Practical cartography. Direct conformal cylindrical projection of the mercator Why is the projection of the mercator most often used?

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Conformal cylindrical Mercator projection is the main and one of the first cartographic projections. One of the first, so is the second to use. Before its appearance, they used the equidistant projection or geographical projection of Marnius of Tire, first proposed in 100 BC (2117 years ago). This projection was neither equal nor conformal. Relatively accurate on this projection, the coordinates of the places closest to the equator were obtained.

Developed by Gerardus Mercator in 1569 for the compilation of maps that were published in his " Atlas». Projection name " conformal”Means that the projection maintains the angles between directions, known as constant headings or bearing angles. All curves on the Earth's surface in the conformal cylindrical Mercator projection are depicted by straight lines.

"... The UTM map projection was developed between 1942 - 1943 in the German Wehrmacht. Its development and appearance was probably carried out in the Abteilung für Luftbildwesen (Aerial Photography Department) of Germany ... since 1947 the US Army has used a very similar system, but with a standard scale factor of 0.9996 on the central meridian, as opposed to German 1.0.

A bit of theory (and history) about the conformal cylindrical Mercator projection

In the Mercator projection, the meridians are parallel, equidistant lines. Parallels are parallel lines, the distance between which near the equator is equal to the distance between the meridians with increasing when approaching the poles. Thus, the scale of distortion to the poles becomes infinite, for this reason the South and North Poles are not depicted on the Mercator projection. Maps in the Mercator projection are limited to areas of 80 ° - 85 ° north and south latitude.

"The Universal Conformal Transverse Mercator (UTM) uses a 2D Cartesian coordinate system ... that is, it is used to determine a location on Earth, regardless of the elevation of the place ...

All lines of constant courses (or points) on Mercator charts are represented by straight segments. Two properties, conformal and straight bearing lines, make this projection uniquely suited for marine navigation applications: courses and heading are measured using a wind rose or protractor, and the corresponding directions are easily transferred from point to point on the map using a parallel ruler or a pair of navigational transports. for drawing lines.

The title and explanation given by Mercator on his world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata: “ New, expanded and revised description of the Land for use by sailors”Indicates that it has been specially conceived for maritime use.

Transverse Mercator projection.

Although the method for constructing the projection is not explained by the author, Mercator probably used the graphical method, transferring some of the lines of points previously drawn on the globe to a rectangular grid of coordinates (a grid formed by lines of latitude and longitude), and then adjusted the distance between the parallels so that these lines became straight, which created the same angle with the meridian, as on a globe.

The development of the Cartographic Conformal Mercator projection represented a major breakthrough in marine cartography in the 16th century. However, its appearance was far ahead of its time, since the old navigation and geodetic methods were not compatible with its use in navigation.

Two main problems prevented its immediate application: the inability to determine longitude at sea with sufficient accuracy, and the fact that magnetic rather than geographic directions were used in marine navigation. Only almost 150 years later, in the middle of the 18th century, after the marine chronometer was invented and the spatial distribution of magnetic declination became known, the cartographic Mercator conformal projection was fully adopted in maritime navigation.

The Gauss-Kruger cartographic conformal projection is synonymous with the transverse Mercator projection, but in the Gauss-Kruger projection, the cylinder does not unfold around the equator (as in the Mercator projection), but around one of the meridians. The result is a conformal projection that does not maintain correct directions.

The central meridian is located in the region that can be selected. Along the central meridian, distortions of all properties of objects in the region are minimal. This projection is most suitable for mapping areas stretching from north to south. The Gauss-Kruger coordinate system is based on the Gauss-Kruger projection.

The Gauss-Kruger map projection is completely similar to the universal transverse Mercator projection, the width of the zones in the Mercator projection is 6 °, while in the Gauss-Kruger projection the width of the zones is 3 °. The Mercator projection is convenient for sailors, the Gauss-Kruger projection for ground forces in the limited territories of Europe and South America. In addition, the Mercator projection 2-dimensional accuracy of determining latitude and longitude on the map does not depend on the altitude of the place, while the Gauss-Kruger projection is 3-dimensional, and the accuracy of determining the latitude and longitude is constantly dependent on the altitude of the place.

Until the end of World War II, this cartographic problem was especially acute, since it complicated the issues of interaction between the fleet and ground forces in the conduct of joint actions.

Equatorial Mercator projection.

Is it possible to combine these two systems into one? It is possible that it was produced in Germany in the period from 1943 to 1944.

The Universal Conformal Transverse Mercator (UTM) uses a 2D Cartesian coordinate system to provide a location on the Earth's surface. Like the traditional latitude and longitude method, it represents a horizontal position, that is, it is used to determine a location on Earth, regardless of the altitude of the location.

The history of the emergence and development of the UTM map projection

However, it differs from this method in several respects. The UTM system is not just a map projection. The UTM system divides the Earth into sixty zones, each with six degrees of longitude, and uses an intersecting transverse Mercator projection in each zone.

Most of the published US publications do not point to the original source of the UTM system. The NOAA website claims the system was developed by the US Army Corps of Engineers, and published material that does not claim origin appears to be based on this estimate.

“Scale distortion increases in each UTM zone as the boundaries between UTM zones get closer. However, it is often convenient or necessary to measure a number of locations on the same grid when some of them are located in two adjacent zones ...

However, a series of aerial photographs found in the Bundesarchiv-Militärarchiv (military unit of the German Federal Archives) appear to be from 1943 to 1944 labeled UTMREF in logical coordinate letters and numbers, as well as displayed in accordance with the transverse Mercator projection. This find perfectly indicates that the UTM map projection was developed between 1942 and 1943 in the German Wehrmacht. Its development and appearance was probably carried out in the Abteilung für Luftbildwesen (Aerial Photography Department) in Germany. Subsequently, from 1947, the US Army used a very similar system, but with a standard scale factor of 0.9996 on the central meridian, in contrast to the German 1.0.

For areas within the United States, the 1866 Clarke ellipsoid was used. For the rest of the Earth, including Hawaii, the International Ellipsoid was used. The WGS84 ellipsoid is now commonly used to model the Earth in the UTM coordinate system, meaning that the current UTM ordinate at a given point can be up to 200 meters away from the old system. For different geographic regions, for example: ED50, NAD83, other coordinate systems can be used.

Prior to the development of the universal transverse Mercator projection coordinate system, several European countries demonstrated the usefulness of conformal mappings (preserving local angles) cartography for their territories during the interwar period.

Calculating the distances between two points on these maps could be done easily in the field (using the Pythagorean theorem), compared to the possible use of trigonometric formulas required by a grid based on a latitude and longitude system. In the postwar years, these concepts were expanded into the Universal Transverse Mercator / Universal Polar Stereographic Coordinate System (UTM / UPS), which is a global (or universal) coordinate system.

