Mercator projection with two principal parallels. Practical cartography

Projections in cartography

For a long time, travelers and navigators have been compiling maps, depicting the studied territories in the form of drawings and diagrams. Historical research shows that cartography appeared in primitive society even before the advent of writing. In the modern era, thanks to the development of means of data transmission and processing, such as computers, the Internet, satellite and mobile communications, the most important component information resources geoinformation remains, i.e. data on the position and coordinates of various objects in the geographic space surrounding us.

Modern maps are compiled in in electronic format using Earth remote sensing devices, satellite global positioning system (GPS or GLONASS), etc. However, the essence of cartography remains the same - it is an image of objects on a map that allows you to uniquely identify them by determining the position by referencing to one or another system of geographical coordinates . It is not surprising, therefore, that one of the main and most common cartographic projections today is the Mercator conformal cylindrical projection, which was first used to create maps four and a half centuries ago.

The work of ancient land surveyors did not go beyond geodetic measurements and calculations for placing milestones along the route of the future road or marking the boundaries of land plots. But a lot of data gradually accumulated - distances between cities, obstacles on the way, the location of water bodies, forests, landscape features, borders of states and continents. Maps captured more and more territories, became more detailed, but their error also increased.

Since the Earth is a geoid (a figure close to an ellipsoid), to depict the surface of the geoid of the Earth on a map, it is necessary to unfold, project this surface onto a plane in one way or another. Methods for displaying a geoid on a flat map are called map projections. There are several types of projections, and each of them introduces its own distortions of lengths, angles, areas or shapes of figures into a flat image.

How to make an accurate map?

It is impossible to completely avoid distortions when building a map. However, you can get rid of any one type of distortion. So called equal area projections preserve areas, but at the same time distort angles and shapes. Equal-area projections are convenient to use in economic, soil and other small-scale thematic maps - in order to use them to calculate, for example, the areas of territories exposed to pollution, or to manage forestry. An example of such a projection is Albers Equal Area Conic Projection, developed in 1805 by the German cartographer Heinrich Albers.

Equangular projections are projections without distortion of angles. Such projections are convenient for solving navigation problems. The angle on the ground is always equal to the angle on such a map, and a straight line on the ground is represented by a straight line on the map. This allows navigators and travelers to chart a route and follow it accurately using compass readings. However linear scale maps with such a projection depends on the position of the point on it.

The oldest conformal projection is considered to be the stereographic projection, which was invented by Apollonius of Perga around 200 BC. This projection is still used to this day for maps of the starry sky, in photography for displaying spherical panoramas, in crystallography for depicting point groups of symmetry of crystals. But the use of this projection in navigation would be difficult due to too large linear distortion.

Mercator projection

In 1569, the Flemish geographer Gerhard Mercator (the Latinized name of Gerard Kremer) developed and first applied in his atlas (the full name is “Atlas, or Cosmographic Discourses on the Creation of the World and the View of the Created”) conformal cylindrical projection, which was later named after him and became one of the main and most common map projections.

To construct a cylindrical Mercator projection, the earth's geoid is placed inside the cylinder so that the geoid touches the cylinder at the equator. The projection is obtained by conducting rays from the center of the geoid to the intersection with the surface of the cylinder. If after that the cylinder is cut along the axis and deployed, then a flat map of the Earth's surface will be obtained. Figuratively, this can be represented as follows: the globe is wrapped in a sheet of paper along the equator, a lamp is placed in the center of the globe, and images of continents, islands, rivers, etc. projected by the lamp, are displayed on the sheet of paper. sheet, we would have a finished map.

The poles in such a projection are located at an infinite distance from the equator, and therefore cannot be depicted on a map. In practice, the map has upper and lower latitude limits - up to about 80 ° N and S.

The parallels and meridians of the cartographic grid are depicted on the map as parallel straight lines, and they are always perpendicular. The distances between the meridians are the same, but the distance between the parallels is equal to the distance between the meridians near the equator, but increases rapidly when approaching the poles.

