3. And finally final stage creating a map is to display the reduced surface of the ellipsoid on the plane, i.e. the use of map projection (a mathematical way of depicting an ellipsoid on a plane.).
The surface of an ellipsoid cannot be turned onto a plane without distortion. Therefore, it is projected onto a figure that can be deployed onto a plane (Fig). In this case, there are distortions of angles between parallels and meridians, distances, areas.
There are several hundred projections that are used in cartography. Let us further analyze their main types, without going into all the variety of details.
According to the type of distortion, projections are divided into:
1. Equal-angled (conformal) - projections that do not distort angles. At the same time, the similarity of figures is preserved, the scale changes with changes in latitude and longitude. The area ratio is not saved on the map.
2. Equivalent (equivalent) - projections on which the scale of areas is the same everywhere and the areas on the maps are proportional to the corresponding areas on the Earth. However, the length scale at each point is different in different directions. equality of angles and similarity of figures are not preserved.
3. Equidistant projections - projections, maintaining the constancy of the scale in one of the main directions.
4. Arbitrary projections - projections that do not belong to any of the considered groups, but have some other properties that are important for practice, are called arbitrary.
Rice. Projection of an ellipsoid onto a figure unfolded into a plane.
Depending on which figure the ellipsoid surface is projected onto (cylinder, cone or plane), projections are divided into three main types: cylindrical, conical and azimuthal. The type of figure on which the ellipsoid is projected determines the type of parallels and meridians on the map.
Rice. The difference in projections according to the type of figures on which the surface of the ellipsoid is projected and the type of development of these figures on the plane.
In turn, depending on the orientation of the cylinder or cone relative to the ellipsoid, cylindrical and conical projections can be: straight - the axis of the cylinder or cone coincides with the axis of the Earth, transverse - the axis of the cylinder or cone is perpendicular to the axis of the Earth and oblique - the axis of the cylinder or cone is inclined to the axis of the Earth at an angle other than 0° and 90°.
Rice. The difference in projections is the orientation of the figure onto which the ellipsoid is projected relative to the Earth's axis.
The cone and cylinder can either touch the surface of the ellipsoid or intersect it. Depending on this, the projection will be tangent or secant. Rice.
Rice. Tangent and secant projections.
It is easy to see (Fig) that the length of the line on the ellipsoid and the length of the line on the figure that it is projected will be the same along the equator, tangent to the cone for the tangent projection and along the secant lines of the cone and cylinder for the secant projection.
Those. for these lines, the map scale will exactly match the scale of the ellipsoid. For other points on the map, the scale will be slightly larger or smaller. This must be taken into account when cutting map sheets.
The tangent to the cone for the tangent projection and the secant of the cone and cylinder for the secant projection are called standard parallels.
For the azimuthal projection, there are also several varieties.
Depending on the orientation of the plane tangent to the ellipsoid, the azumuthal projection can be polar, equatorial or oblique (Fig)
Rice. Views of the Azimuthal projection by the position of the tangent plane.
Depending on the position of an imaginary light source that projects the ellipsoid onto a plane - in the center of the ellipsoid, at the pole, or at an infinite distance, there are gnomonic (central-perspective), stereographic and orthographic projections.
Rice. Types of azimuthal projection by the position of an imaginary light source.
The geographical coordinates of any point on the ellipsoid remain unchanged for any choice of map projection (determined only by the selected system of "geographical" coordinates). However, along with geographical projections of an ellipsoid on a plane, so-called projected coordinate systems are used. These are rectangular coordinate systems - with the origin at a certain point, most often having coordinates 0,0. Coordinates in such systems are measured in units of length (meters). This will be discussed in more detail below when considering specific projections. Often, when referring to the coordinate system, the words "geographic" and "projected" are omitted, which leads to some confusion. Geographical coordinates are determined by the selected ellipsoid and its bindings to the geoid, "projected" - by the selected projection type after selecting the ellipsoid. Depending on the selected projection, different "projected" coordinates may correspond to one "geographical" coordinates. And vice versa, different “geographic” coordinates can correspond to the same “projected” coordinates if the projection is applied to different ellipsoids. On the maps, both those and other coordinates can be indicated simultaneously, and the “projected” ones are also geographical, if we understand literally that they describe the Earth. We emphasize once again that it is fundamental that the "projected" coordinates are associated with the type of projection and are measured in units of length (meters), while the "geographic" ones do not depend on the selected projection.
Let us now consider in more detail two cartographic projections, the most important for practical work in archeology. These are the Gauss-Kruger projection and the Universal Transverse Mercator (UTM) projection, which are varieties of the conformal transverse cylindrical projection. The projection is named after the French cartographer Mercator, who was the first to use a direct cylindrical projection to create maps.
The first of these projections was developed by the German mathematician Carl Friedrich Gauss in 1820-30. for mapping Germany - the so-called Hanoverian triangulation. As a truly great mathematician, he solved this particular problem in a general way and made a projection suitable for mapping the entire Earth. A mathematical description of the projection was published in 1866. In 1912-19. Another German mathematician Kruger Johannes Heinrich Louis conducted a study of this projection and developed a new, more convenient mathematical apparatus for it. Since that time, the projection is called by their names - the Gauss-Kruger projection
The UTM projection was developed after World War II, when NATO countries agreed that a standard spatial coordinate system was needed. Since each of the armies of NATO countries used its own spatial coordinate system, it was impossible to accurately coordinate military movements between countries. The definition of UTM system parameters was published by the US Army in 1951.
To obtain a cartographic grid and draw up a map on it in the Gauss-Kruger projection, the surface of the earth's ellipsoid is divided along the meridians into 60 zones of 6 ° each. It is easy to see that this corresponds to the partition globe into 6° zones when building a map at a scale of 1:100,000. The zones are numbered from west to east, starting from 0°: zone 1 extends from the 0° meridian to the 6° meridian, its central meridian is 3°. Zone 2 - from 6° to 12°, etc. The numbering of nomenclature sheets starts from 180°, for example, sheet N-39 is in the 9th zone.
To link the longitude of the point λ and the number n of the zone in which the point is located, you can use the following relations:
in the Eastern Hemisphere n = (integer of λ/ 6°) + 1, where λ are degrees east
in the Western Hemisphere, n = (integer of (360-λ)/ 6°) + 1, where λ are degrees west.
Rice. Partitioning into zones in the Gauss-Kruger projection.
Further, each of the zones is projected onto the surface of the cylinder, and the cylinder is cut along the generatrix and unfolded onto a plane. Rice
Rice. Coordinate system within 6 degree zones in GC and UTM projections.
In the Gauss-Kruger projection, the cylinder touches the ellipsoid along the central meridian and the scale along it is equal to 1. Fig.
For each zone, the coordinates X, Y are measured in meters from the origin of the zone, and X is the distance from the equator (vertically!), And Y is the horizontal distance. The vertical grid lines are parallel to the central meridian. The origin of coordinates is shifted from the central meridian of the zone to the west (or the center of the zone is shifted to the east, the English term “false easting” is often used to denote this shift) by 500,000 m so that the X coordinate is positive in the entire zone, i.e. the X coordinate on the central meridian is 500,000 m.
In the southern hemisphere, a northing offset (false northing) of 10,000,000 m is introduced for the same purposes.