The Transverse Mercator projection is a variant of the Mercator projection that was originally developed by the Flemish geographer and cartographer Gerardus Mercator in 1570. This projection is conformal, meaning that angles are preserved and therefore allows small regions to be formed. However, it distorts distance and area.

The UTM system divides the Earth between latitude 80 ° south and latitude 84 ° north into 60 zones, each zone 6 ° longitude wide. Zone 1 covers longitudes from 180 ° to 174 ° W (west longitude); the numbering zone increases eastward towards zone 60, which spans longitudes from 174 ° to 180 ° E (east longitude).

Each of the 60 zones uses a transverse Mercator projection that can map a more north-south area with low distortion. By using narrow zones of 6 ° longitude (up to 800 km) wide, and decreasing the scale factor along the central meridian of 0.9996 (reduction 1: 2500), the amount of distortion is kept below 1 part 1000 within each zone. Scale distortion increases to 1.0010 at the boundaries of the zone along the equator.

In each zone, the scale factor of the central meridian reduces the diameter of the transverse cylinder to produce an intersecting projection with two standard or true scale lines, about 180 km on each side, and roughly parallel to the central meridian (Arc cos 0.9996 = 1.62 ° at the equator) ... The scale is less than 1 inside the standard lines and greater than 1 outside, but the overall distortion is kept to a minimum.

Scale distortion increases in each UTM zone as the boundaries between UTM zones approach. However, it is often convenient or necessary to measure a number of locations on the same grid when some of them are located in two adjacent zones.

Around the boundaries of large-scale maps (1: 100,000 or more), the coordinates for both adjacent UTM zones are usually printed within a minimum distance of 40 km on either side of the zone boundary. Ideally, the coordinates of each position should be measured on a graticule for the zone in which they are located, and the scale factor of the still relatively small near-field boundaries can be overlapped by measurements to the adjacent zone for some distance, when necessary.

Latitude stripes are not part of the UTM system, but rather part of a military reference system (MGRS). They are, however, sometimes used.

Ellipsoidal Mercator projection.

Each zone is segmented into 20 latitudinal bands. Each latitudinal strip is 8 degrees high, and begins with lettering with " C"At 80 ° S (south latitude), increasing in the English alphabet to the letter" X", Skipping the letters" I" and " O"(Because of their similarity to the numbers one and zero). The last latitude of the range, " X”Is extended by an additional 4 degrees so that it ends at 84 ° N, thus encompassing the northernmost part on Earth.

Conclusion on the map projection (UTM / UPS) of Mercator

Strip width " A" and " B"Do exist, like stripes" Y" and " Z". They cover the western and eastern sides of the Antarctic and Arctic regions, respectively. It is convenient to remember mnemonically that any letter before " N"In alphabetical order - the zone is in the southern hemisphere, and any letter after the letter" N»- when the zone is in the northern hemisphere.

Combination of zone and latitudinal strip - defines the zone of the coordinate grid. The zone is always recorded first, followed by the latitude strip. For example, the situation in Toronto, Canada will be in zone 17 and latitudinal zone “ T", Thus the full reference is the grid area" 17T". Grid zones are used to define the boundaries of irregular UTM zones. They are also an integral part of the military reference grid. The method is also used to simply add N or S after the zone number to indicate the northern or southern hemisphere (to the horizontal coordinates, along with the zone number, everything is necessary to determine the position, with the exception of which hemisphere).

In hiking and cycling trips, a topographic map is an indispensable companion for a researcher. One of the tasks cartography(one of the disciplines of such a science as geodesy) is an image of the curved surface of the Earth (figure of the Earth) on a flat map. To solve this problem, you must choose ellipsoid- the shape of a three-dimensional body, approximately corresponding to the earth's surface, datum- the starting point of the coordinate system (center of the ellipsoid) and the initial meridian (eng. prime meridian) and projection- a way of depicting the surface of this body on a plane.

Ellipsoids and datums

At different times, various options for representing the Earth's surface in the form of a sphere or an ellipsoid were used to build maps. .

Representing the Earth as a sphere with a radius of 6378137 meters (or 6367600 meters) allows you to determine the coordinates of any point on the earth's surface in the form of two numbers - latitude $ \ phi $ and longitude $ \ lambda $:

For earth ellipsoid as the (geographic) latitude, the concept is used geodetic latitude(eng. geodetic latitude) φ is the angle formed by the normal to the surface of the earth's ellipsoid at a given point and by the plane of its equator , moreover normal does not pass through the center of the ellipsoid excluding the equator and poles:

Longitude value (eng. longitude) λ depends on the choice of the initial (zero) meridian for the ellipsoid.
The radius of the major (equatorial) semiaxis is usually used as the parameters of the ellipsoid a and compression f .
The compression $ f = ((a-b) \ over a) $ determines the flattening of the ellipsoid at the poles.

One of the first ellipsoids was Bessel ellipsoid(Bessel ellipsoid, Bessel 1841), determined from measurements in 1841 by Friedrich Bessel ( Friedrich Wilhelm Bessel), with the length of the major semiaxis a= 6377397.155 m and compression f = 1:299,152815 ... It is currently used in Germany, Austria, Czech Republic and some Asian and European countries.

datum Potsdam (PD)

Previously for building maps in projection UTM used international ellipsoid (International ellipsoid 1924, Hayford ellipsoid) with the length of the major (equatorial) semiaxis a= 6378388 m and compression f = 1:297,00 proposed by the American surveyor John Fillmore Hayford ( in 1910.

John Fillmore Hayford

datum ED 50 (European Datum 1950)

• ellipsoid - International ellipsoid 1924
• Greenwich prime meridian)

To carry out work on the entire territory of the USSR since 1946 (Resolution of the Council of Ministers of the USSR No. 760 dated April 7, 1946), a geodetic coordinate system was used SK-42 (Pulkovo 1942) based on Krasovsky ellipsoid with the length of the major (equatorial) semiaxis a= 6378245 m and compression f= 1:298,3 ... This reference ellipsoid is named after the Soviet astronomer-geodesist Feodosiy Nikolaevich Krasovsky. The center of this ellipsoid is shifted with respect to the center of mass of the Earth by about 100 meters for maximum correspondence with the Earth's surface on the European territory of the USSR.

datum Pulkovo 1942 (Pulkovo 1942)

• ellipsoid - Krasovsky ( Krassowsky 1940)
• prime meridian - Greenwich meridian ( Greenwich prime meridian)

Currently (including in the system Gps) widely used ellipsoid WGS84 (World Geodetic System 1984) with the length of the semi-major axis a= 6378137 m, squeezing f = 1:298,257223563 and eccentric e = 0,081819191 ... The center of this ellipsoid coincides with the center of mass of the Earth.

datum WGS84 (EPSG: 4326)

• ellipsoid - WGS84
• Prime Meridian - reference meridian (IERS Reference Meridian (International Reference Meridian)) passing 5.31 ″ east of the Greenwich meridian. It is from this meridian that the longitude in the system is counted Gps(eng. GPS longitude)
  

Center of coordinate system WGS84 coincides with the center of mass of the Earth, the axis Z coordinate system is aimed at reference pole (eng. IERS Reference Pole (IRP)) and coincides with the axis of rotation of the ellipsoid, the axis X passes along the line of intersection of the prime meridian and the plane passing through the point of origin and perpendicular to the axis Z, axis Y perpendicular to the axis X.

