Cartographic map projections and explanation. Classification of map projections. Classification of projections depending on the orientation of the auxiliary cartographic surface
All cartographic projections are classified according to a number of characteristics, including the nature of distortion, the type of meridians and parallels of the normal cartographic grid, and the position of the pole of the normal coordinate system.
1. Classification of map projections
by the nature of the distortions:
a) equiangular or conformal They leave corners and the shape of contours without distortion, but have significant distortion of areas. An elementary circle in such projections always remains a circle, but its dimensions change greatly. Such projections are especially convenient for determining directions and laying out routes along a given azimuth, which is why they are always used on navigation maps.
These projections can be described by equations in characteristics of the form:
m=n=a=b=m
q=90 0 w=0 m=n
Rice. Distortions in conformal projection. World map in Mercator projection
b) equal in size or equivalent- preserve areas without distortion, but their angles and shapes are significantly distorted, which is especially noticeable in large areas. For example, on a world map, the polar regions appear greatly flattened. These projections can be described by equations of the form R = 1.
Rice. Distortions in equal area projection. World map in Mercator projection
c) equidistant (equidistant).
In these projections, the linear scale along one of the main directions is constant and is usually equal to the main scale of the map, i.e.
or A= 1, or b= 1;
d) arbitrary.
They do not save any angles or areas.
2. Classification of map projections by construction method
Auxiliary surfaces in the transition from an ellipsoid or ball to a map can be a plane, a cylinder, a cone, a series of cones, and some other geometric shapes.
1) Cylindrical projections— The projection of a ball (ellipsoid) is carried out on the surface of a tangent or secant cylinder, and then its lateral surface is turned into a plane.
In these projections, the parallels of normal grids are straight parallel lines, the meridians are also straight lines orthogonal to the parallels. The distances between meridians are equal and always proportional to the difference in longitude
Rice. View of a map grid of a cylindrical projection
Conditional projections - projections for which it is impossible to select simple geometric analogues. They are built based on any given conditions, for example, the desired type of geographic grid, a particular distribution of distortions on the map, a given type of grid, etc., obtained by transforming one or more similar projections.
Pseudocylindrical projections: parallels are depicted by straight parallel lines, meridians - by curved lines, symmetrical relative to the average rectilinear meridian, which is always orthogonal to parallels (used for maps of the world and the Pacific Ocean).
Rice. View of the map grid of pseudocylindrical projection
We assume that the geographic pole coincides with the pole of the normal coordinate system 
A) Normal (straight) cylindrical - if the axis of the cylinder coincides with the axis of rotation of the Earth, and its surface touches the ball along the equator (or cuts it along parallels) . Then the meridians of the normal grid appear in the form of equidistant parallel lines, and parallels - in the form of lines perpendicular to them. Such projections have the least distortion in tropical and equatorial regions.
b) transverse cylindrical projection - the cylinder axis is located in the equatorial plane. The cylinder touches the ball along the meridian, there are no distortions along it, and therefore, in such a projection it is most advantageous to depict territories stretching from north to south.
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c) oblique cylindrical - the axis of the auxiliary cylinder is located at an angle to the equatorial plane . It is convenient for elongated areas oriented northwest or northeast.
2) Conical projections - the surface of a ball (ellipsoid) is projected onto the surface of a tangent or secant cone, after which it is, as it were, cut along a generatrix and unfolded into a plane.
Distinguish:
· normal (straight) conical projection when the axis of the cone coincides with the axis of rotation of the Earth. Meridians are straight lines diverging from a pole point, and parallels are arcs of concentric circles. An imaginary cone touches the globe or cuts it in the region of mid-latitudes, therefore, in such a projection it is most convenient to map the territories of Russia, Canada, and the USA, elongated from west to east in mid-latitudes.
· transverse conical - the axis of the undead cone is in the equatorial plane
· oblique conical— the axis of the cone is inclined to the plane of the equator.
Pseudoconic projections- those in which all parallels are depicted as arcs of concentric circles (as in normal conical circles), the middle meridian is a straight line, and the remaining meridians are curves, and their curvature increases with distance from the middle meridian. Used for maps of Russia, Eurasia, and other continents.