The scale in this projection is not constant, it increases from the equator to the poles as the inverse cosine of latitude, but the vertical and horizontal scales are always equal.

The equality of the vertical and horizontal scales ensures the equiangularity of the projection - the angle between two lines on the ground is equal to the angle between the image of these lines on the map. Thanks to this, the shape of small objects is well displayed. But area distortion increases towards the polar regions. For example, even though Greenland is only one-eighth the size of South America, in the Mercator projection it appears larger. Large area distortions make the Mercator projection unsuitable for general geographic maps of the world.

A line drawn between two points on the map in this projection crosses the meridians at the same angle. This line is called rhumb or loxodromia. It should be noted that this line does not describe the shortest distance between points, but in the Mercator projection it is always depicted as a straight line. This fact makes the projection ideal for navigation needs. If a navigator wishes to sail, for example, from Spain to the West Indies, all he has to do is draw a line between two points, and the navigator will know which compass heading to keep in order to sail to his destination.

Accurate to the centimeter

To use the Mercator projection (as, indeed, any other), it is necessary to determine the coordinate system on earth's surface and correctly choose the so-called reference ellipsoid- an ellipsoid of revolution, approximately describing the shape of the Earth's surface (geoid). Since 1946, Krasovsky's ellipsoid has been used as such a reference ellipsoid for local maps in Russia. In most European countries, the Bessel ellipsoid is used instead. The most popular ellipsoid today, designed for compiling global maps, is the world geodetic system of 1984 WGS-84. It defines a three-dimensional coordinate system for positioning on the earth's surface relative to the center of mass of the earth, the error is less than 2 cm. The classical conformal cylindrical Mercator projection is applied to the corresponding ellipsoid. For example, the Yandex.Maps service uses the elliptical WGS-84 Mercator projection.

AT recent times In connection with the rapid development of mapping web services, another version of the Mercator projection has become widespread - based on a sphere, and not an ellipsoid. This choice is due to simpler calculations that can be quickly performed by clients of these services right in the browser. This projection is often called "spherical Mercator". This version of the Mercator projection is used by Google Maps services, as well as 2GIS.

Another well-known variant of the Mercator projection is Gauss-Kruger conformal projection. It was introduced by the outstanding German scientist Carl Friedrich Gauss in 1820-1830. for mapping Germany - the so-called Hanoverian triangulation. In 1912 and 1919 it was developed by the German surveyor L. Kruger.

In fact, it is a transverse cylindrical projection. The surface of the earth's ellipsoid is divided into three- or six-degree zones bounded by meridians from pole to pole. The cylinder touches the middle meridian of the zone, and it is projected onto this cylinder. In total, 60 six-degree or 120 three-degree zones can be distinguished.

In Russia for topographic maps scale 1:1000000 apply six-degree zones. For topographic plans scales 1:5000 and 1:2000, three-degree zones are used, the axial meridians of which coincide with the axial and boundary meridians of the six-degree zones. When surveying cities and territories for the construction of large engineering structures, private zones with an axial meridian in the middle of the object can be used.

multidimensional map

Modern information technologies make it possible not only to plot the contours of an object on a map, but also to change its appearance depending on the scale, to associate it with geographic location many other attributes, such as the address, information about the organizations located in this building, the number of floors, etc., making the electronic map multidimensional, multi-scale, integrating several reference databases in it at the same time. To process this array of information and present it in a user-friendly form, rather complex software products, the so-called geoinformation systems, the development and support of which can only be carried out by fairly large IT companies with the necessary experience. But, despite the fact that modern electronic maps bear little resemblance to their paper predecessors, they are still based on cartography and one or another way of displaying the earth's surface on a plane.

To illustrate the methods of modern cartography, we can consider the experience of the Data East company (Novosibirsk), which develops software in the field of geoinformation technologies.

The projection that is chosen for building an electronic map depends on the purpose of the map. For public cards and for navigation charts, as a rule, the Mercator projection with the WGS-84 coordinate system is used. For example, this coordinate system was used in the project "Mobile Novosibirsk", created by order of the mayor's office of the city of Novosibirsk for the city municipal portal.