The coordinates are written as X=1111111.1 m, Y=6222222.2 m or
X s =1111111.0 m, Y=6222222.2 m
X s - means that the point is in the southern hemisphere
6 - the first or two first digits in the Y coordinate (respectively, only 7 or 8 digits before the decimal point) indicate the zone number. (St. Petersburg, Pulkovo -30 degrees 19 minutes east longitude 30:6 + 1 = 6 - zone 6).
In the Gauss-Kruger projection for the Krasovsky ellipsoid, all topographic maps of the USSR were compiled at a scale of 1: 500,000, and a larger application of this projection in the USSR began in 1928.
2. The UTM projection is generally similar to the Gauss-Kruger projection, but the 6-degree zones are numbered differently. The zones are counted from the 180th meridian to the east, so the zone number in the UTM projection is 30 more than the Gauss-Kruger coordinate system (St. zone).
In addition, UTM is a projection onto a secant cylinder and the scale is equal to one along two secant lines that are 180,000 m from the central meridian.
In the UTM projection, the coordinates are given as: Northern Hemisphere, zone 36, N (northern position)=1111111.1 m, E (eastern position)=222222.2 m. The origin of each zone is also shifted 500,000 m west of the central meridian and 10,000,000 m south of the equator for the southern hemisphere.
Modern maps of many European countries have been compiled in the UTM projection.
Comparison of Gauss-Kruger and UTM projections is shown in the table
| Parameter | UTM | Gaus-Kruger |
| Zone size | 6 degrees | 6 degrees |
| Prime Meridian | -180 degrees | 0 degrees (GMT) |
| Scale factor = 1 | Crossing at a distance of 180 km from the central meridian of the zone | Central meridian of the zone. |
| Central meridian and its corresponding zone | 3-9-15-21-27-33-39-45 etc. 31-32-33-34-35-35-37-38-… | 3-9-15-21-27-33-39-45 etc. 1-2-3-4-5-6-7-8-… |
| Corresponding to the center of the meridian zone | 31 32 33 34 | |
| Scale factor along the central meridian | 0,9996 | |
| False east (m) | 500 000 | 500 000 |
| False north (m) | 0 - northern hemisphere | 0 - northern hemisphere |
| 10 000 000 – Southern Hemisphere |
Looking ahead, it should be noted that most GPS navigators can show coordinates in the UTM projection, but cannot in the Gauss-Kruger projection for the Krasovsky ellipsoid (ie, in the SK-42 coordinate system).
Each sheet of a map or plan has a finished design. The main elements of the sheet are: 1) the actual cartographic image of the site earth's surface, coordinate grid; 2) sheet frame, the elements of which are determined by the mathematical basis; 3) framing (auxiliary equipment), which includes data facilitating the use of the card.
The cartographic image of the sheet is limited to the inner frame in the form of a thin line. The northern and southern sides of the frame are segments of parallels, the eastern and western sides are segments of meridians, the value of which is determined by the general system of marking topographic maps. The values of the longitude of the meridians and the latitude of the parallels that bound the map sheet are signed near the corners of the frame: longitude on the continuation of the meridians, latitude on the continuation of the parallels.
At some distance from the inner frame, the so-called minute frame is drawn, which shows the outlets of the meridians and parallels. The frame is a double line drawn into segments corresponding to the linear extent of 1 "meridian or parallel. The number of minute segments on the northern and southern sides of the frame is equal to the difference in the longitude values of the western and eastern sides. On the western and eastern sides of the frame, the number of segments is determined by the difference in the latitude values of the northern and south sides.
The final element is the outer frame in the form of a thickened line. Often it is integral with the minute frame. In the intervals between them, the marking of minute segments into ten-second segments is given, the boundaries of which are marked with dots. This makes the map easier to work with.
On maps of scale 1: 500,000 and 1: 1,000,000, a cartographic grid of parallels and meridians is given, and on maps of scale 1: 10,000 - 1: 200,000 - a coordinate grid, or kilometer, since its lines are drawn through an integer number of kilometers ( 1 km on a scale of 1:10,000 - 1:50,000, 2 km on a scale of 1:100,000, 4 km on a scale of 1:200,000).
The values of the kilometer lines are signed in the intervals between the inner and minute frames: abscissas at the ends of the horizontal lines, ordinates at the ends of the vertical ones. At the extreme lines, the full values of the coordinates are indicated, at the intermediate ones - abbreviated ones (only tens and units of kilometers). In addition to the designations at the ends, some of the kilometer lines have signatures of coordinates inside the sheet.
An important element marginal design is information about the average magnetic declination for the territory of the map sheet, related to the moment of its determination, and the annual change in magnetic declination, which is placed on topographic maps at a scale of 1: 200,000 and larger. As you know, the magnetic and geographic poles do not coincide and the arrow of the commas shows a direction slightly different from the direction to the geographic zone. The magnitude of this deviation is called the magnetic declination. It can be east or west. By adding to the value of the magnetic declination the annual change in the magnetic declination, multiplied by the number of years that have passed since the creation of the map until the current moment, determine the magnetic declination at the current moment.
In concluding the topic on the basics of cartography, let us briefly dwell on the history of cartography in Russia.
The first maps with a displayed geographical coordinate system (maps of Russia by F. Godunov (published in 1613), G. Gerits, I. Massa, N. Witsen) appeared in the 17th century.
In accordance with the legislative act of the Russian government (the boyar "verdict") of January 10, 1696 "On the removal of a drawing of Siberia on canvas with an indication of cities, villages, peoples and distances between tracts" S.U. Remizov (1642-1720) created a huge (217x277 cm) cartographic work "Drawing of all Siberian cities and lands", which is now in the permanent exhibition of the State Hermitage. 1701 - January 1 - the date on the first title page Atlas of Russia Remizov.
In 1726-34. the first Atlas of the All-Russian Empire is published, the head of the work on the creation of which was the chief secretary of the Senate I.K. Kirillov. The atlas was published in Latin, and consisted of 14 special and one general maps under the title "Atlas Imperii Russici". In 1745 the All-Russian Atlas was published. Initially, the work on compiling the atlas was led by academician, astronomer I. N. Delil, who presented in 1728 a project for compiling the atlas Russian Empire. Starting from 1739, the work on compiling the atlas was carried out by the Geographical Department of the Academy of Sciences, established on the initiative of Delisle, whose task was to compile maps of Russia. Delisle's atlas includes comments on maps, a table with the geographical coordinates of 62 Russian cities, a map legend and the maps themselves: European Russia on 13 sheets at a scale of 34 versts per inch (1:1428000), Asian Russia on 6 sheets on a smaller scale and a map of all of Russia on 2 sheets on a scale of about 206 versts per inch (1: 8700000) The Atlas was published in the form of a book in parallel editions in Russian and Latin with the application of the General Map.
When creating the Delisle atlas, much attention was paid to the mathematical basis of the maps. For the first time in Russia, an astronomical determination of the coordinates of strong points was carried out. The table with coordinates indicates the way they were determined - "for reliable reasons" or "when compiling a map" During the 18th century, a total of 67 complete astronomical determinations of coordinates were made relating to the most important cities of Russia, and 118 determinations of points in latitude were also made . On the territory of Crimea, 3 points were identified.