Polyconic projections- projections obtained as a result of projecting a ball (ellipsoid) onto a set of cones. In normal polyconic projections, parallels are represented by arcs of eccentric circles, and meridians are curves symmetrical with respect to the right middle meridian. Most often, these projections are used for world maps.
3) Azimuthal projections — the surface of the globe (ellipsoid) is transferred to a tangent or secant plane. If the plane is perpendicular to the axis of rotation of the Earth, then it turns out normal (polar) azimuthal projection . In these projections, parallels are depicted as single-center circles, meridians - as a bunch of straight lines with a vanishing point coinciding with the center of the parallels. The polar regions of our and other planets are always mapped in this projection.
a - normal or polar projection onto the plane; V - grid in transverse (equatorial) projection;
G - grid in oblique azimuthal projection.
Rice. Azimuthal projection map grid view
If the projection plane is perpendicular to the equatorial plane, then it turns out transverse (equatorial) azimuthal projection. It is always used for hemispheric maps. And if the design is carried out on a tangent or secant auxiliary plane located at any angle to the equatorial plane, then it turns out oblique azimuthal projection.
Among azimuthal projections, several varieties are distinguished, differing in the position of the point from which the ball is projected onto the plane.
Pseudo-azimuth projections - modified azimuthal projections. In polar pseudo-azimuth projections, parallels are concentric circles, and meridians are curved lines symmetrical about one or two straight meridians. Transverse and oblique pseudo-azimuth projections have a general oval shape and are usually used for maps of the Atlantic Ocean or the Atlantic Ocean together with the Arctic Ocean.
4) Polyhedral projections — projections obtained by projecting a ball (ellipsoid) onto the surface of a tangent or secant polyhedron. Most often, each face is an equilateral trapezoid.
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3) Classification of map projections according to the position of the pole of the normal coordinate system
Depending on the pole position of the normal system R o, all projections are divided into the following:
a) straight or normal- pole of the normal system R o coincides with the geographic pole ( φ o= 90°);
b) transverse or equatorial- pole of the normal system R o lies on the surface in the equatorial plane ( φ o = 0°);
c) oblique or horizontal- pole of the normal system R o located between the geographic pole and the equator (0°< φ o<90°).
In direct projections, the main and normal grids coincide. There is no such coincidence in oblique and transverse projections.
Rice. 7. Position of the pole of the normal system (P o) in an oblique map projection
MAP PROJECTION AND ITS TYPES
Justification for choosing the topic of the paragraph
For our work we chose the topic “Map Projections”. Currently, this topic is practically not discussed in geography textbooks; information about various map projections can only be seen in the 6th grade atlas. We believe that students will be interested in knowing the principles by which various projections of geographic maps are selected and constructed. Questions about map projections are often raised in Olympiad assignments. They also appear on the Unified State Exam. In addition, atlas maps, as a rule, are built in different projections, which raises questions among students. Cartographic projection is the basis for constructing maps. Thus, knowledge of the basic principles of constructing map projections will be useful to students when choosing the professions of a pilot, sailor, and geologist. In this regard, we consider it appropriate to include this material in a geography textbook. Since at the 6th grade level 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 material about sources of geographic 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 from 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 will notice that the degree network looks different on different maps. On some maps, parallels and meridians intersect at right angles and form a grid of parallel and perpendicular lines. On other maps, meridians fan out from one melancholy, and parallels are represented as arcs. On a map of Antarctica, the meridians look like snowflakes, and the parallels extend from the center in concentric circles.
CREATING MAPS
The creation of cartographic works is carried out by the cartography section of cartography. 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 map projections - a method of transition from a real, geometrically complex earth's surface to the map plane. To do this, they first move on to a 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 depicting 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 a 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, the axis of the cone must exactly coincide with the axis of the Earth. On the resulting map, parallels are depicted as circular arcs, 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 constructing a map of Russia.