An alternative to an ellipsoid WGS84 is an ellipsoid PZ-90 used in the system GLONASS, with the length of the major semiaxis a= 6378136 m and compression f = 1:298,25784 .

Datum conversions

In the simplest version of the transition between datums Pulkovo-1942 and WGS84 it is necessary to take into account only the displacement of the center of the Krasovsky ellipsoid with respect to the center of the ellipsoid WGS84:
recommended in GOST 51794-2001 —
dX= +00023.92 m; dY= –00141.27 m; dZ= –00080.91 m;
recommended in World Geodetic System 1984. NIMA, 2000 —
dX= +00028 m; dY= –00130 m; dZ= –00095 m.
It should be noted that the averaged values ​​of the coefficients are given above, which, for a more accurate transformation, must be calculated for each point on the earth's surface individually. For example, for Poland, neighboring Belarus, these parameters are as follows:
dX= +00023 m; dY= –00124 m; dZ= –00082 m (according to )
This transformation is called three-parameter.
With a more accurate transformation ( transformation of Molodensky) it is necessary to take into account the difference between the shapes of the ellipsoids, determined by two parameters:
da- the difference between the lengths of the major semiaxes, df- difference between compression ratios (difference in flattening). Their values ​​are the same for GOST and NIMA:
da= - 00108 m; df= + 0.00480795 ⋅ 10 -4 m.

When navigating between datums ED 50 and WGS84 conversion parameters are as follows:
da= - 00251 m; df= - 0.14192702 ⋅ 10 -4 m;
for Europe dX= -87 m; dY= –96 m; dZ= -120 m (according to User's Handbook on Datum Transformations involving WGS-84, 3rd edition, 2003 ).

A set of the specified five parameters ( dX, dY, dZ, da, df) can be entered into a navigator or navigation program as a characteristic of the datum used by the user.

Projection

The way the three-dimensional earth's surface is displayed on a two-dimensional map is determined by the chosen map projection.
The most popular ( normal) cylindrical Mercator and such a variety as transverse cylindrical Mercator (Transverse mercator).

Unlike the normal Mercator projection known for centuries, which is especially good for depicting equatorial regions, the transverse projection differs in that the cylinder onto which the planet's surface is projected is rotated by 90 °:

Cylindrical Mercator

Spherical Mercator

For a spherical projection, the following formulas are used to convert the latitude $ \ phi $ and longitude $ \ lambda $ of a point on the surface of the earth's sphere (in radians) into rectangular coordinates $ x $ and $ y $ on the map (in meters):
$ x = (\ lambda - (\ lambda) _0) \ cdot R $;
$ y = arcsinh (\ tan (\ phi)) \ cdot R = \ ln ((\ tan (((\ phi \ over 2) + (\ pi \ over 4))))) \ cdot R $
(logarithmic tangent formula) ,
where $ R $ is the radius of the sphere, $ (\ lambda) _0 $ is the longitude of the prime meridian.
The scale factor $ k $ is the ratios of the distance over the map grid (eng. grid distance) to the local (geodetic) distance (eng. geodetic distance):
$ k = (1 \ over (\ cos \ phi)) $.
The reverse translation is implemented using the following formulas:
$ \ lambda = (x \ over R) + (\ lambda) _0 $;
$ \ phi = (\ pi \ over 2) - 2 \ arctan (e ^ (- y \ over R)) $.
An important feature of the Mercator projection for navigation is that rumba line(eng. rhumb lines) or loxodrome (eng. loxodrome) on it is represented by a straight line.
A loxodrome is an arc that crosses the meridians at the same angle, i.e. path with constant ( loxodromic) track angle.
Track angle, PU(eng. heading) is the angle between the north direction of the meridian at the measurement point and the direction of the track line, counted clockwise from the direction to the geographical north (0 ° is used to indicate the direction of movement to the north, 90 ° - to the east).
Loxodromes are spirals making an unlimited number of turns, approaching the poles.

It should be noted that the loxodrome is not the shortest route between two points - orthodromic, arc large circle connecting these points .

Web Mercator

A variant of the mercator spherical projection is used by many map services, for example, OpenStreetMap, Google Maps, Bing Maps.

V OpenStreetMap the world map is a square with the coordinates of the points along the axes x and y lying between -20,037,508.34 and 20,037,508.34 m. As a result, such a map does not show the areas lying north of 85.051129 ° north latitude and south of 85.051129 ° south latitude. This latitude value $ \ phi_ (max) $ is the solution to the equation:
$ \ phi_ (max) = 2 \ arctan (e ^ \ pi) - (\ pi \ over 2) $.
Like any map compiled in the Mercator projection, it is characterized by area distortions, which are most pronounced when comparing the Greenland and Australia depicted on the map:

When drawing a map in OpenStreetMap coordinates (latitude and longitude) on an ellipsoid in the system WGS84 are projected onto the map plane as if these coordinates were defined on a sphere with a radius R = a= 6 378 137 m(reprojection) - spherical representation of ellipsoidal coordinates (" spherical development of ellipsoidal coordinates"). This projection, called Web Mercator) matches EPSG (European Petroleum Survey Group) code 3857 (" WGS 84 / Pseudo-Mercator«).
Re-projection from EPSG: 4326 v EPSG: 3857($ \ phi, \ lambda \ rightarrow x, y $) is implemented according to the above formulas for the usual spherical Mercator projection.
On such a map, the direction to the north always corresponds to the direction to the upper side of the map, the meridians are equidistant vertical lines.
But such a projection, unlike the spherical or elliptical Mercator projection, is not p avocular ( conformal), the rumba lines in it are not straight. Rumba line (loxodrome) Is a line crossing the meridians at a constant angle.
The advantage of the considered projection is the simplicity of calculations.