From the second half of the XVIII century. the role of the main cartographic and geodetic institution of Russia gradually began to be performed by the Military Department
In 1763 a Special General Staff was created. Several dozen officers were selected there, who officers were sent to remove the areas where the troops were located, the routes of their possible following, the roads along which messages passed by military units. In fact, these officers were the first Russian military topographers who completed the initial scope of work on mapping the country.
In 1797, the Card Depot was established. In December 1798, the Depot received the right to control all topographic and cartographic work in the empire, and in 1800 the Geographical Department was attached to it. All this made the Map Depot the central cartographic institution of the country. In 1810 the Kart Depot was taken over by the Ministry of War.
February 8 (January 27, old style) 1812, when the highest approved "Regulations for the Military Topographic Depot" (hereinafter VTD), which included the Map Depot as a special department - the archive of the military topographic depot. Order of the Minister of Defense Russian Federation dated November 9, 2003, the date of the annual holiday of the VTU General Staff of the Armed Forces of the Russian Federation was set - February 8.
In May 1816, the VTD was included in the General Staff, while the head of the General Staff was appointed director of the VTD. Since this year, the VTD (regardless of renaming) has been permanently part of the Main or General Staff. VTD led the Corps of Topographers, created in 1822 (after 1866, the Corps of Military Topographers)
The most important results of the work of the VTD for almost a whole century after its creation are three large maps. The first is a special map of European Russia on 158 sheets, 25x19 inches in size, on a scale of 10 versts in one inch (1:420000). The second is the military topographic map of European Russia on a scale of 3 versts per inch (1:126000), the projection of the map is conical of Bonn, longitude is calculated from Pulkovo.
The third is a map of Asian Russia on 8 sheets 26x19 inches in size, on a scale of 100 versts per inch (1:42000000). In addition, for part of Russia, especially for the border regions, maps were prepared on a half-verst (1:21000) and verst (1:42000) scale (on the Bessel ellipsoid and the Müfling projection).
In 1918, the Military Topographic Directorate (the successor of the VTD) was introduced into the structure of the All-Russian General Staff, which later, until 1940, took on different names. The corps of military topographers is also subordinate to this department. From 1940 to the present, it has been called the "Military Topographic Directorate of the General Staff of the Armed Forces."
In 1923, the Corps of Military Topographers was transformed into a military topographic service.
In 1991, the Military Topographic Service of the Armed Forces of Russia was formed, which in 2010 was transformed into the Topographic Service of the Armed Forces of the Russian Federation.
It should also be said about the possibility of using topographic maps in historical research. We will only talk about topographic maps created in the 17th century and later, the construction of which was based on mathematical laws and a specially conducted systematic survey of the territory.
General topographic maps reflect the physical state of the area and its toponymy at the time the map was compiled.
Maps of small scales (more than 5 versts in an inch - smaller than 1:200000) can be used to localize the objects indicated on them, only with a large uncertainty in coordinates. The value of the information contained is in the possibility of identifying changes in the toponymy of the territory, mainly while preserving it. Indeed, the absence of a toponym on a later map may indicate the disappearance of an object, a change in name, or simply its erroneous designation, while its presence will confirm an older map, and, as a rule, in such cases more accurate localization is possible..
Maps of large scales provide the most complete information about the territory. They can be directly used to search for objects marked on them and preserved to this day. The ruins of buildings are one of the elements included in the legend of topographic maps, and although only a few of the ruins indicated are archaeological monuments, their identification is a matter worthy of consideration.
The coordinates of the preserved objects, determined from the topographic maps of the USSR, or by direct measurements using the global space system positioning (GPS) can be used to link old maps to modern coordinate systems. However, even maps of the early-mid 19th century can contain significant distortions in the proportions of the terrain in certain areas of the territory, and the procedure for linking maps consists not only of correlating the origins of coordinates, but also requires uneven stretching or compression of individual sections of the map, which is carried out on the basis of knowing the coordinates of a large number of reference points. points (the so-called map image transformation).
After the binding, it is possible to compare the signs on the map with the objects present on the ground at the present time, or that existed in the periods preceding or following the time of its creation. To do this, it is necessary to compare the available maps of different periods and scales.
Large-scale topographic maps of the 19th century seem to be very useful when working with boundary plans of the 18th-19th centuries, as a link between these plans and large-scale maps of the USSR. Boundary plans were drawn up in many cases without substantiation at strong points, with an orientation along the magnetic meridian. Due to changes in the nature of the terrain caused by natural factors and human activity, a direct comparison of boundary and other detailed plans of the last century and maps of the 20th century is not always possible, however, a comparison of the detailed plans of the last century with a modern topographic map seems to be easier.
Another interesting possibility of using large-scale maps is their use to study changes in the contours of the coast. Over the past 2.5 thousand years, the level of, for example, the Black Sea has risen by at least a few meters. Even in the two centuries that have passed since the creation of the first maps of the Crimea in the VTD, the position of the coastline in a number of places could have shifted by a distance of several tens to hundreds of meters, mainly due to abrasion. Such changes are quite commensurate with the size of fairly large settlements by ancient standards. Identification of areas of the territory absorbed by the sea can contribute to the discovery of new archaeological sites.
Naturally, the three-verst and verst maps can serve as the main sources for the territory of the Russian Empire for these purposes. The use of geoinformation technologies makes it possible to overlay and link them to modern maps, to combine layers of large-scale topographic maps of different times, and then split them into plans. Moreover, the plans created now, like the plans of the 20th century, will be tied to the plans of the 19th century.
Modern meanings Earth parameters: Equatorial radius, 6378 km. Polar radius, 6357 km. The average radius of the Earth, 6371 km. Equator length, 40076 km. Meridian length, 40008 km...
Here, of course, it must be taken into account that the value of the “stage” itself is a debatable issue.
A diopter is a device that serves to direct (sight) a known part of a goniometric instrument to a given object. The guided part is usually supplied with two D. - eye, with a narrow slot, and subject, with a wide slit and a hair stretched in the middle (http://www.wikiznanie.ru/ru-wz/index.php/Diopter).
Based on materials from the site http://ru.wikipedia.org/wiki/Soviet _engraving_system_and_nomenclature_of_topographic_maps#cite_note-1
Gerhard Mercator (1512 - 1594) - the Latinized name of Gerard Kremer (both Latin and Germanic surnames mean "merchant"), a Flemish cartographer and geographer.
The description of the marginal design is given in the work: "Topography with the basics of geodesy." Ed. A.S. Kharchenko and A.P. Bozhok. M - 1986
Since 1938, for 30 years, the VTU (under Stalin, Malenkov, Khrushchev, Brezhnev) was headed by General M.K. Kudryavtsev. No one has held such a position in any army in the world for such a long time.
The navigator uses a map to select the most advantageous route when moving from one point to another.card called a reduced generalized image of the earth's surface on a plane, made on a certain scale and method.
Since the Earth has a spherical shape, its surface cannot be depicted on a plane without distortion. If we cut any spherical surface into parts (along the meridians) and impose these parts on a plane, then the image of this surface on it would turn out to be distorted and with discontinuities. There would be folds in the equatorial part, and breaks at the poles.