Map projections
On cylindrical projections obtaining a normal mesh 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 unfolded onto a plane. The grid is obtained from mutually perpendicular straight lines of parallels and meridians.
On azimuthal projections a normal mesh is obtained immediately on the projection plane. To do 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 in the center.
Conditional projections are built according to some predetermined conditions. This category cannot be classified with 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 this, we will inevitably end up with a tear in the image. However, we do not see these gaps on the map, and even when transferring the image to the surface of a cylinder, cone or plane, the image turns out to be uniform. What's the matter?
By projecting points from the surface of the globe onto the surface of a future map, we obtain distorted images. If we imagine projecting the Earth's surface onto a plane in the form of a shadow, which is obtained when highlighting an object from the center of the Earth, then the further the object is from the place of direct contact of the map surface with the ball, the more its image will change.
Based on the nature of distortion, all projections are divided into equiangular, equal-area and arbitrary.
On conformal projections Angles on the ground between any directions are equal to angles on the map between the same directions, that is, they (the angles) do not have distortions. The scale depends only on the position of the point and does not depend on the direction. An angle on the ground is always equal to an angle on the map, a line that is straight on the ground is a straight line on the map. Infinitesimal figures on the map, due to the property of equiangularity, will be similar to the same figures on Earth. But the linear dimensions on the maps of this projection will have distortions. Imagine a perfectly round lake. No matter where it is located on the resulting map, its shape will remain round, but the dimensions can change significantly. The river bed 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 The areas are not distorted, their proportionality is maintained. But the angles and shapes are greatly distorted. When its outline is transferred to the map at the point of contact between the ball and the surface of the future map, its image will be just as round. At the same time, the further it is located from the line of contact, the more its outlines will stretch out, although the area of the lake will remain unchanged.
On arbitrary projections Both angles and areas are distorted, the similarity of the figures is not preserved, but they have some special properties that are not inherent in other projections, which is why 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 are created using the aerial photography method, which allows you to quickly obtain a topographic map of a vast territory. Many photographs (aerial photographs) of the area are taken from a flying airplane using special photographic devices. Then these aerial photographs are processed using special devices. Before becoming a map, a series of aerial photographs goes through a long and complex process in production.
Ellipsoid
All small-scale general geographic 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 geographic maps and globes, which serves to count the geographic coordinates of points on the earth’s surface - longitudes and latitudes.
Ellipsoid- 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 mesh- 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. Which branch of cartography deals with the creation of projections? 4. What determines the nature of distortions on the map?
Work at home
1. Fill out a table in your notebook showing the characteristics of various map projections.
2. Determine in what projections the atlas maps are built. Which type of projection was used most often? Why?
A task for the curious
Using additional sources of information, find in which projection the map of the hemispheres is constructed.
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)". 192 p. MOSCOW. EDUCATION. 1985
Map projection. Types of distortions in map projections. Classification of projections
Map projection is a mathematically defined method of displaying the surface of the earth's ellipsoid on a plane. It establishes a functional relationship between the geographic coordinates of points on the surface of the earth's ellipsoid and the rectangular coordinates of these points on the plane, i.e.
X= ƒ 1 (B, L) And Y= ƒ 2 (IN,L).
Cartographic projections are classified by the nature of distortion, by the type of auxiliary surface, by the type of normal grid (meridians and parallels), by the orientation of the auxiliary surface relative to the polar axis, etc.
By nature of distortion The following projections are distinguished:
1. equiangular, which convey the magnitude of angles without distortion and, therefore, do not distort the shapes of infinitesimal figures, and the length scale at any point remains the same in all directions. In such projections, distortion ellipses are depicted as circles of different radii (Fig. 2 A).
2. equal in size, in which there are no area distortions, i.e. The ratios of areas of areas on the map and the ellipsoid are preserved, but the shapes of infinitesimal figures and length scales in different directions are greatly distorted. Infinitesimal circles at different points of such projections are depicted as equal-area ellipses having different elongations (Fig. 2 b).