In the specified projection, the map can be drawn with a rectangular grid of coordinates (in terms of longitude and latitude).
Map snapping (comparison of rectangular coordinates on the map and geographic coordinates on the ground) can be carried out by $ N $ points with known coordinates. For this, it is necessary to solve a system of $ 2 N $ equations of the form
$ X = \ rho _ (\ lambda) \ lambda - X_0 $, $ Y = arcsinh (\ tan (\ phi)) \ cdot \ rho _ (\ phi) - Y_0 $.
To solve the system of equations and determine the values ​​of the parameters $ X_0 $, $ Y_0 $, $ \ rho _ (\ lambda) $, $ \ rho _ (\ phi) $, you can use, for example, the mathematical package Mathcad.
To check the correctness of the map snapping, you can determine the ratio of the lengths of the sides of the rectangle of the constructed mesh. If the horizontal and vertical sides of the rectangle correspond to the same angular length in longitude and latitude, then the ratio of the length of the horizontal side (arc of parallel - small circle) to the length of the vertical side (arc of the meridian - great circle) should be equal to $ \ cos \ phi $, where $ \ phi $ - geographic latitude of the place.

Elliptical Mercator

Elliptical Mercator ( EPSG: 3395 — WGS 84 / World Mercator) is used, for example, by services Yandex maps,Space images.
For an elliptical projection, the following formulas are used to convert the latitude $ \ phi $ and longitude $ \ lambda $ of a point on the surface of the earth's sphere (in radians) into rectangular coordinates $ x $ and $ y $ on the map (in meters):
$ x = (\ lambda - (\ lambda) _0) \ cdot a $;
$ y = a \ ln (\ tan ((\ pi \ over 4) + (\ phi \ over 2)) (((1 - e \ sin (\ phi)) \ over (1 + e \ sin (\ phi )))) ^ (e \ over 2)) $ ,
where $ a $ is the length of the semi-major axis of the ellipsoid, $ e $ is the eccentricity of the ellipsoid, $ (\ lambda) _0 $ is the longitude of the prime meridian.
The scale factor $ k $ is given by:
$ k = ((\ sqrt ((1 - (e ^ 2) (((\ sin \ phi)) ^ 2)))) \ over (\ cos \ phi)) $.
The reverse translation is implemented using the following formulas:
$ \ lambda = (x \ over a) + (\ lambda) _0 $;
$ \ phi = (\ pi \ over 2) - 2 \ arctan (e ^ (- y \ over a) (((1 - e \ sin (\ phi)) \ over (1 + e \ sin (\ phi) ))) ^ (e \ over 2)) $.
Latitude is calculated using an iterative formula; as a first approximation, the latitude value calculated using the formula for the spherical Mercator projection should be used.

Transverse cylindrical Mercator

The most commonly used are two types of the transverse-cylindrical Mercator projection - the Gauss-Kruger projection (eng. Gauss - Krüger) (became widespread in the territory of the former USSR) and the universal transverse Mercator projection (eng. Universal transverse mercator (UTM)).
For both projections, the cylinder onto which the projection occurs spans the earth's ellipsoid along a meridian called the central (axial) meridian ( English central meridian, longitude origin) zones. Zone(eng. zone) is an area of ​​the earth's surface bounded by two meridians with a 6 ° longitude difference. There are 60 zones in total. Zones completely cover the Earth's surface between latitudes 80 ° S and 84 ° N.
The difference between the two projections is that the Gauss-Kruger projection is a projection onto a tangent cylinder, and the universal transverse Mercator projection is a projection onto a secant cylinder (to avoid distortion on the extreme meridians):

Gauss-Kruger projection

The Gauss-Kruger projection was developed by German scientists Karl Gauss and Louis Kruger.
In this projection, the zones are numbered from west to east, starting at the 0 ° meridian. For example, zone 1 extends from the 0 ° meridian to the 6 ° meridian, and its central meridian is 3 °.
In the Soviet system of marking and nomenclature of topographic maps, zones are called columns and are numbered from west to east, starting from the 180 ° meridian.
For example, Gomel and its surroundings belong to the zone 6 (column 36 ) with a central meridian of 33 °.
Zones / columns are divided by parallels into rows (every 4 °), which are denoted by capital Latin letters from A before V starting from the equator to the poles.
For example, Gomel and its surroundings belong to the series N... Thus, the full name of a sheet of a 1: 1,000,000 (10 km in 1 cm) map depicting Gomel looks like N-36... This sheet is divided into larger scale map sheets:


For Belarus and neighboring countries, the schedule is as follows:

To determine the position of a point on a topographic map, a grid of rectangular coordinates is applied to the map X and Y expressed in kilometers. It is formed by a system of lines parallel to the image of the axial meridian of the zone (vertical grid lines, axes X) and perpendicular to it (horizontal grid lines, axes Y).
On a map with a scale of 1: 200,000, the distance between the grid lines is 4 km; on a map with a scale of 1: 100,000 - 2 km.
Coordinate X subscribes on the vertical edges of the map sheet and expresses the distance to the equator, and the coordinate Y is signed on the horizontal edges of the map sheet and consists of the zone number (the first one or two digits of the value) and the position of the point relative to the central meridian of the zone (the last three digits of the value, and the central meridian of the zone is assigned the value of 500 km).


fragment of sheet N36-123 of the Soviet topographic map at a scale of 1: 100,000

For example, on the above fragment of the map, the inscription 6366 near the vertical grid line means: 6 - 6th zone, 366 - distance in kilometers from the axial meridian, conditionally shifted to the west by 500 km, and the inscription 5804 near the horizontal grid line indicates the distance from the equator in kilometers.

Universal Transverse Mercator

Universal transverse Mercator projection ( UTM) was developed by the United States Corps of Engineers ( United States Army Corps of Engineers) in the 1940s.

To build maps in projection UTM previously used ellipsoid International 1924 - net UTM (International), and currently an ellipsoid WGS84 - net UTM (WGS84).
In this projection, the zones are numbered from west to east, starting at the 180 ° meridian.
This system is used by the US and NATO armed forces (eng. United States and NATO armed forces):

Each zone is divided into horizontal stripes every 8 ° latitude. These stripes are indicated by letters, from south to north, starting from the letter C for latitude 80 ° S and ending with the letter X for latitude 84 ° N... Letters I and O omitted to avoid confusion with the numbers 1 and 0. The strip marked with a letter X, occupies 12 ° in latitude.
The zone in this projection is designated by a number (eng. longitude zone) and a letter (latitude channel, eng. latitude zone):


This figure shows two non-standard zones of longitude - zone 32V expanded to cover all of southern Norway, and zone 31V reduced to cover only water.
For Gomel and its environs, the zone is designated as 36U with a central meridian of 33 °:

The zone is covered by a rectangular (kilometer) grid (grid according to the universal transverse Mercator projection, SUPPM):


The side length of the grid square in the above map fragment is 10 km.