For large-scale maps, both zonal conformal projections (Gauss-Kruger) and non-equiangular projections (for example, conic equidistant projection - Equidistant conic).

Today, maps are created with extensive use of aerial photography and satellite photos. For high-quality work on maps, Data East created an archive satellite images covering the territories of Novosibirsk, Kemerovo, Tomsk, Omsk regions, Altai Territory, Republics of Altai and Khakassia, other regions of Russia. With the help of this archive, in addition to large-scale maps of the territory, it is possible to make schemes of individual objects and sections under the order. In this case, depending on the territory and the required scale, one or another projection is used.

Since the time of Mercator, cartography has changed radically. The information revolution has affected this area of ​​human activity, probably the most. Instead of volumes of paper maps, now every traveler, tourist, driver has access to compact electronic navigators containing a lot of useful information about geographic features.

But the essence of the maps remained the same - to show us in a convenient and clear form, indicating the exact geographical coordinates, the location of the objects of the world around us.

Literature

GOST R 50828-95. Geoinformation mapping. Spatial data, digital and electronic maps. General requirements. M., 1995.

Kapralov E. G. et al. Fundamentals of geoinformatics: in 2 books. / Proc. allowance for students. universities / Ed. Tikunova V. S. M.: Academy, 2004. 352, 480 s.

Zhalkovsky E. A. et al. Digital cartography and geoinformatics / Brief terminological dictionary. Moscow: Kartgeocenter-Geodesizdat, 1999. 46 p.

Yu. B. Baranov and others. Geoinformatics. Dictionary basic terms. M.: GIS-Association, 1999.

Demers N. N. Geographical information systems. Fundamentals.: Per. from English. M.: Date+, 1999.

Maps courtesy of Data East LLC (Novosibirsk)

Allows you to overlay the contours of countries on other territories, taking into account the compensation of distortions of the Mercator projection. This projection was once created for navigational purposes - to ensure the exact relative position of territories along the north-south and west-east axes. However, it has its drawback - the closer to the poles, the greater the distortion. Other projections also have serious distortions. That is why our perception geographical map is also significantly distorted - say, Greenland on the Mercator projection map occupies an area three times larger than Australia, although in reality it is 3.5 times smaller (!). And the closer to the equator, the smaller the relative size of the countries.

In general, on this site you can do all sorts of curious tricks and watch metamorphoses different countries in overlay. It's even surprising that such a site has not appeared before - the basic idea is so good. Sometimes amazing effects are obtained that break the usual patterns. In addition, the country can be rotated in a circle, in which case projection compensation will also be taken into account.

Let's see some of the effects.
Here, for example, is an overlay on the Indonesian islands of some European countries. See how rather modest France looks on Kalimantan (on the right). The Czech Republic is superimposed on southern Malaysia and Singapore (center), and on the left is Norway on Sumatra. Very long on a European scale, in fact it is only slightly longer than Sumatra.

2. China in Eastern Eurasia. If we fix its western border on the Tallinn-Prague line, then the east (Manchuria) will be east of Novosibirsk, and the Liaodong Peninsula will be somewhere in the Astana region. Hainan will be in central Iran.

3. Australia in Eastern Eurasia. This is where the compensation of the Mercator projection is most clearly seen: it extends from Munich to Chelyabinsk, and even more from south to north. Here you can see what colossal desert territories there are in Australia - no less than the Siberian cold expanses, because more or less only the southeast and a narrow strip to the west are inhabited there.

4. Mexico on Europe. From French Brest almost to Nizhny Novgorod. And Mexican California stretches from Normandy to Venice.

5. Indonesia in Eastern Eurasia. The length of the islands is equivalent to the distance from Northern Ireland to Central Kazakhstan, and Kalimantan alone easily covers the entire Baltic with the Russian North-West.

6. United States in Eastern Eurasia. From Tallinn - more than to Krasnoyarsk!

7. Kazakhstan on Europe. Also, in general, very solid: from the west of France almost to Kharkov. Covers most of continental Europe.