3. arbitrary, in which there are distortions in different proportions of both angles and areas. Among them, equidistant ones stand out, in which the length scale along one of the main directions (meridians or parallels) remains constant, i.e. the length of one of the axes of the ellipse is preserved (Fig. 2 V).
By type of auxiliary surface for design The following projections are distinguished:
1. Azimuthal, in which the surface of the earth's ellipsoid is transferred to a tangent or secant plane.
2. Cylindrical, in which the auxiliary surface is the lateral surface of the cylinder, tangent to the ellipsoid or cutting it.
3. Conical, in which the surface of the ellipsoid is transferred to the lateral surface of the cone, tangent to the ellipsoid or cutting it.
Based on the orientation of the auxiliary surface relative to the polar axis, projections are divided into:
A) normal, in which the axis of the auxiliary figure coincides with the axis of the earth's ellipsoid; in azimuthal projections the plane is perpendicular to the normal, coinciding with the polar axis;
b) transverse, in which the axis of the auxiliary surface lies in the plane of the earth's equator; in azimuthal projections, the normal of the auxiliary plane lies in the equatorial plane;
V) oblique, in which the axis of the auxiliary surface of the figure coincides with the normal located between the earth’s axis and the equatorial plane; in azimuthal projections the plane is perpendicular to this normal.
Figure 3 shows various positions of the plane tangent to the surface of the earth's ellipsoid.
Classification of projections by type of normal grid (meridians and parallels) is one of the main ones. Based on this feature, eight classes of projections are distinguished.
a B C
Rice. 3. Types of projections by orientation
auxiliary surface relative to the polar axis.
A-normal; b-transverse; V- oblique.
1. Azimuthal. In normal azimuthal projections, meridians are depicted as straight lines converging at one point (pole) at angles equal to the difference in their longitudes, and parallels are depicted as concentric circles drawn from a common center (pole). In oblique and most transverse azimuthal projections, meridians, excluding the middle one, and parallels are curved lines. The equator in transverse projections is a straight line.
2. Conical. In normal conical projections, meridians are depicted as straight lines converging at one point at angles proportional to the corresponding differences in longitude, and parallels are depicted as arcs of concentric circles with the center at the point of convergence of the meridians. In oblique and transverse ones there are parallels and meridians, excluding the middle one, there are curved lines.
3. Cylindrical. In normal cylindrical projections, meridians are depicted as equidistant parallel lines, and parallels are depicted as lines perpendicular to them, which in general are not equidistant. In oblique and transverse projections, parallels and meridians, excluding the middle one, have the form of curved lines.
4. Polyconical. When constructing these projections, the network of meridians and parallels is transferred to several cones, each of which unfolds into a plane. Parallels, excluding the equator, are depicted by arcs of eccentric circles, the centers of which lie on the continuation of the middle meridian, which looks like a straight line. The remaining meridians are curves, symmetrical to the middle meridian.
5. Pseudo-azimuth, the parallels of which are concentric circles, and the meridians are curves that converge at the pole point and are symmetrical about one or two straight meridians.
6. Pseudoconic, in which parallels are arcs of concentric circles, and meridians are curved lines symmetrical with respect to the average rectilinear meridian, which may not be depicted.
7. Pseudocylindrical, in which parallels are depicted as parallel straight lines, and meridians as curves, symmetrical with respect to the average rectilinear meridian, which may not be depicted.
8. Circular, whose meridians, excluding the middle one, and parallels, excluding the equator, are depicted by arcs of eccentric circles. The middle meridian and equator are straight lines.
Conformal transverse cylindrical Gauss–Kruger projection. Projection zones. Counting order of zones and columns. Kilometer grid. Determining the zone of a topographic map sheet by digitizing a kilometer grid
The territory of our country is very large. This leads to significant distortions when it is transferred to a plane. For this reason, when constructing topographic maps in Russia, not the entire territory is transferred to the plane, but its individual zones, the length of which in longitude is 6°. To transfer zones, the transverse cylindrical Gauss–Kruger projection is used (used in Russia since 1928). The essence of the projection is that the entire earth's surface is depicted by meridional zones. Such a zone is obtained as a result of dividing the globe by meridians every 6°.