The origin of the coordinate system for each zone is determined by the intersection of the zone's equator and central meridian.
Coordinate E (Easting) on such a grid represents the distance on the map from the central meridian in meters (to the east - positive, to the west - negative), to which + 500,000 meters (eng. False Easting
Coordinate N (Northing) on such a grid represents the distance on the map from the equator in meters (to the north - positive, to the south - negative), and in the southern hemisphere this distance is subtracted from 10,000,000 meters (eng. False Northing) to avoid negative values.
For example, for the lower left corner of the grid square in the above map, the coordinates are written as
36U(or 36+ ) 380000 5810000 ,
where 36 — longitude zone, U — latitude zone, 380000 — easting, 5810000 — northing.

Converting latitude and longitude to coordinates UTM illustrated by the figure:

P- considered point
F- the point of intersection of the perpendicular dropped to the central meridian from the point P, with the central meridian (a point on the central meridian with the same northing as the point under consideration P). Point latitude F(eng. footprint latitude) is denoted as $ \ phi '$.
O- equator
OZ- central meridian
LP- parallel point P
ZP- point meridian P
OL = k 0 S- arc of the meridian from the equator
OF = N — northing
FP = E — easting
GN- north direction of the map grid (eng. grid north)
C- the angle of convergence of the meridians (eng. convergence of meridians) - the angle between the direction to true north (eng. true north) and north of the map grid

When transforming rectangular coordinates ( X, Y) for the Gauss-Kruger projection on the ellipsoid WGS84 to rectangular coordinates ( N, E) for the universal transverse Mercator projection on the same ellipsoid WGS84 it is necessary to take into account the scale factor (eng. scale factor) $ k_0 = 0.9996 $:
$ N = X \ cdot k_0 $;
$ E = Y_0 + Y \ cdot k_0 $,
where $ Y_0 = $ 500,000 meters.

The specified scale factor $ k_0 = 0.9996 $ is only valid for the central meridian of the zone. With distance from the axial meridian, the scale factor changes.

Note. The error in reading coordinates from the map ( georeferencing accuracy) is usually taken equal to ± 0.2 mm. This is precisely the accuracy of the devices used to create an analog map.

Geoid

It should be noted that a more accurate approximation of the surface of our planet is geoid(eng. geoid) Is the equipotential surface of the earth's gravity field, i.e., the surface of the geoid is everywhere perpendicular to the plumb line. But gravity is determined by the vector sum of the gravitational force from the Earth and the centrifugal force associated with the rotation of the Earth, so the potential of gravity does not coincide with the purely gravitational potential.
The geoid coincides with the average level of the World Ocean, relative to which the countdown is conducted altitudes above sea level.
The geoid has a complex shape, reflecting the distribution of masses inside the Earth, and therefore, for solving geodetic problems, the geoid is replaced by an ellipsoid of revolution. The most modern mathematical model of the geoid is EGM2008, which replaced the popular model EGM96.

To be continued.

He never made sea voyages, he made all the discoveries in his office, but his works worthily crown the era of the great geographical discoveries. He brought together all the geographical knowledge accumulated in Europe, created the most accurate maps. A science called cartography originates from Gerard Mercator.

In the XIII-XIV centuries, a compass and nautical charts appeared in Europe, on which the coastline was quite accurately displayed, and the inner land areas were filled with pictures from the life of the peoples who inhabited them, sometimes very far from reality. In the years 1375-1377, Abraham Cresquez compiled the famous Catalan maps.

They reflected all the sailing experience accumulated by that time. Instead of a grid of parallels and meridians, lines were drawn on them, marking the direction indicated by the arrow of the compass: it was possible to navigate along them in distant voyages. In 1409, Manuel Chrysoporus translated Ptolemy's Geography, rediscovering it for his contemporaries.

The sea voyages of Columbus, Vasco da Gama, Magellan gave a lot of new facts that did not fit into the previous geographical concepts. They demanded comprehension and design in the form of a new geography, which made it possible to carry out long-distance trade and military campaigns. This task was completed by Gerard Mercator, the famous geographer, the author of a new cartography.

This amazing map was drawn in 1538 by Gerhard Mercator, a highly respected cartographer who lived in the 16th century. His work is quite famous, and you can still buy the Mercator atlas in the store. He was the first to use the word "Atlas" for a collectionkart. And his works in geography were just as important todevelopment of science, like Copernicus in astronomy. By the way, hewas friends and collaborated with a famous alchemist, magician andastrologer John Dee. He was a good expert in mathematicsand even taught her at one time. Developed a waymass production of globes.

Gerhard Mercator was known for periodically updating his work and creating new, more detailed atlases of the world as more and more shores were opened to mariners, and more and more accurate data came to him. In one such update, his 1538 world map (shown in the picture above) was replaced with a new one in 1569. And surprisingly, the 1538 map was not only more accurate than the later one, but also contained correct measurements of geographic longitude.

To understand the significance of this fact, it must be said that calculating longitude is much more complex than calculating latitude, which can be determined by observing the stars and the Sun. Calculating longitude requires solving the equation “Distance = speed times time” and, more importantly, an accurate clock. Determination of longitude was at one time called "the greatest problem of maritime navigation" and in the 1700s in England a special Committee on Longitude was even created to solve this problem. In 1714, Sir Isaac Newton appeared before the Committee and explained that the true root of the problem is that "the clock needed to measure this precision has not yet been invented." The Queen of England then set a reward of 200 thousand pounds to a person who could build such a watch, and finally, in 1761, a certain Garrison received this award and put forward his prototype chronometer, which then "opened the world to a new era of sea travel." During the 19th century, maps were updated with correct longitude measurements.

However, the Mercator map was marked with accurate longitude values ​​as early as 1538 - 223 years before it was discovered. Where did he get this information from? Obviously, Mercator himself did not have any knowledge of longitude at that time and should have received this information from some other source, since subsequent maps were marked with incorrect values ​​- which means that their source was considered more reliable. These maps are fraught with a big mystery - if a person of deep antiquity never made a round-the-world trip and did not have any knowledge about geographical longitude, then how did these maps come into being? We do not know the answer to this question.

World map, 1531:

Gerard Mercator was born on March 5, 1512 in the city of Rüpelmond (modern Belgium), in the area that was then part of the Netherlands. He was the seventh child in a rather poor family. When Gerard was 14 or 15 years old, his father died, and the family was left without a livelihood. Gerard's tutor is his relative, the priest Gisbert Kremer. Thanks to him, Gerard is educated at the gymnasium of the small town of Bois-de-Dunes. Although this gymnasium had a spiritual orientation, it also taught the classical ancient languages ​​and the beginnings of logic. At this time, Gerard changes his German surname Kremer, which means "shopkeeper", to the Latin Mercator - "merchant", "merchant".