8. Iran in Northern Europe: from Norwegian Lofoten to Kazan :)

9. Vietnam on European Russia. Vertically, it is equivalent to the distance of train No. 7 Leningrad - Sevastopol, but also nothing horizontally: from Moscow to Chelyabinsk, moreover, it is curved.

Other interesting comparisons.

10. Kamchatka and Great Britain. Quite small: from Cape Lopatka to Palana.

11. Estonia as a third of Liberia, which is small in principle.

12. Austria, Hungary, Belgium in Madagascar.

Let's now look at the equivalents of Russia.

13. Russia on Australia. If Perth is in the Makhachkala region, then Melbourne is somewhere near Barnaul. Solid. But still, Rossiyushka stretches almost to Fiji.

14. Russia in Africa. Kuban in the region of South Africa (Novorossiysk as Cape Town) - Kamchatka reaches the south of Anatolia, approximately where Antalya is.

15. Russia in South America. If Tierra del Fuego is about where Chechnya is, then Kamchatka is in the Colombian region, and Chukotka comes north of the Panama Canal. Do you see how colossal our country is? More than a whole continent.

16. Russia in North America. San Francisco in the Crimea region - Chukotka is almost near Ireland. Here you can clearly see the size of the ocean expanses of the North Atlantic, by the way.

17. Luxembourg in St. Petersburg. It's not that small :)

18. In this territory (Bangladesh, marked in blue) - 168 million people live !!! Can you imagine the population density? And this is not a comfortable temperate climate, but a humid tropical jungle and the channels of the Ganges and the Brahmaputra...

19. And for dessert - Chile along the Trans-Siberian Railway. As you can see, it covers the distance from Moscow to Baikal in a narrow strip.

Here are some interesting comparisons :)

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

Developed by Gerard Mercator in 1569 for the compilation of maps that were published in his " Atlas». The name of the projection equiangular' means that the projection maintains angles between directions, known as constant courses or rhumb angles. All curves on the surface of the Earth in the conformal cylindrical Mercator projection are shown as straight lines..

"... 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) of Germany ... since 1947, the US Army used a very similar system, but with a standard scale factor of 0.9996 on the central meridian, as opposed to the German 1.0.

A little 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, increasing as they approach 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 2-dimensional Cartesian coordinate system... that is, it is used to determine a location on Earth, regardless of the height of the place...

All lines of constant courses (or rhumbs) on the Mercator maps are represented by straight segments. Two properties, equiangularity and straight lines of bearings, make this projection uniquely suited for marine navigation applications: courses and headings are measured using a wind rose or protractor, and corresponding directions are easily transferred from point to point on a chart using a parallel ruler or a pair of navigational protractors for drawing lines.

The name and explanation given by Mercator on his world map Nova et Aucta Orbis Terrae Descriptio ad Usum Navigantium Emendata: " New, supplemented and corrected description of the Earth for use by sailors” indicates that it was specifically conceived for use in maritime navigation.

Transverse Mercator projection.

Although the method of constructing the projection is not explained by the author, Mercator probably used a graphical method, transferring some of the rhombus lines 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 the globe.

The development of the Mercator conformal map projection represented a major breakthrough in nautical cartography in the 16th century. However, its appearance was far ahead of its time, as the old navigational and surveying methods were not compatible with its use in navigation.

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

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

The central meridian is located in the region that can be selected. On the central meridian, the 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 map projection of Gauss-Kruger is completely similar to the universal transverse Mercator, the zone width in the Mercator projection is 6°, while in the Gauss-Kruger projection the zone width is 3°. The Mercator projection is convenient to use for sailors, the Gauss-Kruger projection for ground forces in limited areas of Europe and South America. In addition, the Mercator projection is a 2-dimensional accuracy of determining latitude and longitude on the map does not depend on the height of the place, while the Gauss-Kruger projection is 3-dimensional, and the accuracy of determining latitude and longitude is constantly dependent on the height of the place.

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

Equatorial Mercator projection.