In Fig. Figure 2.23 shows a cylinder tangent to an ellipsoid, the axis of which is perpendicular to the minor axis of the ellipsoid.
When constructing a zone on a separate tangent cylinder, the ellipsoid and the cylinder have a common line of tangency, which runs along the middle meridian of the zone. When moving to a plane, it is not distorted and retains its length. This meridian, passing through the middle of the zone, is called axial meridian.
When the zone is projected onto the surface of the cylinder, it is cut along its generatrices and unfolded into a plane. When unfolded, the axial meridian is depicted without distortion of the straight line RR′ and it is taken as an axis X. Equator HER' also depicted by a straight line perpendicular to the axial meridian. It is taken as an axis Y. The origin of coordinates in each zone is the intersection of the axial meridian and the equator (Fig. 2.24).
As a result, each zone is a coordinate system in which the position of any point is determined by flat rectangular coordinates X And Y.
The surface of the earth's ellipsoid is divided into 60 six-degree longitude zones. The zones are counted from the Greenwich meridian. The first six-degree zone will have a value of 0°–6°, the second zone 6°–12°, etc.
The 6° wide zone adopted in Russia coincides with the column of sheets of the State Map at a scale of 1:1,000,000, but the zone number does not coincide with the number of the column of sheets of this map.
Check zones is underway from Greenwich meridian, A check columns – from meridian 180°.
As we have already said, the origin of coordinates of each zone is the point of intersection of the equator with the middle (axial) meridian of the zone, which is depicted in the projection by a straight line and is the abscissa axis. Abscissas are considered positive north of the equator and negative south. The ordinate axis is the equator. The ordinates are considered positive to the east and negative to the west of the axial meridian (Fig. 2.25).
Since the abscissas are measured from the equator to the poles, for the territory of Russia, located in the northern hemisphere, they will always be positive. The ordinates in each zone can be either positive or negative, depending on where the point is located relative to the axial meridian (in the west or east).
To make calculations convenient, it is necessary to get rid of negative ordinate values within each zone. In addition, the distance from the axial meridian of the zone to the extreme meridian at the widest point of the zone is approximately 330 km (Fig. 2.25). To make calculations, it is more convenient to take a distance equal to a round number of kilometers. For this purpose, the axis X conditionally assigned to the west 500 km. Thus, the point with coordinates is taken as the origin of coordinates in the zone x = 0, y = 500 km. Therefore, the ordinates of points lying west of the zone’s axial meridian will have values less than 500 km, and those of points lying east of the axial meridian will have values of more than 500 km.
Since the coordinates of the points are repeated in each of the 60 zones, the ordinates are ahead Y indicate the zone number.
To plot points by coordinates and determine the coordinates of points on topographic maps, there is a rectangular grid. Parallel to the axes X And Y draw lines through 1 or 2 km (taken at map scale), and therefore they are called kilometer lines, and the grid of rectangular coordinates is kilometer grid.
3. And finally, the final stage of creating a map is to display the reduced surface of the ellipsoid on a plane, i.e. the use of cartographic projection (a mathematical method of depicting the surface of 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 expanded onto a plane (Fig). In this case, distortions of angles between parallels and meridians, distances, and areas occur.
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. Conformal (conformal) – projections that do not distort angles. At the same time, the similarity of the figures is preserved, the scale changes with changes in latitude and longitude. Area ratios are not saved on the map.
2. Equal area (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 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 that maintain constant scale in one of the main directions.
4. Arbitrary projections - projections that do not belong to any of the groups considered, but have some other properties that are important for practice, are called arbitrary.
Rice. Projecting an ellipsoid onto a figure unfolded into a plane.
Depending on the shape on which the surface of the ellipsoid is projected (cylinder, cone or plane), projections are divided into three main types: cylindrical, conical and azimuthal. The type of figure onto which the ellipsoid is projected determines the appearance of parallels and meridians on the map.
