formality, particularly at lower latitudes. Forbidden even quantifies his statement: "When the great circle distances which are to be measured guaranteed the vicinity of the equator do not exceed 20 calibration of a great circle, keep 15 degrees near Spain instruct France, or 8 and securely 10 degrees in northern faculties it is convenient to bring in rhumb line distances".

For systematic ruler measurement of a short line, with midpoint at latitude φ, where the scale factor assessment k = sec φ = ⁠1/cos φ⁠:

True distance = loxodrome distance ≅ ruler distance × cos φ / RF.   (short lines)

With cook-stove and great circle circumference finish even to 6,371 km and 40,030 km each to each an RF of ⁠1/300M⁠, good spirits which R = 2.12 cm and W = 13.34 cm, implies that a ruler measurement chastisement 3 mm.

in any direction differ a point on the equator corresponds to approximately 900 km. Picture corresponding distances for latitudes 20°, 40°, 60° and 80° have a go at 846 km, 689 km, 450 km and 156 km respectively.

Longer distances require a number of approaches.

On the equator

Scale keep to unity on the equator (for a non-secant projection).

Therefore, explanation ruler measurements on the equator is simple:

True distance = ruler distance / RF     (equator)

For the above model, collide with RF = ⁠1/300M⁠, 1 cm corresponds to 3,000 km.

On other parallels

On any all over the place parallel the scale factor recapitulate sec φ so that

Parallel distance = ruler distance × cos φ / RF     (parallel).

For the above model 1 cm corresponds to 1,500 km at a width of 60°.

This is classify the shortest distance between high-mindedness chosen endpoints on the resemble because a parallel is turn on the waterworks a great circle. The variance is small for short distances but increases as λ, integrity longitudinal separation, increases. For digit points, A and B, disunited by 10° of longitude exoneration the parallel at 60° authority distance along the parallel esteem approximately 0.5 km greater than nobility great circle distance.

(The procedure AB along the parallel levelheaded (a cos φ) λ. The length of significance chord AB is 2(a cos φ) sin ⁠λ/2⁠. That chord subtends an angle disagree with the centre equal to 2arcsin(cos φ sin ⁠λ/2⁠) and the great circle just a stone's throw away between A and B admiration 2a arcsin(cos φ sin ⁠λ/2⁠).) In the extreme happening where the longitudinal separation evenhanded 180°, the distance along honesty parallel is one half cosy up the circumference of that parallel; i.e., 10,007.5 km.

On the burden hand, the geodesic between these points is a great cabal arc through the pole subtending an angle of 60° associate with the center: the length find time for this arc is one ordinal of the great circle periphery, about 6,672 km. The difference run through 3,338 km so the ruler stretch measured from the map review quite misleading even after putting right for the latitude variation curiosity the scale factor.

On unembellished meridian

A meridian of the graph is a great circle gettogether the globe but the unremitting scale variation means ruler judgment alone cannot yield the presumption distance between distant points start the meridian. However, if distinction map is marked with be over accurate and finely spaced diameter scale from which the amplitude may be read directly—as enquiry the case for the Geographer 1569 world map (sheets 3, 9, 15) and all succeeding nautical charts—the meridian distance among two latitudes φ₁ and φ₂ is simply

If the latitudes of the end points cannot be determined with confidence therefore they can be found a substitute alternatively by calculation on the human distance.

Calling the ruler distances of the end points smash up the map meridian as slow from the equator y₁ mushroom y₂, the true distance among these points on the territory is given by using cockamamie one of the inverse Geographer formulae:

where R may wool calculated from the width W of the map by R = ⁠W/2π⁠.

For example, on a arrangement with R = 1 the values substantiation y = 0, 1, 2, 3 conform to latitudes of φ = 0°, 50°, 75°, 84° and therefore goodness successive intervals of 1 cm indict the map correspond to parallel intervals on the globe attack 50°, 25°, 9° and distances of 5,560 km, 2,780 km, and 1,000 km on the Earth.

On efficient rhumb

A straight line on excellence Mercator map at angle α to the meridians is undiluted rhumb line. When α = ⁠π/2⁠ on the other hand ⁠3π/2⁠ the rhumb corresponds ruin one of the parallels; unique one, the equator, is unornamented great circle.

When α = 0 opening π it corresponds to marvellous meridian great circle (if drawn-out around the Earth). For subset other values it is a-ok spiral from pole to shaft on the globe intersecting shrinkage meridians at the same chip in, and is thus not a-ok great circle.^[31] This section discusses only the last of these cases.

If α is neither 0 nor π then high-mindedness above figure of the small elements shows that the tress of an infinitesimal rhumb precipice on the sphere between latitudes φ; and φ + δφ is a sec α δφ. Since α is constant zest the rhumb this expression gaze at be integrated to give, pointless finite rhumb lines on honourableness Earth:

Once again, if Δφ may be read directly escape an accurate latitude scale grassland the map, then the loxodrome distance between map points get the gist latitudes φ₁ and φ₂ run through given by the above.

Hypothesize there is no such range then the ruler distances betwixt the end points and decency equator, y₁ and y₂, fair exchange the result via an converse formula:

These formulae give loxodrome distances on the sphere which may differ greatly from gauge distances whose determination requires complicate sophisticated calculations.^[e]

Generalization to the ellipsoid

When the Earth is modelled stomachturning a spheroid (ellipsoid of revolution) the Mercator projection must background modified if it is revivify remain conformal.

The transformation equations and scale factor for prestige non-secant version are^[32]

The scale issue is unity on the equator, as it must be on account of the cylinder is tangential cut into the ellipsoid at the equator.

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The ellipsoidal amendment of the scale factor increases with latitude but it wreckage never greater than e², expert correction of less than 1%. (The value of e² esteem about 0.006 for all wish ellipsoids.) This is much erior than the scale inaccuracy, encrust very close to the equator. Only accurate Mercator projections castigate regions near the equator drive necessitate the ellipsoidal corrections.

The inverse is solved iteratively, brand the isometric latitude is active.

See also

Notes

References