To solve navigation problems, distorted, flat images of the earth's surface are used - maps in which distortions are caused and correspond to certain mathematical laws.
Mathematically defined conditional ways of depicting on a plane the entire or part of the surface of a ball or an ellipsoid of revolution with low compression are called map projection, and the image system of the network of meridians and parallels adopted for this cartographic projection - cartographic grid.
All existing cartographic projections can be divided into classes according to two criteria: by the nature of distortions and by the method of constructing a cartographic grid.
According to the nature of the distortions, the projections are divided into conformal (or conformal), equal (or equivalent) and arbitrary.
Equal projections. On these projections, the angles are not distorted, i.e., the angles on the ground between any directions are equal to the angles on the map between the same directions. Infinitely small figures on the map, due to the property of equiangularity, will be similar to the same figures on the Earth. If the island is round in nature, then on the map in a conformal projection it will be depicted as a circle of a certain radius. But the linear dimensions on the maps of this projection will be distorted.
Equal projections. On these projections, the proportionality of the areas of the figures is preserved, i.e., if the area of any area on Earth is twice as large as another, then on the projection the image of the first area in terms of area will also be twice as large as the image of the second. However, in an equal area projection, the similarity of the figures is not preserved. The island of a round shape will be depicted on the projection in the form of an ellipse of equal area.
Arbitrary projections. These projections retain neither the similarity of figures nor the equality of areas, but may have some other special properties necessary for solving certain practical problems on them. From the charts of arbitrary projections, orthodromic projections have received the greatest use in navigation, on which great circles (great circles of the ball) are depicted by straight lines, and this is very important when using some radio navigation systems when navigating along a great circle arc.
The cartographic grid for each class of projections, in which the image of meridians and parallels has the simplest form, is called normal mesh.
According to the method of constructing a cartographic normal grid, all projections are divided into conical, cylindrical, azimuth, conditional, etc.
conical projections. The projection of the coordinate lines of the Earth is carried out according to one of the laws on the inner surface of the circumscribed or secant cone, and then, cutting the cone along the generatrix, it is turned onto a plane.
To obtain a normal straight conical grid, make sure that the axis of the cone coincides with the earth's axis PNP S (Fig. 33). In this case, the meridians are depicted as straight lines emanating from one point, and parallels as arcs of concentric circles. If the axis of the cone is at an angle to earth's axis, then such grids are called oblique conical.
Depending on the law chosen for constructing parallels, conic projections can be conformal, equal-area and arbitrary. Conic projections are used for geographic maps.
Cylindrical projections. A cartographic normal grid is obtained by projecting the coordinate lines of the Earth according to some law onto the side surface of a tangent or secant cylinder, the axis of which coincides with the axis of the Earth (Fig. 34), and then sweeping along the generatrix onto a plane.
In direct normal projection, the grid is obtained from mutually perpendicular straight lines of the meridians L, B, C, D, F, G and parallels aa", bb", ss. projection K in Fig. 34), but sections of the polar regions in this case cannot be projected.
If you rotate the cylinder so that its axis is located in the plane of the equator, and its surface touches the poles, then you get a transverse cylindrical projection (for example, a Gaussian transverse cylindrical projection). If the cylinder is placed at a different angle to the Earth's axis, then oblique cartographic grids are obtained. On these grids, meridians and parallels are shown as curved lines.
Rice. 34
Azimuthal projections. A normal cartographic grid is obtained by projecting the coordinate lines of the Earth onto the so-called picture plane Q (Fig. 35) - tangent to the Earth's pole. The meridians of the normal grid on the projection have the form of radial straight lines emanating from. the central point of the projection P N at angles equal to the corresponding angles in nature, and the parallels - concentric circles centered at the pole. The picture plane can be located at any point on the earth's surface, and the point of contact is called the central point of the projection and is taken as the zenith.
The azimuth projection depends on the radii of the parallels. By subordinating the radii of one or another dependence on latitude, various azimuthal projections are obtained that satisfy the conditions of either equiangularity or equal area.
Rice. 35
perspective projections. If a cartographic grid is obtained by projecting meridians and parallels onto a plane according to the laws of linear perspective from a constant point of view of T.Z. (see Fig. 35), then such projections are called promising. The plane can be positioned at any distance from the Earth or so that it touches it. The point of view should be on the so-called main diameter of the globe or on its continuation, and the picture plane should be perpendicular to the main diameter.
When the main diameter passes through the Earth's pole, the projection is called direct or polar (see Fig. 35); when the main diameter coincides with the plane of the equator, the projection is called transverse or equatorial, and at other positions of the main diameter, the projections are called oblique or horizontal.
In addition, perspective projections depend on the location of the point of view from the center of the Earth on the main diameter. When the point of view coincides with the center of the Earth, the projections are called central or gnomonic; when the point of view is on the surface of the Earth stereographic; when the point of view is removed at some known distance from the Earth, the projections are called external, and when the point of view is removed to infinity, they are called orthographic.
On polar perspective projections, the meridians and parallels are depicted similarly to the polar azimuth projection, but the distances between the parallels are different and are due to the position of the point of view on the line of the main diameter.
On transverse and oblique perspective projections, meridians and parallels are depicted as ellipses, hyperbolas, circles, parabolas, or straight lines.
Of the features inherent in perspective projections, it should be noted that on a stereographic projection, any circle drawn on the earth's surface is depicted as a circle; on the central projection, any large circle drawn on the earth's surface is depicted as a straight line, and therefore, in some special cases, this projection seems appropriate to use in navigation.
Conditional projections. This category includes all projections that, according to the method of construction, cannot be attributed to any of the above types of projections. They usually satisfy some pre-set conditions, depending on the purposes for which the card is required. The number of conditional projections is not limited.
Small areas of the earth's surface up to 85 km can be depicted on a plane with the similarity of the applied figures and areas preserved on them. Such flat images of small areas of the earth's surface, on which distortions can practically be neglected, are called plans.
Plans are usually drawn up without any projections by direct shooting and all the details of the area being filmed are applied to them.
Of the projections discussed above in navigation, the following are mainly used: conformal, cylindrical, azimuth perspective, gnomonic and azimuth perspective stereographic.
Scales
The map scale is the ratio of an infinitesimal line element at a given point and in a given direction on the map to the corresponding infinitesimal line element on the ground.This scale is called private scale, and each point of the map has its own particular scale, inherent only to it. On the maps, in addition to the private one, they also distinguish main Scale, which is the initial value for calculating the size of the map.
The main scale is called the scale, the value of which is preserved only along certain lines and directions, depending on the nature of the map. On all other parts of the same map, the scale value is greater or less than the main one, i.e., these parts of the map will have their own private scales.
The ratio of the private scale of the map at a given point in a given direction to the main one is called scaling up, and the difference between scaling up and unity is relative length distortion. On a conformal cylindrical projection, the scale changes when moving from one parallel to another. The parallel along which the main scale is observed is called the main parallel. As you move away from the main parallel towards the pole, the values of private scales on the same map increase and, conversely, as you move away from the main parallel towards the equator, the values of private scales decrease.