Can these two systems be combined 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 2-dimensional Cartesian coordinate system to provide a definition of 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 height 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 American published publications do not indicate the original source of the UTM system. The NOAA website claims that 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.

"Distortion of scale 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...

However, a series of aerial photographs found in the Bundesarchiv-Militärarchiv (military part of the German Federal Archives) appear to be from 1943 - 1944 with the inscription UTMREF logically derived coordinate letters and numbers, and also displayed in accordance with the transverse Mercator projection. This find is an excellent indication that the UTM map projection was developed between 1942 and 1943 by the German Wehrmacht. Its development and appearance was probably carried out in the Abteilung für Luftbildwesen (Aerial Photography Department) of Germany. Further from 1947, the US Army used a very similar system, but with a standard scale factor of 0.9996 on the central meridian, as opposed to the German 1.0.

For areas within the United States, an 1866 Clarke ellipsoid was used. For other regions 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 differ by up to 200 meters from the old system. For different geographical regions, for example: ED50, NAD83 other coordinate systems can be used.

Prior to the development of the universal transverse coordinate system of the Mercator projection, some European countries demonstrated the utility of grid based conformal mappings (preserving local angles) of 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), as opposed to possibly using the trigonometric formulas required by a grid based system of latitude and longitude. In the post-war 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 is a variant of the Mercator projection, which was originally developed by the Flemish geographer and cartographer Gerardus Mercator in 1570. This projection is conformal, meaning that angles are preserved and therefore allow small regions to be formed. However, it distorts distance and area.

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

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

In each zone, the central meridian scaling factor 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 of them, but the overall distortion is kept to a minimum.

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.

Around the boundaries of large-scale maps (1:100,000 or more), the coordinates for both adjoining 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 the grid for the zone in which they are located, and the scale factor of the still relatively small boundaries of the near zone can be overlapped by measurements to the adjacent zone by some distance when necessary.

The Latitude Bands are not part of the UTM system, but rather part of the Military Reference Reference System (MGRS). They are, however, sometimes used.

Ellipsoidal Mercator projection.

Each zone is segmented into 20 latitude bands. Each latitude band is 8 degrees high, and begins in capital letters 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 resemblance to the digits one and zero). The last latitude of the range, " X”, is extended by an additional 4 degrees, so that it ends at 84 ° north latitude, thus covering the northernmost part on Earth.

Mercator Map Projection Conclusion (UTM/UPS)

Band width " A" and " B" do exist, as do the 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.

The combination of zone and latitude band - defines the zone of the coordinate grid. The zone is always written first, followed by the latitude band. For example, a position in Toronto, Canada would be in zone 17 and latitude zone " T", thus, the full reference grid zone " 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 plan coordinates along with the zone number is all that is needed to determine the position, except which hemisphere).

Look at this map and tell me which area is larger: Greenland marked in white or Australia marked in orange? It seems that Greenland is at least three times larger than Australia.

But, looking into the directory, we will be surprised to read that the area of ​​Australia is 7.7 million km 2, and the area of ​​Greenland is only 2.1 million km 2. So Greenland seems so big only on our map, but in reality it is about three and a half times smaller than Australia. Comparing this map with a globe, you can see that the farther the territory is from the equator, the more it is stretched.

The map that we are considering was built using a map projection, which was invented in the 16th century by the Flemish scientist Gerard Mercator. He lived in an era when new trade routes were being laid across the oceans. Columbus discovered America in 1492, and the first circumnavigation of the world under the leadership of Magellan took place in 1519-1522 - when Mercator was 10 years old. Open lands had to be mapped, and for this it was necessary to learn how to depict a round Earth on a flat map. And the cards had to be made in such a way that it was convenient for the captains to use them.

And how does the captain use the map? He charts a course for her. Navigators of the 13th-16th centuries used portolans - maps that depicted the Mediterranean basin, as well as the coasts of Europe and Africa lying beyond Gibraltar. Such maps were marked with a grid of rhumbs - lines of constant direction. Let the captain need to sail in the open sea from one island to another. He applies a ruler to the map, determines the course (for example, "to the south-south-east") and gives the order to the helmsman to keep this course according to the compass.