He graduates from high school very quickly, in three and a half years, and almost immediately continues his studies at the University of Louvain, again thanks to the support of Gisbert Kremer. Louvain was the largest scientific and educational center in the Netherlands, it housed 43 gymnasiums, and its university, founded in 1425, was the best in Northern Europe. The city turned into a center of humanistic education and free thought thanks to Erasmus of Rotterdam (1465-1536), who lived for some time in Louvain.

It was during his university years that Mercator developed a special interest in the natural sciences, especially in astronomy and geography. He begins to read the works of ancient authors, trying to find out how the earth works. Subsequently, he wrote: "When I became addicted to the study of philosophy, I really liked the study of nature, because it explains the causes of all things and is the source of all knowledge, but I turned only to a particular question - to the study of the structure of the world." Convinced of the insufficiency of his knowledge in the field of mathematics, especially geometry, he begins to study it on his own. The textbook that existed at that time clearly does not satisfy him, and he reads the first seven books of Euclid's Principles in the original.

"When I became addicted to the study of philosophy, I really liked the study of nature, because it is the source of all knowledge, but I turned only to the study of the structure of the world." From a letter by G. Mercator

After graduation, Mercator receives a Master of Arts degree (licentiate) and stays in Louvain. Without losing touch with the university, he listens to lectures on the planets of Professor Gemma Frisius, one of the outstanding people of that time. A brilliant astronomer, mathematician, cartographer and physician, Frisius blazed new trails in science and practice. He penned works on cosmography and geography, he made globes and astronomical instruments. Mercator becomes his student and assistant. Starting with engraving, he then moves on to more complex ones - to the manufacture of globes, astrolabes and other astronomical instruments. The instruments designed and manufactured by him, thanks to their accuracy, bring him fame almost immediately.

At the same time, Mercator is involved in the development of the mathematical foundations of cartography. The main problem was that, due to the spherical shape of the Earth, its surface could not be depicted on a plane without distortion, and it was necessary to find a way in which the images of the oceans and continents on the map would look the most similar. At the age of 25, Mercator presents his first independent cartographic work: a map of Palestine, published in Louvain.

The following year, he publishes a map of the world in a double heart-shaped projection, made with great care and taking into account the latest geographic information. On this map, the name America is for the first time extended to both continents of the New World, and America itself is depicted as separated from Asia, contrary to the then widespread misconception. All of Mercator's works are subordinate to a single plan and are closely interrelated: in the explanatory text to the map, he says that the world shown on the map will subsequently be considered in detail.

In 1541, Mercator constructs a celestial globe depicting stars and constellation figures, whichwhich became one of the best for that time. It rotated freely around an axis passing through the poles and fixed inside a massive copper ring. OA distinctive feature of this globe was a grid of curved lines applied to its surface, designed to facilitate maritime navigation. These lines allow us to believe that when creating the globe by Mercator, the development of the famous cartographic projection, later named after him, was basically completed.

The Mercator cartographic projection increases the size of the polar countries, but it makes it easy to determine the desired direction - this is of great importance in navigation.

Thanks to his work on the manufacture of maps and astronomical instruments, Mercator is becoming more and more famous, the fame of him even reaches the King of Spain Charles V. But wide popularity also attracts the attention of the Inquisition. There are reports that Mercator freely discusses inconsistencies in the teachings of Aristotle and in the Bible, and besides, he is in constant travel, which in itself always looks suspicious in the eyes of the inquisitors. In 1544 he goes to prison. Numerous intercessions do not lead to success, and only after the intervention of Charles V, after spending four months in prison, Mercator regains freedom.

Fearing persecution, he moved to Duisburg, where he breathes more freely, but the working conditions are much worse. This city is remote from the sea and from trade routes, and it is more difficult to obtain information about the latest discoveries, to obtain new blueprints and maps here than in Louvain. However, the geographer Abraham Ortelius rescues him: close correspondence is struck between colleagues, thanks to which Mercator receives the necessary information.

In Duisburg, he continues to work on map publishing. Now he works alone, on his shoulders lies the compilation and drawing and engraving of maps, the compilation of inscriptions and legends, as well as taking care of the sale of maps. Work on the creation of a comprehensive work on cosmography, which absorbed him entirely, began in 1564. Mercator conceived a cartographic work, including the sections "Creation of the World", "Description of Celestial Objects", "Earth and Seas", "Genealogy and History of States", "Chronology".

Due to the sphericity of the Earth, its surface cannot be depicted on a plane absolutely accurately. On the maps compiled by Mercator, the outlines of the oceans and continents are presented with the least distortion.

In 1569, Mercator publishes a map of the World, which he called "The new and most complete image of the globe, tested and adapted for use in navigation." It was made on 18 sheets; in its manufacture, a new method of depicting a grid of parallels and meridians was used, which later received the name of the Mercator (or cylindrical) projection. When drawing up a map, he set himself the task of showing the globe on a plane so that the images of all points on the earth's surface correspond to their true position, and the outlines of countries, if possible, are not distorted. Another goal was to depict the world known to the ancients - that is, the Old World - and the place it occupied on Earth. Mercator wrote that with the discovery of new continents, the achievements of the ancients in the study of the Old World appeared more clearly and vividly before the whole world, the image of which is presented with the fullest possible completeness on the map.

By 1571, Mercator completed the work, which he called "Atlas, or cartographic considerations about the creation of the world and the type of creation." Maps were attached to the Atlas. Since then, the word "atlas" has become a household name for a collection of maps. The Atlas was published only in 1595, a year after the death of Gerard Mercator.

John Dee's 1582 map. On it we see almost the same image of Arctida as on the Mercator map of 1569, but without painting in different colors of different territories and without applying names. Arctida "pygmies" here protrude to the south even more, but the coastal area, separated by a mountain range, is absent here at all. America has left the Fourth Arctida very far, so that the ocean in this place is very wide, and the narrowest point is in the strait that makes contact with Asia. So the tendency for the separation of the Arctids from the continents is carried out here to the greatest extent.

The brave sailors, whose large exploration voyages have opened up the world, are iconic figures in European history. Columbus found the New World in 1492; The Cape of Good Hope was discovered in 1488; and Magellan set aside to sail around the world in 1519. However, there is one difficulty with this confident assertion of European prowess: it may not be true.

It seems more likely that the world and all of its continents were discovered by a Chinese admiral named Zheng He, whose fleets roamed the oceans between 1405 and 1435. His deeds, which are well documented in Chinese historical records, were written about in a book that appeared in China in about 1418, entitled "Marvelous Visions of the Star Raft".