Rice. The difference in projections is based on 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 based on 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 notice (Fig.) that the length of the line on the ellipsoid and the length of the line on the figure which it is projected will be the same along the equator, tangent to the cone for a tangent projection and along the secant lines of the cone and cylinder for a secant projection.
Those. for these lines, the map scale will exactly correspond to the ellipsoid scale. 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 a cone for a tangent projection and the secants of a cone and cylinder for a secant projection are called standard parallels.
There are also several varieties for azimuthal projection.
Depending on the orientation of the plane tangent to the ellipsoid, the azumuthal projection can be polar, equatorial or oblique (Fig.)
Rice. Types of Azimuthal projection based on the position of the tangent plane.
Depending on the position of the imaginary light source that projects the ellipsoid onto the plane - at the center of the ellipsoid, at the pole, or at an infinite distance, gnomonic (central perspective), stereographic and orthographic projections are distinguished.
Rice. Types of azimuthal projection based on the position of an imaginary light source.
The geographic coordinates of any point on the ellipsoid remain unchanged for any choice of map projection (they are determined only by the selected “geographical” coordinate system). However, along with geographical ones, so-called projected coordinate systems are used for projections of an ellipsoid onto a plane. These are rectangular coordinate systems - with the origin of coordinates 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 talking about coordinate systems, the words “geographic” and “projected” are omitted, which leads to some confusion. Geographic coordinates are determined by the selected ellipsoid and its references to the geoid, “projected” - by the selected projection type after selecting the ellipsoid. Depending on the selected projection, different “projected” coordinates may correspond to the same “geographical” coordinates. And on the contrary, the same “projected” coordinates can correspond to different “geographical” ones if the projection is applied to different ellipsoids. Maps can indicate both these and other coordinates at the same time, and “projected” ones are also geographical, if we take it literally that they describe the Earth. We emphasize once again that the fundamental thing is that “projected” coordinates are associated with the type of projection and are measured in units of length (meters), while “geographical” coordinates do not depend on the selected projection.
Let us now consider in more detail the two cartographic projections that are most important for practical work in archeology. These are the Gauss-Kruger projection and the Universal Transverse Mercator (UTM) projection, a type of equiangular transverse cylindrical projection. The projection is named after the French cartographer Mercator, who was the first to use direct cylindrical projection when creating 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 form 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. From now on, 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 NATO armies used its own spatial coordinate system, it was impossible to accurately coordinate military movements between countries. The UTM system definition was published by the US Army in 1951.
To obtain a cartographic grid and draw a map from it in the Gauss-Kruger projection, the surface of the earth's ellipsoid is divided along the meridians into 60 zones of 6° each. As is easy to see, this corresponds to the division of the globe into 6° zones when constructing a map at a scale of 1:100000. 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 connect the longitude of a 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 part of λ/ 6°) + 1, where λ – degrees east longitude
in the Western Hemisphere n = (integer part of (360-λ)/ 6°) + 1, where λ is degrees of western longitude.
Rice. Division into zones in the Gauss-Kruger projection.
Then 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 X, Y coordinates are measured in meters from the origin of the zone, with X the distance from the equator (vertically!), and Y 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 throughout the entire zone, i.e. the X coordinate on the central meridian is 500,000 m.
In the southern hemisphere, for the same purposes, a false northing of 10,000,000 m is introduced.
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 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).
All topographic maps of the USSR at a scale of 1:500000 were compiled in the Gauss–Kruger projection for the Krasovsky ellipsoid, and the larger use 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. Petersburg, Pulkovo -30 degrees 19 minutes east longitude 30:6+1+30=36 - 36 zone).
In addition, UTM is a projection onto a secant cylinder and the scale is equal to one along two secant lines spaced 180,000 m from the central meridian.
In the UTM projection, coordinates are given in the form: 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 are compiled in the UTM projection.