Mercator's idea was to keep the principle of plotting a course on a ruler and on a world map. That is, if you keep a constant direction on the compass, then the path on the map will be straight. But how to do that? This is where mathematics comes to the rescue. Mentally cut the globe into narrow strips along the meridians, as shown in the figure. Each such strip can be deployed on a plane without much distortion, after which it will turn into a triangular figure - a “wedge” with curved sides.

However, the globe in this case turns out to be dissected, and the map should be solid, without cuts. To achieve this, we divide each wedge into "almost squares". To do this, from the lower left point of the wedge, we draw a segment at an angle of 45 ° to the right side of the wedge, from there we draw a horizontal cut to the left side of the wedge - we cut off the first square. From the point where the cut ends, we again draw a segment at an angle of 45 ° to the right side, then a horizontal one to the left, cutting off the next “almost square”, and so on. If the original wedge was very narrow, the "near-squares" will not differ much from real squares, since their sides will be almost vertical.

Let's do the final steps. Let's straighten the "almost squares" to a real square shape. As we understood, the distortions can be made as small as desired by reducing the width of the wedges into which we cut the globe. We will lay out the squares adjacent to the equator on the globe in a row. On them we lay all the other squares in order, stretching them before that to the size of the equatorial squares. Get a grid of squares of the same size. True, in this case, parallels equidistant on the map will no longer be equidistant on the globe. After all, the farther the original square on the globe was from the equator, the greater the increase it underwent when transferred to the map.

However, the angles between the directions with such a construction will remain undistorted, because each square has practically only changed in scale, and the directions do not change with a simple increase in the picture. And that's exactly what Mercator wanted when he came up with his projection! The captain can plot his course on the map along the ruler and guide his ship along this course. In this case, the ship will sail along a line that runs at the same angle to all meridians. This line is called loxodromia .

Loxodrome swimming is very convenient because it does not require any special calculations. True, the loxodrome is not the shortest line between two points on the earth's surface. Such a shortest line can be determined by pulling a thread on the globe between these points.

Artist Evgeny Panenko

Mercator projection

The conformal cylindrical projection was first proposed and applied in 1569 by the Dutch cartographer Mercator.

To derive the formulas for this projection, we first determine the scale along the parallels in the simplest of the cylindrical projections in the so-called square projection. In this projection, the meridians and parallels drawn through the same number of degrees in longitude and latitude form a grid of squares on the map, and the lengths along all meridians and the equator are preserved (equidistant projection).

Let PC0A0 and PD0B0 (Fig. 1) be meridians on a globe of radius R with an infinitesimal longitude difference, and let the straight lines

Rice. 1. Two meridians and two parallels on the globe and on the map in a cylindrical projection

SA and DB - the corresponding meridians on the map in a square projection.

Then an infinitesimal segment С0D0 of an arbitrary parallel with latitude and radius r on the globe will correspond to an infinitesimal segment CD on the map, and the scale along the parallel

CD = AB = A0 B0 ,

Where A0B0 is the arc of the equator.

Since the ratio of the arcs of the circles is equal to the ratio of their radii, then

From OS 0FROM", where OS 0FROM"= We have

Consequently,

It can be seen from the formula that the scale along the parallel in the square projection varies from unity to infinity, and it is equal to unity at the equator (at = 0°), and infinity at the pole point (at = 90°). The pole in a square projection will be depicted as a straight line segment equal in length to the equator.