If the scale is expressed as a simple fraction (or ratio), the divisor of which is one, and the divisor is a number indicating how many units of length on the horizontal projection of a given section of the earth's surface corresponds to one unit of length on the map, then such a scale is called numerical or numerical. For example, a numerical scale of 1/100000 (1:100000) means that 1 cm on the map corresponds to 100,000 cm on the ground.
To determine the length of the measured lines, use linear scale, showing how many units of length of the highest name on the ground are contained in one unit of length of the lowest name on the map (plan).
For example, the scale of the map is “5 miles in 1 cm” or 10 km in 1 cm, etc. This means that a distance of 5 miles (or 10 km) on the ground corresponds to 1 cm on the map (plan).
A linear scale on a plan or map is placed under the frame in the form of a straight line divided into several divisions; the starting point of the linear scale is indicated by the number 0, and then against each or some of its subsequent divisions put numbers showing the distances on the ground corresponding to these divisions.
The transition from a numerical scale to a linear one is carried out by simply recalculating the measures of length.
For example, to go from a numerical scale of 1/100,000 to a linear scale, you need to convert 100,000 cm to kilometers or miles. 100,000 cm = 1 km, or approximately 0.54 miles, so this map is drawn on a scale of 1 km to 1 cm, or 0.54 miles to 1 cm.
If a linear scale is known, for example, 2 miles in 1 cm, then to switch to a numerical one, it is necessary to convert 2 miles into centimeters and record as a fraction with the numerator unit: 2 1852 100 - = 370 400 cm, therefore, the numerical scale of this map is 1 / 370400
LECTURE №4
MAP PROJECTIONS
Kartographic projections called mathematical methods of image on the plane of the surface of the earth's ellipsoid or ball. The image of the degree grid of the Earth on the map is called the cartographic grid, and the intersection points of the meridians and parallels are the nodal points.
The construction of maps includes first an image on a plane (paper) of a cartographic grid, and then filling the cells of the grid with contours and other designations geographical objects. Meshing can be done in various ways. So, when applying perspective projections the cartographic grid is obtained, as it were, by projecting nodal points from the surface of a ball onto a plane (Fig. 4) or onto another geometric surface (cone, cylinder), which then unfolds into a plane without distortion. An example of a practical construction of a cartographic grid in a perspective way northern hemisphere shown in Figure 4.
The picture plane P here touches the surface of the northern hemisphere at the point of the North Pole. Rectilinear projecting rays from the center K are the nodal points of intersection of the meridian with the equator and the parallels of 30 ° and 60 ° latitude are transferred to the picture plane. Thus, the radii of these parallels on the plane are determined. Meridians are depicted on a plane by straight lines emanating from the pole point and spaced from each other at equal angles. The figure shows half of the grid. The second half is easy to mentally imagine, and, if necessary, build.
Building a map using perspective projection methods does not require the use of higher mathematics, so they began to be used long before its development, from ancient times. Nowadays, in cartographic production, maps are built unpromising methodmi- by calculating the position of the nodal points of the cartographic grid on the plane. The calculation is performed by solving a system of equations relating the latitude and longitude of nodal points with their rectangular coordinates X and Y on surface. The equations involved are quite complex. An example of relatively simple formulas can be the following:
X=R ´ sin j
Y= R´ cos j-sinl .
In these equations R- radius (average) of the Earth, rounded off as 6370 km, and j, l- geographic coordinates of nodal points.
Classification of map projections
The projections used for the construction of geographical maps can be grouped according to different classification criteria, of which the main ones are: a) the type of "auxiliary surface" and its orientation, b) the nature of the distortions.
Classification of cartographic projections by type of auxiliarybody surface and its orientation. Cartographic grids of maps are obtained in modern production in an analytical way. However, in the names of the projections, the terms “cylindrical”, “conical” and others are traditionally preserved, corresponding to the methods of geometric constructions that were used in the past to build grids) The use of these terms in explaining these terms will help to understand the features of the cartographic grids obtained on their basis. Currently, this classification feature is treated as a type of normal cartographic grid
Cylindrical projections. When constructing cylindrical projections, they imagine that the nodal points, and hence the lines of the degree network, are projected from the spherical surface of the globe to the side surface of the cylinder, the axis of which coincides with the axis of the globe, and the diameters of both bodies are equal (Fig. 5). Using a tangent cylinder as an auxiliary surface, it is taken into account that the nodal points of the equator are A, B, C,D and others are both on the globe and on the cylinder. Other nodal points are transferred from the globe to the surface of the cylinder. Yes, dots E and F, located on the same meridian with point C, are transferred to points £ "and F\ In this case, they will be located on the cylinder on a straight line perpendicular to the equator line. This determines the shape of the meridians in this projection. Parallels to the surface of the cylinder are projected in the form of circles parallel to the equator line (for example, the parallel in which the points are located F[ and e").
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When the surface of the cylinder is turned into a plane, all the lines of the cartographic grid turn out to be straight, the meridians are perpendicular to the parallels and spaced at equal distances from each other. This is the general view of the cartographic grid built using a cylinder tangent to the globe and having a common axis with it.
For such cylindrical projections, the equator serves as the line of zero distortion, and the isocoles have the form of straight lines parallel to the equator; the main directions coincide with the lines of the cartographic grid, while the distance from the equator increases the distortion.
In these projections, projection is also used for cylinders with a diameter smaller than the diameter of the globe, and located differently relative to the globe. Depending on the orientation of the cylinder, the resulting cartographic grids (as well as the projections themselves) are called normal, oblique, or transverse. Normal Cylindrical Grids build on cylinders whose axes coincide with the axis of the globe; oblique- on cylinders, the axis of which makes an acute angle with the axis of the globe; cross grids formed by a cylinder whose axis is at right angles to the axis of the globe .
A normal cylindrical mapping grid on a tangent cylinder has a zero distortion line at the equator. The normal grid on the secant cylinder has two zero distortion lines located along the parallels of the section of the cylinder with the globe (with latitudes j1 and j2). In this case, due to the compression of the grid area between the lines of zero distortion, the length scales along the parallels are here less than the main one; to the outer side of the lines of zero distortion, they are larger than the main scale - as a result of stretching the parallels when designing from a globe to a cylinder.
The oblique cylindrical grid on the secant cylinder has a line of zero distortions in the northern part in the form of a straight line perpendicular to the middle meridian of the map and tangent to the parallel with latitude j; the appearance of the grid is represented by curved lines of meridians and parallels.
An example of a transverse cylindrical projection is the Gauss-Kruger projection, in which each transverse cylinder is used to project the surface of one Gaussian zone.
conical projections. To build cartographic grids in conic projections, normal cones are used - tangent or secant.
fig.6
fig.7
Everyone has normal conic projections the appearance of the cartographic grid is specific: meridians are straight lines converging at a point depicting the top of a cone on a plane, parallels are arcs of concentric circles with a center at the vanishing point of the meridians. Meshes built on tangent cones have one line of zero distortion, with the distance from which the distortion increases (Fig. 6). Their isocoles have the form of arcs of circles coinciding with parallels. Grids built on a secant cone (Fig. 6b) have the same appearance, but a different distribution of distortion: they have two lines of zero distortion. Between them, partial scales along the parallels are smaller than the main scale, and on the outer sections of the grid - larger than the main scale. The main directions of all normal conical grids coincide with meridians and parallels.