Comparison of Gauss-Kruger and UTM projections is given in the table
| Parameter | UTM | Gaus-Kruger |
| Zone size | 6 degrees | 6 degrees |
| Prime Meridian | -180 degrees | 0 degrees (Greenwich) |
| Scale coefficient = 1 | Secant at a distance of 180 km from the central meridian of the zone | Central meridian of the zone. |
| The 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-… |
| Zone corresponding to the center of the merdian | 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 show coordinates in the Gauss-Kruger projection for the Krasovsky ellipsode (i.e. 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 a section of the earth's surface, a coordinate grid; 2) a sheet frame, the elements of which are determined by a mathematical basis; 3) border design (auxiliary equipment), which includes data that facilitates the use of the card.
The cartographic image of the sheet is limited by an internal frame in the form of a thin line. The northern and southern sides of the frame are segments of parallels, the eastern and western are segments of meridians, the meaning of which is determined by the general system of layout of topographic maps. The values of the longitude of the meridians and the latitude of the parallels limiting 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, a so-called minute frame is drawn, which shows the exits of the meridians and parallels. The frame is a double line drawn into segments corresponding to the linear length 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 northern latitude and south sides.
The final element is the outer frame in the form of a thick line. Often it is integral with the minute frame. In the intervals between them, minute segments are marked into ten-second segments, the boundaries of which are marked with dots. This makes working with the map easier.
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 at scale 1: 10,000 - 1: 50,000, 2 km at scale 1: 100,000, 4 km at scale 1: 200,000).
The values of kilometer lines are signed in the intervals between the inner and minute frames: abscissas at the ends of horizontal lines, ordinates at the ends of vertical ones. The full coordinate values are indicated at the extreme lines, and abbreviated coordinates are indicated at the intermediate lines (only tens and units of kilometers). In addition to the markings at the ends, some of the kilometer lines have coordinate signatures inside the sheet.
An important element of the border design is information about the average magnetic declination for the territory of the map sheet, relating to the time 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 shows a direction slightly different from the direction to the geographic zone. The magnitude of this deviation is called magnetic declination. It can be eastern or western. 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 from the creation of the map to the current moment, determine the magnetic declination at the current moment.
To conclude the topic about the basics of cartography, let us briefly dwell on the history of cartography in Russia.
The first maps with a displayed geographic 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 (boyar “sentence”) of January 10, 1696 “On making a drawing of Siberia on canvas with indications thereof 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”, now on permanent display at the State Hermitage. 1701 - January 1 – the date on the first title page of Remizov’s Atlas of Russia.
In 1726-34. The first Atlas of the All-Russian Empire is published, the head of 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 map 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 and astronomer I. N. Delisle, who in 1728 presented a project for compiling an atlas of the Russian Empire. Beginning in 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. The Delisle Atlas includes comments on the maps, a table with geographic 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 book form in parallel editions in Russian and Latin with the appendix 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, astronomical determination of the coordinates of strongholds was carried out. The table with coordinates indicates the method of their determination - “on reliable grounds” or “when compiling a map.” During the 18th century, a total of 67 complete astronomical determinations of coordinates related to the most important cities of Russia were made, and 118 determinations of points by latitude were also made . 3 points were identified on the territory of Crimea.
From the second half of the 18th century. The role of the main cartographic and geodetic institution of Russia gradually began to be played by the Military Department
In 1763, the Special General Staff was created. Several dozen officers were selected there, and the officers were sent to survey the areas where the troops were located, their possible routes, and the roads along which communications by military units passed. In fact, these officers were the first Russian military topographers who completed the primary 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 topographical and cartographic work in the empire, and in 1800 the Geographical Department was annexed to it. All this made the Map Depot the central cartographic institution of the country. In 1810, the Card Depot came under the jurisdiction of the War Ministry.
February 8 (January 27, old style) 1812, when the “Regulations for the Military Topographical Depot” (hereinafter VTD) was approved by the highest authorities, into which the Map Depot was included as a special department - the archive of the military topographical depot. By order of the Minister of Defense of the Russian Federation dated November 9, 2003, the date of the annual holiday of the VTU General Staff of the Russian Armed Forces was set as February 8.
In May 1816, the VTD was introduced into the General Staff, and 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. The 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 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 a military topographic map of European Russia on a scale of 3 versts per inch (1:126000), Bonn conical map projection, longitude calculated from Pulkovo.