A map on a stone from the city of Ica, Peru, the mainland is divided into 4 parts by rivers - in my opinion it is similar to Hyperborea, if this is so - then you have an ancient map before you, the age of the stones is dated from several million to tens of millions of years! since among the stones found (there are more than 15,000 in total) there are images of dinosaurs, and as petson the island at the top there is a teremok.

When solving navigation problems, it becomes necessary to display the ship's course line (loxodrome) on a nautical chart, measure and plot the angle and directions. Based on these tasks, the following requirements are imposed on the cartographic projection of the sea chart:

The loxodrome on the map should be depicted as a straight line;
- the angles measured on the ground should be equal to the corresponding angles laid on the map, that is, the projection should be conformal.

These requirements are met by a straight conformal cylindrical projection developed in 1569 by the Dutch cartographer Gerard Kremer (Mercator).

1. The earth is taken as a sphere and a conditional globe is considered, the scale of which is equal to the main scale.
2. Coordinate lines (meridians and parallels) are projected onto the cylinder.
3. The axis of the cylinder coincides with the axis of the conventional globe.
4. The cylinder touches the conditional globe along the equator line.
5. Meridians and parallels of the conventional globe are projected onto the surface of the cylinder in such a way that their projections remain in the planes of meridians and parallels.
6. After cutting the cylinder along the generatrix and unfolding it into a plane, a cartographic grid is formed - mutually perpendicular straight lines: meridians and parallels.

7. The cylinder touches the symbolic globe along the equator, therefore the circle Ao1 at the equator on the map is represented by the circle A1.
8. When projecting parallels, their stretching occurs, and the farther the parallel is from the equator (the greater the geographical latitude), the greater the stretching: the circles Ao2 and Ao3 on the map are depicted by ellipses A2, A3, that is, the resulting projection is not conformal.
9. For the ellipses A2 and Az to turn into circles A2 "A3", it is necessary to extend the meridian at each point in proportion to the extension of the parallel at this point.
The greater the latitude, the more the parallel is stretched, and therefore, the more the meridian should be stretched.
10. As a result, the same circles on the globe, located at different parallels, will appear on the map as circles of different sizes, increasing with the geographical latitude.

The graphical representation on the map of one minute of the meridian arc (nautical mile) increases with latitude.

Therefore, when measuring and plotting distances, it is necessary to use that part of the linear scale of the map, in the latitude of which the ship is sailing.

The projection obtained in this way is:
- straight line - the axis of the cylinder coincides with the axis of rotation of the Earth;
- Conformal - an elementary circle on the earth's surface is depicted in a circle on the map (the similarity of figures is preserved);
- cylindrical - the cartographic grid (meridians and parallels) represents mutually perpendicular straight lines.

The projection equation for the ball is:

X = R ln tan (45 "+ φ / 2); y = R λ;

When obtaining the projection, the main scale corresponded to the main scale of the conditional globe, that is, when projected onto the cylinder, there were no distortions on the line along which the cylinder touched the globe - at the equator.

When making maps in this projection, it turned out to be insufficiently convenient. Therefore, for each latitudinal zone, a projection line was chosen, on which there are no distortions - the main parallel. The parallel where the scale is equal to the main scale is called the main parallel. The latitude of the main parallel of this map is indicated in the title of the map.

When the ship is moving at a constant true heading, the heading line crosses each meridian at the same angle and on the earth's surface this line turns out to be of double curvature, called loxodromia(which in translation from Greek means "oblique run").

Sailing on the loxodrome is convenient, since the course of the vessel remains constant, and this simplifies all calculations associated with laying. The main properties of a loxodrome passing through two points can be identified from its equation:

From this equation it follows that at K = 0 ° or K = 180 ° tan K = 0, then λ2 - λ1 = 0, therefore, at the true courses of 0 or 180 °, the longitude of the points does not change and the loxodrome coincides with the meridian, turning into an arc of a great circle, and in this case passes through the earth's poles.

If the equation is written in the form

and take K - 90 ° or K = 270 °, then at these values ​​tan K = ~. Since the difference in longitudes λ2 - λ1 located in the numerator cannot be equal to infinity, the denominator must be zero, and it can be zero at 45 ° + φ1 / 2 = 45 ° + φ2 / 2, i.e. when φ1 = φ2.

Rice. 36

Therefore, at K = 90 ° or K = 270 °, the latitude of the points does not change and the loxodrome coincides with the parallel or at φ2 = φ1 = = 0 - with the equator.

For all true courses other than 0 - 180 ° and 90 - 270 °, the loxodrome in a spiral approach to one of the poles, but never reaches it (Fig. 36).

The length of the loxodrome segment traveled by a vessel on a given course is not the shortest distance on the earth's surface. The shortest distance on the earth's surface when a ship moves from one point to another will be a great circle, called orthodromy(which in translation from Greek means "straight run").

Orthodromy with each meridian makes variable angles. Therefore, sailing along the orthodrome requires a preliminary calculation of both its position and the courses that the ship is guided by in the great circle (see § 46).

Requirements for nautical charts

When choosing a projection for building a particular map, one always proceeds from the requirements of ensuring the solution of the tasks for which it is intended.

The cartographic projection of nautical navigational charts should be the most convenient for their use at sea, that is, for solving the main problems of ensuring the safety of navigation in the most simple ways and techniques.

Based on this, the cartographic projection of nautical navigational charts must meet the following requirements. So that: the line of the path of the ship going on a constant course, that is, the loxodrome, is depicted as a straight line;

The magnitude of the angles measured from the ship between different landmarks on the terrain corresponded to the values ​​of the angles between the same landmarks on the map, that is, the projection of the map should be conformal; the scale within the limits of the map varied within the smallest possible limits, i.e., the distortions of the lengths on the map did not exceed the errors of graphic constructions and measurements on the map, performed with the help of a padding tool.

The maps that satisfy these requirements were built according to the projection proposed in 1569 by the Dutch cartographer Gerard Kremer, known as Mercator, therefore this projection is called mercatorial. The Mercator projection is a conformal cylindrical projection, on it the earth's meridians and parallels are depicted by straight, mutually perpendicular lines, and the loxodromy is a straight line that makes the same angle with the meridians.

Mathematical substantiation of the principle of mercator projection

Let's imagine that the image of the Earth is made in the form of a globe (Fig. 37), the meridians on it are made of steel elastic wires fixed at the poles, and the parallels are made of stretching material, fastened to the meridians.

Rice. 37

We paint the meridians and parallels with paint and release the fastenings of the wire meridians at the poles. Then the meridians will straighten, and the parallels will stretch and, as it were, will be imprinted on the inner surface of the cylinder. Now let's cut the cylinder along the generatrix (along one of the meridians); a rectangular grid (traces of parallels and meridians) will be applied on it, in which the length of the meridians remains unchanged, and each parallel is stretched to the length of the equator. In this case, the parallel close to the equator will stretch less, and with increasing latitude, the stretching of the parallels increases more and more significantly. The island K of a round shape, which was on the globe, will be projected in the form of an oval on the unfolded plane of the cylinder. To preserve the similarity of the image on the globe and its projection on the plane, it is necessary to respectively stretch along the length and meridians.