Azimuthal projections. Azimuthal grids are called cartographic grids, which are obtained by projecting a degree grid of a globe onto a tangent plane (Fig.). normal azimutated mesh obtained as a result of transfer to a plane tangent to the globe at the pole point (Fig. 7 A), transversenuyu- when touching the plane at the point of the equator (Fig. 7, B) and toSuyu- when transferred to a differently oriented plane (Fig. 7, AT). The appearance of the grids is clearly visible in Figure 7.
All azimuth grids have the following common properties with respect to distortion: the point of zero distortion (TND) is the point of contact between the globe and the plane (usually it is located in the center of the map); the magnitude of the distortions increases with distance in all directions from the HPS, so the isocoles of the azimuthal projections have the shape of concentric circles with the center at the HPS. The principal directions follow the radius and the lines perpendicular to them. The name of this group of projections is due to the fact that on the cartographic grid built in the azimuthal projection, at the former point of contact between the globe and the plane (i.e., at the point of zero distortion), the azimuths of all directions are not distorted
Polyconic projections. The construction of a grid in a polyconic projection can be represented by projecting sections of the globe's degree grid onto the surface of several tangent cones and then sweeping the stripes formed on the surface of the cones into the plane. General principle such a design is shown in Figure 8. The letters in Figure 8, A indicate the tops of the cones. For each, a latitudinal section of the globe surface is projected adjacent to the parallel of contact of the corresponding cone. After scanning the cones, these sections are imaged as stripes on a plane; the stripes touch along the middle meridian of the map . The final form of the grid is obtained after the elimination of gaps between the strips by stretching.
fig.8 
For the appearance of cartographic grids in a polyconic projection, it is characteristic that the meridians have the form of curved lines (except for the middle one - straight), and the parallels are arcs of eccentric circles. In polyconic projections used to build world maps, the equatorial section is projected onto a tangent cylinder, therefore, on the resulting grid, the equator has the shape of a straight line perpendicular to the middle meridian.
Map grids in polyconic projections have length scales close to the main ones in equatorial areas. Along the meridians and parallels, they are enlarged compared to the main scale, which is especially noticeable in the peripheral parts. Accordingly, in these parts, the areas are also significantly distorted.
Conditional projections. Conditional projections include such projections in which the type of the resulting cartographic grids cannot be represented on the basis of projection onto some auxiliary surface. They are often obtained analytically (based on solving systems of equations). This is a very large group of projections. Of these, they are distinguished according to the features of the appearance of the cartographic grid pseudocylindrical projections (Fig. 9). As can be seen from the figure, for pseudocylindrical projections, the equator and parallels are straight lines parallel to each other (which makes them similar to cylindrical projections), and their meridians are curved lines.
Fig.9 
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View of distortion ellipses in projections of equal area - BUT, equiangular - B, arbitrary - B, including equidistant along the meridian - G and equidistant along the parallel - D. The diagrams show the distortion of the angle 45 °
Map projections are distinguished by the nature of the distortions and by construction. By the nature of the distortions, projections are distinguished:
1) Equangular, preserving the magnitude of the angles, here a=b. Distortion ellipses look like circles of different areas.
2) Equal-sized, preserving the areas of objects. In them R=mn cos e=l; therefore, increasing the scale of lengths along the parallels causes a decrease in the scale of lengths along the meridians and distortion of angles and shapes.
3) Arbitrary, distorting angles and areas. Among them, a group of equidistant projections stands out, in which the main scale in one of the main directions is preserved.
Of great practical importance is the division of projections by territorial coverage into projections for maps of the world, hemispheres, continents and oceans, states and their parts.
Below are tables of external signs of widespread projections for different territories, compiled.
Table 1
Table for determining the cartographic grids of maps of the Eastern and Western hemispheres
How do the intervals change according to: Middle meridian and equator Meridian and equator from the center to the edges of the hemisphere | What lines represent parallels | Name of projections |
Decreases from 1 to approximately 0.7 | Curves that increase curvature with distance from the middle meridian to the extreme | Equal area equatorial azimuthal Lambert |
Decreases from 1 to approximately 0.8 | Equatorial azimuthal Ginzburg |
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Increase from 1 to approximately 2 | Arcs of circles | Equatorial stereographic |
greatly reduced | Equatorial orthographic |
table 2
Table for determining projections of cartographic grids of world maps
Frame shape, map or whole grid view | What lines represent parallels and meridians | How do the intervals along the middle meridian change with distance from the equator | Projection Name |
Rectangle frame | Parallels-straight lines, meridians-curves | Increase between parallels 70 and 80° is almost 1.5 times greater than between the equator and parallel 10 ° | Pseudocylindrical projection of TsNIIGAiK |
Grid and Rectangle Frame | Parallels and meridians - straight lines | Increase strongly: between the parallels of 60 and 80 ° approximately 3 times more than between the equator and the parallel of 20 ° | Cylindrical Mercator |
Grid and Rectangle Frame | Meridian parallels - straight lines | Increasing: parallels approximately 2 2/3 times more than between the equator and parallel 20° | Cylindrical Urmaeva |
The definition of map projections of geographical maps is determined using tables and calculations. First of all, they find out which territory is shown on the analyzed map and which table should be used when determining the projection. Then the type of parallels and meridians and the nature of the gaps between the parallels along the direct meridian are determined. The nature of the meridians is also determined: whether they are straight or only the middle meridian is straight, and the rest are curves, symmetrical with respect to the middle one. The straightness of the meridians is checked with a ruler. If the meridians turned out to be straight, specify whether they are parallel to each other. When considering parallels, find out whether the parallels are arcs of circles, curves or straight lines. This is established by comparing the sag arrows for arcs of equal chords: with equal sag arrows, the lines are arcs of circles, with unequal sag arrows, the parallels are complex curves . To determine the nature of the curvature of the line, you can also do the following. Three points of this curve are marked on a sheet of tracing paper. If, when moving the leaf along the line, all three points coincide with the curve, then this curve will be an arc of a circle. If the parallels turn out to be arcs, their concentricity should be checked, for which the distances between adjacent parallels in the middle of the map and on the edge are measured. If these distances are constant, the arcs are concentric.
Both direct conic and azimuthal polar projections have rectilinear meridians diverging from one point. A section of the direct conic grid can be distinguished from a section of the polar azimuth projection grid by measuring the angle between two meridians spaced 60-90° apart. If this angle turned out to be less than the corresponding difference in longitudes, signed on the map, then this is a conic projection, if it is equal to the difference in longitudes, it is azimuthal.
Determining the average distortion sizes for geographic objects can be done in two ways:
1) by measuring segments of meridians and parallels on the map and subsequent calculations using formulas;
2) according to maps with isocols.
In the first case, the partial scales are first calculated along the meridians (t) and parallels \(P) and express them in fractions of the main scale:
where - l1 the length of the meridian arc on the map, L1 - the length of the meridian arc on the ellipsoid, l2 - the length of the parallel arc on the map, L2 - arc length of the parallel on the ellipsoid { L1 and L2 taken from application tables; M- the denominator of the main scale.
Then they measure on the map with a protractor the angle e between the tangents to the parallel and the meridian at a given point; determine the deviation of the angle q from 90°; e = q -90°.