The third is a map of Asian Russia on 8 sheets measuring 26x19 inches, on a scale of 100 versts per inch (1:42000000). In addition, for part of Russia, especially for the border areas, maps were prepared in half-verst (1:21000) and verst (1:42000) scales (on the Bessel ellipsoid and the Müfling projection).
In 1918, the Military Topographical Directorate (successor to the VTD) was introduced into the newly created All-Russian General Staff, which subsequently took on different names until 1940. The corps of military topographers is also subordinate to this department. From 1940 to the present, it has been called the “Military Topographical 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 Topographical Service of the Armed Forces of Russia was formed, which in 2010 was transformed into the Topographical 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 per inch - finer than 1:200000) can be used to localize the objects indicated on them, only with great uncertainty in the coordinates. The value of the information contained in the possibility of identifying changes in the toponymy of the territory, mainly in its preservation. 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.
Large-scale maps provide the most complete information about the territory. They can be directly used to search for objects marked on them that have survived to the present day. Building ruins are one of the elements included in the legend of topographic maps, and although few of the designated ruins are archaeological sites, their identification is a matter worthy of consideration.
The coordinates of surviving objects, determined from topographic maps of the USSR, or by direct measurements using the global space positioning system (GPS), can be used to link old maps to modern coordinate systems. However, even maps of the early to mid-19th century may 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 coordinate origins, but requires uneven stretching or compression of individual sections of the map, which is carried out on the basis of knowledge of the coordinates of a large number of reference points. points (the so-called transformation of the map image).
After the binding has been carried out, it is possible to compare the signs on the map with objects present on the ground at the present time, or that existed in periods preceding or subsequent to the time of its creation. To do this, it is necessary to compare existing 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 justification at strong points, with orientation along the magnetic meridian. Due to changes in the nature of the area 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 detailed plans of the last century with a contemporary topographic map seems simpler.
Another interesting possibility of using large-scale maps is their use to study changes in coastal contours. Over the past 2.5 thousand years, the level of, for example, the Black Sea has increased by at least several meters. Even over the two centuries that have passed since the creation of the first maps of 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 settlements that were quite large by ancient standards. Identification of areas of territory absorbed by the sea may contribute to the discovery of new archaeological sites.
Naturally, the main sources on the territory of the Russian Empire for these purposes can be three-verst and verst maps. The use of geographic information technologies makes it possible to overlay and link them to modern maps, combine layers of large-scale topographic maps from different times and further 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 values of the Earth's parameters: Equatorial radius, 6378 km. Polar radius, 6357 km. The average radius of the Earth is 6371 km. Length of the equator, 40076 km. Meridian length, 40008 km...
Here, of course, we must take into account that the size of the “stage” itself is a debatable issue.
A diopter is a device used to direct (sight) a known part of a goniometer instrument towards a given object. The guided part is usually equipped with two D. - ocular, with a narrow slot, and substantive, 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_system_of_engraving_and_nomenclature_of_topographic_maps#cite_note-1
Gerhard Mercator (1512 - 1594) is the Latinized name of Gerard Kremer (both Latin and Germanic surnames mean "merchant"), a Flemish cartographer and geographer.
A description of the frame design is given from 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.
Cartographic projections- these are mathematical methods of depicting the surface of the globe (ellipsoid) on a plane.
Most accurately, the shape of the Earth is conveyed by the globe, because it is as spherical as our planet. But globes take up a lot of space, they are difficult to take on the road, and cannot be included in a book. They have a very small scale; they cannot show in detail a small area of the earth's surface.
There are many map projections. The most common - azimuthal, cylindrical, conical. Depending on the type of map projection, the greatest distortions may be in one place or another on the map, and the degree network may look different.
Which projection to choose depends on the purpose of the map, the size of the depicted territory and the latitude at which it is located. For example, for countries elongated in mid-latitudes, such as Russia, it is convenient to use a conical projection, for polar regions an azimuthal projection, and for maps of the world, individual continents, and oceans, a cylindrical projection is often used.