Now, in order to make the scale along the meridians equal to the scale along the parallels (m=n), i.e., to switch from a square projection to an conformal one (from distortion ellipses to circles), it is necessary to stretch the meridians of the square projection at each point as many times as times the parallels of this projection are increased in relation to the corresponding parallels of the globe, i.e., in Times. Therefore, in order to transform, in the first approximation, a square cartographic grid into a cartographic grid of a conformal projection, it is necessary to multiply the segments of the meridian OA, AB, BC, etc. (Fig. 2) respectively

Rice. 2. Converting a square projection to an conformal cylindrical one

by 1, 2, 3, etc., where 1,2, 3 are, respectively, the latitudes of the midpoints of these segments. Then the meridian segment OS1 in the conformal projection, corresponding to the segment OS in the square projection, will be represented by the expression

OS1 = OA1 + A1 B1, + B1C1 \u003d OA 1 + AB 2 + BC 3 ,

And since the segments

OA = AB = BC,

OS 1 \u003d OA (1 +2 +3).

Meridian segment OS 1 will be determined more precisely, the smaller the segments that make up it are taken, since the stretching of the meridians must be continuous from the equator to this parallel.

The most accurate result will be obtained when the meridian segment D in the Mercator projection will consist of the sum of an infinitely large number of infinitesimal quantities

,

Where Dx- an infinitesimal segment of the meridian in a square projection,

DD- the infinitesimal segment of the meridian corresponding to it in the conformal Mercator projection. But due to the constancy of the scale along the meridians in a square projection, the segment

The sum of infinitesimal quantities in higher mathematics is called an integral. To take the integral of both parts of the equality means to take the sum of the infinitesimal values ​​of these parts of the equality within certain limits.

Integral of expression within the latitude value from 0 to Let's write like this

As a result of integration on the left side of the equality, we obtain the meridian segment D; the right side of the equality is a tabular integral equal to

Thus, the meridian segment

,

where C is the constant of integration.

The value C must be constant for all latitudes, so it can be easily determined by taking = 0°. At = 0°, the parallel corresponds to the equator, for which D = 0, i.e.

Consequently,

Passing from the natural logarithm to the decimal and expressing D in the main map scale and in centimeters, we will have the final working formula for calculating the meridian segment D in the conformal cylindrical projection for the ball

(29)

Where Mod=0,4343.

The formula shows that the meridian segment D for the pole (= 90°) is equal to infinity, i.e. the pole is not shown on the map in this projection.

Taking the Earth as an ellipsoid, we will have the formula

(30)

Where a is the radius of the equator of the earth's ellipsoid (expressed in meters),

U is the same value as in the formula (22) of the conformal conic projection.

The distances between the meridians in the conformal projection, as well as in the square projection, are determined by the formula

Where expressed in radian measure. Taking the Earth as an ellipsoid and expressing it on the main scale of the map and in centimeters, we will have

This formula is often written as

(31)

Where At- distance from the middle meridian of the map to the determined one,

° is the difference between the longitudes of the mean and defined meridians, expressed in degrees, °=57°.3.

It is obvious that distortions in a conformal cylindrical projection on a tangent cylinder will be expressed by the formulas

(32)

To calculate the meridian segments D, ordinates y and scales in a conformal cylindrical projection on a secant cylinder, the working formulas will look like

(34)

(35)

(37)

Where r0 is the radius of the section parallel with latitude 0 on the earth's ellipsoid,

r-radius of the parallel with the latitude on the earth's ellipsoid, which is used to determine the scale,

main map scale,

° - the difference between the longitudes of the mean and determined meridians, expressed in degrees.

Map grid in the Mercator projection

To build a cartographic grid in the Mercator projection and plot control points on the map being compiled, it is necessary to know the rectangular coordinates (meridian segment D and ordinate y) of the intersection points of the meridians and parallels and control points.

The average D value for the latitude argument is selected from special tables compiled by the Hydrographic Department of the Navy, and the y value is calculated using formula (35).

The point of intersection of the middle meridian and the main parallel of the sea basin for which charts are compiled is taken as the origin of coordinates on sea charts. This parallel is the parallel of the section, and the scale along it is equal to one.

Knowing the rectangular coordinates of the vertices of the corners of the frame of the map sheet, the dimensions of the sides of this frame are found as the difference between the meridian segments D for the southern and northern parallels and the difference between the values ​​of y for the western and eastern meridians. According to the found dimensions of the sides, a rectangle is built (the inner frame of the sheet), which will be the basis for constructing intermediate meridians and parallels of the map, as well as for drawing strong points.