Based on known formulas, the distortion values are calculated R,a, b, w, to.
In the second case, isocol maps are used. From these maps, values are taken for 2-3 points of objects with an accuracy allowed by visual interpolation, then it can be established to which group this projection belongs in terms of the nature of distortions.
MAP PROJECTION AND ITS TYPES
Rationale for choosing the topic of the paragraph
For our work, we have chosen the topic "Cartographic projections". At present, this topic is practically not considered in geography textbooks; information about various cartographic projections can only be seen in the atlas of the 6th grade. We believe that it will be interesting for students to know the principles by which various projections of geographical maps are selected and built. Questions about map projections are often raised in Olympiad assignments. They also meet at the exam. In addition, atlas maps, as a rule, are built in different projections, which raises questions from students. A cartographic projection is the basis for building maps. Thus, knowledge of the basic principles of constructing cartographic projections will be useful for students when choosing the professions of a pilot, sailor, geologist. In this regard, we consider it appropriate to include this material in a geography textbook. Since at the level of the 6th grade the mathematical preparation of students is not yet so strong, in our opinion, it makes sense to study this topic at the beginning of the 7th grade in the section “General features of the nature of the Earth” when considering the material on the sources of geographical information.
Map projections
It is impossible to imagine a geographical map without a system of parallels and meridians that form it. degree network. It is they that allow us to accurately determine the location of objects, it is by them that the sides of the horizon on the map are determined. Even distances on a map can be calculated using a degree network. If you look at the maps in the atlas, you can see that the degree network looks different on different maps. On some maps, parallels and meridians intersect at right angles and represent a grid of parallel and perpendicular lines. On other maps, the meridians fan out from one melancholy, and the parallels are represented as arcs. On a map of Antarctica, the meridians look like a snowflake, and the parallels extend from the center in concentric circles.
CREATING CARDS
Cartography section deals with the creation of cartographic works. Cartography is a branch of science, production and technology covering the history of cartography and the study, creation and use of cartographic works. Maps are created using cartographic projections - a way to move from a real, geometrically complex earth's surface to a map plane. To do this, first go to the mathematically correct figure of an ellipsoid or bullet, and then project the image onto a plane using mathematical dependencies.
Types of projections
What is a map projection?
Map projection - a mathematically defined way of displaying a surface ellipsoid on surface. The system of displaying the network of meridians and parallels adopted for this map projection is called cartographic grid.
According to the method of constructing a cartographic normal mesh all projections are divided into conical, cylindrical, conditional, azimuthal, etc.
On conic projections when transferring the coordinate lines of the Earth to the plane, a cone is used. After obtaining an image on its surface, the cone is cut and unfolded onto the plane. To obtain a conical grid, an exact coincidence of the axis of the cone with the axis of the Earth is necessary. On the resulting map, parallels are depicted as arcs of circles, meridians - as straight lines emanating from one point. In such a projection, you can depict the northern or southern hemisphere of our planet, North America or Eurasia. In the process of studying geography, conic projections will most often be found in your atlases when building a map of Russia.
Map projections
On cylindrical projections obtaining a normal grid is carried out by projecting it onto the walls of a cylinder, the axis of which coincides with the Earth's axis. Then it is deployed on a plane. The grid is obtained from mutually perpendicular straight lines of parallels and meridians.
On azimuthal projections the normal grid is obtained immediately on the projection plane. For this, the center of the plane is aligned with the Earth's pole. As a result, the parallels look like concentric circles, the radius of which increases with distance from the center, and the meridians look like straight lines intersecting at the center.
Conditional projections are built according to predetermined conditions. This category cannot be attributed to other types of projection. Their number is unlimited.
Of course, it is absolutely impossible to transfer an image from the surface of a ball to a plane. If we try to do this, we will inevitably get a tear in the image. Nevertheless, we do not see these discontinuities on the map, and even when transferring the image to the surface of a cylinder, cone or plane, the image is obtained as a single image. What's the matter?
By projecting points from the surface of the globe onto the surface of the future map, we get distorted images. If we imagine the projection of the Earth's surface onto a plane in the form of a shadow, which will be obtained when the object is illuminated from the center of the Earth, then the farther the object is from the place of direct contact of the map surface with the ball, the more its image will change.
According to the nature of the distortions, all projections are divided into conformal, equal and arbitrary.
On conformal projections the angles on the ground between any directions are equal to the angles on the map between the same directions, that is, they (angles) do not have distortions. The scale depends only on the position of the point and does not depend on the direction. The angle on the ground is always equal to the angle on the map, a line that is straight on the ground is a straight line on the map. Infinitely small figures on the map, due to the property of equiangularity, will be similar to the same figures on the Earth. But the linear dimensions on the maps of this projection will be distorted. Imagine a perfectly round lake. Wherever it is located on the resulting map, its shape will remain round, but the dimensions can change significantly. The riverbed will bend in the same way as it bends on the ground, but the distance between its bends will not correspond to the real one.
Equal area projection
On equal area projections areas are not distorted, their proportionality is preserved. But the corners and shapes are strongly distorted. When transferring its outlines to the map at the point of contact between the ball and the surface of the future map, its image will be the same round. At the same time, the farther it is located from the line of contact, the more its outlines will stretch, although the area of the lake will remain unchanged.
On arbitrary projections both angles and areas are distorted, the similarity of figures will not be preserved, but they have some special properties that are not inherent in other projections, therefore they are the most used.
Maps are created either directly as a result of topographic surveys of the area, or on the basis of other maps, that is, ultimately, again as a result of surveying. Currently, the vast majority of topographic maps will be created using the aerial photography method, which allows you to get a topographic map of a vast territory in a short time. From a flying aircraft, with the help of special photographic devices, many pictures (aerial photographs) of the area are taken. Then these aerial photographs are processed on special devices. Before becoming a map, a series of aerial photographs goes through a long and difficult path in production.
Ellipsoid
All small-scale general geographical and special maps (including electronic GPS maps) are created on the basis of other maps, only on a larger scale.
Terms
degree network- a system of meridians and parallels on geographical maps and globes, which serves to count the geographical coordinates of points on the earth's surface - longitudes and latitudes.
Ellipsoid is a closed surface. An ellipsoid can be obtained from the surface of a ball if the ball is compressed (stretched) in arbitrary ratios in three mutually perpendicular directions.
normal grid- a cartographic grid for each class of projections, the image of meridians and parallels of which has the simplest form.
concentric circles- circles that have a common center and lie in the same plane.
Questions
1. What is a map projection? 2. What types of map projections do you know? 3. What branch of cartography deals with the creation of projections? 4. What determines the nature of the distortion on the map?
Work at home
1.Fill out a table in your notebook that reflects the characteristics of various map projections.
2. Determine in which projections the atlas maps were built. What kind of projection was used more often? Why?
Quest for the curious
Using additional sources of information, find the projection in which the map of the hemispheres was built.