To prove this statement, consider Fig. 38, where we denote the radius of the parallel пп by r, the latitude of this parallel cp, the radius of the globe R.

Rice. 38

From the triangle nOe, in which the side is Oe = r, we get r = R - cos φ, a R = r * 1 / cos φ or R = r - sec φ. Multiplying both sides of the equality by 2nd, we get 2ПR = 2Пtr * sec φ.

Consequently, each parallel on a cylindrical projection map is stretched by an amount proportional to the secant of its latitude. Therefore, to preserve the similarity of the figures on the map to the figures on the ground, the meridian segments must be stretched in proportion to sec φ, which will achieve the conformal projection.

Meridian parts

Distances along the meridian from the equator to these parallels on the mercator map, expressed in linear units, are called meridional parts. They are designated with the letter D.

For convenience, the meridional parts are expressed by the length of the equatorial arc in I, called the equatorial mile.

Table 26 (МТ-63), the length of the meridional parts is calculated in relation to the Krasovsky ellipsoid.

The values ​​in the table are calculated for latitudes from 0 to 89 ° 59 "through 1" latitude with an accuracy of 0.1 equatorial mile. To determine the value of the meridional parts at intermediate values ​​of the minute of latitude (for tenths of 1 "), simple interpolation is used.

Example. Find the meridional part for the parallel 50 ° 18 ", 5.

Solution. According to the table. 26 (MT-6.3) we find:

The distance along the meridian on the mercator projection between two parallels, expressed in equatorial miles, is called difference of meridional parts(RMCH) and is designated AD.

The difference between the meridional parts of two parallels is equal to the algebraic difference of the meridional parts of these parallels

Example. Determine the difference between the meridional parts of the parallels cp1 = 63 ° 40 "N and cp2 = 66 ° 20" N.

Solution. According to the table. 26 (MT-63) we find:

Example. Determine the difference between the meridional parts of the parallels cp1 = 5 ° 12 "N and cp2 = 3 ° 28.5.

Solution. According to the table. 2 6 (MT-63) we have:

The meridional parts are used in the construction of a cartographic grid of nautical charts in the mercator projection, and the difference between the meridional parts is included in one of the basic formulas of written reckoning (see Chapter VII).

The difference between the meridional parts of two parallels, spaced 1 "apart, will give us the length of the segment representing one equatorial minute on the map of the Mercator projection at a given latitude. This difference in the meridional parts is nothing more than the image of one nautical mile on the map of the Mercator projection. The mercator mile is used as a linear scale unit to measure latitudes and distances on a mercator projection map.

Since the nautical mile, as indicated earlier, has a constant value on the surface of the Earth, it is depicted on the nautical map of the mercator projection by segments of various lengths, depending on the latitude of the place to which it belongs.

Solution. 1) We select the meridional parts for latitudes 39 ° 30 "and 40 ° 30" according to the table. 26 (MT-63):

Hence, the mercator mile at latitude 40 ° is 78.0 / 60 = 1.3 equiv. miles.

2) choose the meridional parts for latitudes 69 ° 30 "and 70 ° 30":

Therefore, at cp = 70 ° the mercator mile is 175.4 / 60 = 2.923 equiv. miles. This example shows that the ratio of the length of the mercator mile at cp = 70 ° to its length cp = 40 ° is 2.923 / 1.3 = 2.248, i.e., the mercator mile at cp = 70 ° is depicted by a segment 2.248 times larger, than in cp = 40 °.

Therefore, when measuring the distances between any points on a nautical navigation chart, it is necessary to always take distances of one mile or several miles from the side frame of the chart in the same latitude at which the points are located. In practice, to measure distances on the mercator projection map, the length of the mercator mile is used, which corresponds to the average latitude of the measured line.

The main and private scales of maps of the mercator projection

The main scale on a mercator map is the scale referred to the equator (if the projection is built on the surface of a cylinder tangent to it) or to the parallel of the section, called the main parallel (if the projection is built on the surface of the secant cylinder).

The private scale in the mercator projection is constant in all directions, not only at a given point, but also at all points belonging to the same parallel.

Outside the equator or main parallel, the numerical value of the private scale will differ from the main scale, changing more and more as you move north or south of the equator or main parallel.

If the projection is built on the surface of a tangent cylinder, then at the equator the scale increase is c = 1, and since each parallel is equal to the equator (stretched by sec φ times), then on each parallel c = sec φ.

For example, at 30 ° latitude, the magnification will be 1.5 times, at 60 ° latitude - 2 times, and at 80 ° latitude - 5.75 times.

When constructing a projection on the surface of the secant cylinder on the main (secant) parallel, the scale increases with = 1.

In such a projection, all parallels become equal to the main one, and at the same time all parallels that are closer to the pole than the main one are stretched as many times as the secant of the latitude sec φ of the given parallel is greater than the secant of the latitude of the main parallel ses cpg.p. Therefore, on these parallels, the scale up is c> 1. The parallels located to the equator are reduced by as many times as sec φ GP. is greater than sec φ, and, therefore, with
Since the increase in scale is the ratio of the private scale to the main scale c = μ / μ0, then the partial scale μ = cμ0. If x is replaced by the ratio 1 / C (С is the denominator of the private scale), and the main scale μ0 is expressed through 1 / C0 where С0 is the denominator of the main scale, then the denominator of the private scale

headquarters for the points of each parallel when constructing a projection onto the surface of the tangent cylinder is determined from the expression C =
and when constructing on the surface of the secant cylinder C =

Nautical charts, as a rule, cover insignificant areas of the earth's surface, therefore, within the limits of the map, the values ​​of the main and private scales differ little from each other. According to the main scale indicated in the title of the map, the navigator selects maps for solving certain problems.

Ultimate Scale Accuracy

The scale of maps and plans determines the accuracy with which linear measurements can be made on them.

The linear distance on the ground, corresponding to 0.2 mm on a map or plan, is called extreme accuracy of scale. The value of 0.2 mm is adopted because it is approximately equal to the diameter of the indentation obtained on the card when pricked with a compass needle, and corresponds to the minimum value distinguishable by the naked eye. The magnitude of the extreme accuracy of the scale depends on the scale of the map. So, if the scale of the map is 1/100000, then this value will be 20 m.

Consequently, a line drawn on a map of this scale with a sharpened pencil will correspond to a 20 m wide strip on the ground, and on this map we will not be able to distinguish distances less than 20 m.

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