Meridians and parallels in the Mercator projection are depicted as parallel and mutually perpendicular lines, therefore, to construct them, it is enough to determine the meridian segments D. For the points of intersection of the parallels of the map with the X axis and the y-ordinate for the points of intersection of the meridians of the map with the Y axis. When these values ​​are found, determine differences D - Dyu and y - y3 for the indicated points. Here Dyu is the meridian segment of the southern parallel, and the uz is the ordinate of the western meridian. These differences are deposited from the top of the southwestern corner of the frame along the western and southern sides, and lines are drawn through the deposition points parallel to the southern and lateral sides, respectively, which will be the parallels and meridians of the map.

Figure 3 Cartographic grid in the conformal cylindrical projection (Mercator)

On fig. 3 shows a cartographic grid in a conformal cylindrical projection (on a tangent cylinder) for depicting the globe. Scale values ​​in this projection are given in Table 4.

Table 4

Scales in the conformal cylindrical Mercator projection.

Due to the fact that the Mercator projection is conformal, and the meridians are depicted in it as parallel straight lines, it has one remarkable property: a line that intersects all the meridians at the same angle is depicted in this projection as a straight line. This line is called a loxodrome. A moving vessel, if it keeps the same course with the help of a compass, is actually moving along a loxodrome. This property of the Mercator projection led to its widespread use for nautical charts.

Rice. 4. Orthodromia and loxodrome on the map in the Mercator projection

Orthodromia and loxodromia

On a map drawn in the Mercator projection, it is easy and simple to mark the path of the vessel and determine its constant course, that is, the direction in which it must move in order to get from one point to another. The constant course of the ship is determined by measuring the angle between the straight line connecting these points on the map and one of the meridians with a protractor.

However, it should be noted that with a large distance between points A and B (Fig. 4), the loxodrome on the sphere deviates significantly from the orthodrome (the shortest distance between these points), which in the projection

Rice. 5. Orthodromia and loxodrome between New York and Moscow on the map in the Mercator projection.

Mercator is represented by a curved line. In this case, the navigator steers the ship not along one course, but along several, changing the direction of movement at certain points (a and b). In this case, the ship's path will be displayed on the map in the form of broken lines of chords inscribed in the great circle. In relation to the figure, the ship from point A to point AND will go under the azimuth from the point AND to point b - under azimuth, from point b to end point B - under azimuth.

For clarity, it can be indicated (Fig. 5) that between New York and Moscow the length of the great circle is 7507 km, and the loxodrome is 8371 km, i.e. the difference between their lengths is 864 km. The greatest distance of loxodrome points from orthodrome here reaches 1650 km.

The second convenience of the Mercator projection in its application for nautical charts is that it makes it easy, with sufficient accuracy for practice, to determine distances in nautical miles on the map, without resorting to building special scales, but using only divisions (in degrees or minutes) printed on the sides of the card frame. A nautical mile is 1852 m, which is approximately equal to an average length of a meridian arc of one minute.

If, for example, on the map it is required to determine the distance AB in nautical miles (Fig. 42), then, having removed the segment AB with a compass solution, apply the compass to the nearest lateral side of the map frame so that the middle of the segment - point C - is at the average latitude of points A and B (at point C1). The number of meridian minutes calculated within this segment will express the distance AB in nautical miles (in Fig. 6 segment AB \u003d 215 miles).

In conclusion, it should be noted that when compiling topographic and survey topographic maps of various scales, various maps are widely used as cartographic material. Nautical charts, compiled in a conformal cylindrical projection. Therefore, knowledge of the features of this projection is of great practical importance.

Rice. 6. Determining the distance AB in miles on the map in the Mercator projection

Exercise

Calculate the meridian segment D and the “y” ordinate in a conformal cylindrical projection on a tangent cylinder for a point with geographic coordinates = 30°, 35° (from the middle meridian taken as the X axis) at = 1:5000000. Ellipsoid of Krasovsky.

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