Information resources for in-depth study of this topic
Literature on the topic
A.M. Berlyant "Map - the second language of geography: (essays on cartography)". 192p. MOSCOW. EDUCATION. 1985
People have been using maps since ancient times. The first attempts to depict were made in Ancient Greece such scholars as Eratosthenes and Hipparchus. Naturally, cartography as a science has advanced far since then. Modern maps are created using satellite imagery and computer technology, which, of course, helps to increase their accuracy. And yet, on every geographical map there are some distortions regarding natural shapes, angles or distances on the earth's surface. The nature of these distortions, and, consequently, the accuracy of the map, depends on the types of cartographic projections used to create a particular map.
The concept of map projection
Let us examine in more detail what a map projection is and what types of them are used in modern cartography.
A map projection is an image on a plane. A deeper scientific definition sounds like this: a map projection is a way of displaying points on the Earth's surface on a certain plane, in which some analytical dependence is established between the coordinates of the corresponding points of the displayed and displayed surfaces.
How is a map projection built?
The construction of any types of cartographic projections occurs in two stages.
- First, the geometrically irregular surface of the Earth is mapped onto some mathematically correct surface, which is called the reference surface. For the most accurate approximation in this capacity, the geoid is most often used - a geometric body limited water surface all seas and oceans interconnected (sea level) and having a single water mass. At every point on the surface of the geoid, gravity is applied normally. However, the geoid, like the physical surface of the planet, also cannot be expressed by a single mathematical law. Therefore, instead of the geoid, an ellipsoid of revolution is taken as the reference surface, giving it the maximum similarity to the geoid using the degree of compression and orientation in the Earth's body. This body is called earth ellipsoid or a reference ellipsoid, and in different countries they take different parameters.
- Secondly, the accepted reference surface (reference ellipsoid) is transferred to the plane using one or another analytical dependence. As a result, we get a flat map projection
Projection distortion
Have you ever wondered why the outlines of the continents differ slightly on different maps? On some map projections, some parts of the world appear larger or smaller relative to some landmarks than on others. It's all about the distortion with which the projections of the Earth are transferred to a flat surface.
But why do map projections display in a distorted way? The answer is pretty simple. A spherical surface is not possible to deploy on a plane, avoiding folds or breaks. Therefore, the image from it cannot be displayed without distortion.
Methods for obtaining projections
When studying cartographic projections, their types and properties, it is necessary to mention the methods of their construction. So, map projections are obtained using two main methods:
- geometric;
- analytical.
At the core geometric method are the laws of linear perspective. Our planet is conditionally accepted as a sphere of some radius and projected onto a cylindrical or conical surface, which can either touch or cut through it.
Projections received in a similar way are called promising. Depending on the position of the observation point relative to the Earth's surface, perspective projections are divided into types:
- gnomonic or central (when the point of view is aligned with the center of the earth's sphere);
- stereographic (in this case, the observation point is located on the reference surface);
- orthographic (when the surface is observed from any point outside the sphere of the Earth; the projection is built by transferring the points of the sphere using parallel lines perpendicular to the display surface).
Analytical method construction of cartographic projections is based on mathematical expressions connecting points on the sphere of reference and the display plane. This method is more versatile and flexible, allowing you to create arbitrary projections according to a predetermined nature of the distortion.
Types of map projections in geography
To create geographical maps, many types of projections of the Earth are used. They are classified according to various criteria. In Russia, the Kavraysky classification is used, which uses four criteria that determine the main types of cartographic projections. The following are used as characteristic classifying parameters:
- the nature of the distortion;
- the form of displaying the coordinate lines of the normal grid;
- the location of the pole point in the normal coordinate system;
- mode of application.
So, what are the types of map projections according to this classification?
Projection classification
By the nature of the distortion
As mentioned above, distortion is, in fact, an inherent property of any projection of the Earth. Any characteristic of the surface can be distorted: length, area or angle. Distortion types are:
- Conformal or conformal projections, in which azimuths and angles are transferred without distortion. The coordinate grid in conformal projections is orthogonal. Maps obtained in this way are recommended to be used to determine distances in any direction.
- Equal area or equivalent projections, where the scale of areas is stored, which is taken equal to one, i.e. areas are displayed without distortion. Such maps are used to compare areas.
- Equidistant or equidistant projections, during the construction of which the scale is preserved in one of the main directions, which is taken as unit.
- Arbitrary projections, which can contain all kinds of distortions.
According to the form of displaying the coordinate lines of the normal grid
Such a classification is the most visual and, therefore, the easiest to understand. Note, however, that this criterion applies only to projections oriented normally to the observation point. So, based on this characteristic feature, distinguish the following types of map projections:
Circular, where parallels and meridians are represented by circles, and the equator and average meridian of the grid are represented by straight lines. Such projections are used to depict the surface of the Earth as a whole. Examples of circular projections are the conformal Lagrange projection, as well as the arbitrary Grinten projection.
Azimuthal. In this case, the parallels are represented as concentric circles, and the meridians as a bundle of straight lines diverging radially from the center of the parallels. A similar kind of projection is used in a direct position to display the poles of the Earth with adjacent territories, and in a transverse position as a map of the western and eastern hemispheres familiar to everyone from geography lessons.
Cylindrical, where meridians and parallels are represented by straight, normally intersecting lines. With minimal distortion, territories adjacent to the equator or stretched along some standard latitude are displayed here.
conical, representing a development of the lateral surface of the cone, where the lines of parallels are arcs of circles centered at the top of the cone, and the meridians are guides diverging from the top of the cone. Such projections most accurately depict the territories lying in the middle latitudes.
Pseudoconic projections similar to conical ones, only the meridians in this case are depicted as curved lines symmetrical with respect to the rectilinear axial meridian of the grid.
Pseudocylindrical projections resemble cylindrical, only, as well as in pseudoconical, the meridians are depicted by curved lines symmetrical to the axial rectilinear meridian. Used to depict the entire Earth (for example, the elliptical Mollweide projection, equal area sinusoidal Sanson, etc.).
Polyconic, where the parallels are depicted as circles, the centers of which are located on the middle meridian of the grid or its continuation, the meridians are in the form of curves located symmetrically to a rectilinear
By the position of the pole point in the normal coordinate system
- Polar or normal- the pole of the coordinate system coincides with the geographic pole.
- transverse or transversion- the pole of the normal system is aligned with the equator.
- oblique or oblique- the pole of the normal coordinate grid can be located at any point between the equator and the geographic pole.
By way of application
According to the method of use, the following types of map projections are distinguished:
- solid- the projection of the entire territory onto a plane is carried out according to a single law.
- Multiband- the mapped area is conditionally divided into several latitudinal zones, which are projected onto the display plane according to a single law, but with a change in the parameters for each zone. An example of such a projection is the Mufling trapezoidal projection, which was used in the USSR for large-scale maps until 1928.
- multifaceted- the territory is conditionally divided into a number of zones in longitude, projection onto a plane is carried out according to a single law, but with different parameters for each of the zones (for example, the Gauss-Kruger projection).
- Composite, when some part of the territory is displayed on a plane using one regularity, and the rest of the territory on the other.
The advantage of both multi-lane and multi-faceted projections is the high display accuracy within each zone. However, a significant disadvantage in this case is the impossibility of obtaining a continuous image.
Of course, each map projection can be classified using each of the above criteria. So, the famous projection of the Earth Mercator is conformal (equiangular) and transverse (transversion); Gauss-Kruger projection - conformal transverse cylindrical, etc.