Date: 24.10.2015
map projection- a mathematical way of depicting the globe (ellipsoid) on a plane.
For projecting a spherical surface onto a plane use auxiliary surfaces.
By type auxiliary cartographic projection surface is divided into:
Cylindrical 1(auxiliary surface is the side surface of the cylinder), conical 2(lateral surface of the cone), azimuth 3(the plane, which is called the picture plane).
Also allocate polyconical
pseudocylindrical conditional
and other projections.
Orientation auxiliary figures of the projection are divided into:
- normal(in which the axis of the cylinder or cone coincides with the axis of the Earth model, and the picture plane is perpendicular to it);
- transverse(in which the axis of the cylinder or cone is perpendicular to the axis of the Earth model, and the picture plane is or parallel to it);
- oblique, where the axis of the auxiliary figure is in an intermediate position between the pole and the equator.
Cartographic distortion- this is a violation of the geometric properties of objects on the earth's surface (lengths of lines, angles, shapes and areas) when they are displayed on a map.
The smaller the scale of the map, the more significant the distortion. On large scale maps, distortion is negligible.
There are four types of distortions on the maps: lengths, areas, corners and forms objects. Each projection has its own distortions.
According to the nature of distortions, map projections are divided into:
- equiangular, which store the angles and shapes of objects, but distort the lengths and areas;
- equal, in which areas are stored, but the angles and shapes of objects are significantly changed;
- arbitrary, in which the distortions of lengths, areas and angles, but they are evenly distributed on the map. Among them, projections are especially distinguished, in which there are no distortions of lengths either along parallels or along meridians.
Zero Distortion Lines and Points- lines along which there are also points where there are no distortions, since here, when projecting a spherical surface onto a plane, the auxiliary surface (cylinder, cone or picture plane) was tangents to the ball.
Scale indicated on the cards, persists only on lines and at zero-distortion points. It's called the main one.
In all other parts of the map, the scale differs from the main one and is called partial. To determine it, special calculations are required.
To determine the nature and magnitude of distortion on the map, you need to compare the degree grid of the map and the globe.
on the globe all parallels are at the same distance from each other, all meridians are equal and intersect with parallels at right angles. Therefore, all cells of the degree grid between adjacent parallels have the same size and shape, and the cells between the meridians expand and increase from the poles to the equator.
To determine the amount of distortion, distortion ellipses are also analyzed - elliptical figures formed as a result of distortion in a certain projection of circles drawn on a globe of the same scale as the map.
Conformal projection the distortion ellipses are shaped like a circle, the size of which increases depending on the distance from the zero distortion points and lines.
In an equal area projection distortion ellipses have the shape of ellipses, the areas of which are the same (the length of one axis increases, and the second decreases).
Equidistant projection distortion ellipses have the shape of ellipses with the same length of one of the axes.
The main signs of distortion on the map
- If the distances between the parallels are the same, then this indicates that the distances along the meridians are not distorted (equidistant along the meridians).
- Distances are not distorted by parallels if the radii of the parallels on the map correspond to the radii of the parallels on the globe.
- Areas are not distorted if the cells created by the meridians and parallels at the equator are squares, and their diagonals intersect at right angles.
- The lengths along the parallels are distorted, if the lengths along the meridians are not distorted.
- The lengths are distorted along the meridians, if the lengths along the parallels are not distorted.
The nature of distortions in the main groups of cartographic projections
| Map projections | distortion |
| Equangular | Preserve angles, distort areas and lengths of lines. |
| isometric | They preserve areas, distort angles and shapes. |
| Equidistant | In one direction they have a constant length scale, the distortions of angles and areas are in equilibrium. |
| Arbitrary | Distort corners and squares. |
| Cylindrical | There are no distortions along the line of the equator, but they increase with the degree of approach to the poles. |
| conical | There are no distortions along the parallel of contact between the cone and the globe. |
| Azimuthal | There are no distortions in the central part of the map. |
map projection
Map projections can be classified in two main ways:
By the nature of the distortions;
By the form of meridians and parallels of a normal cartographic grid.
A cartographic grid is called normal if the meridians and parallels on the map in a given projection are depicted by simpler lines than the coordinate lines of any other spherical coordinate system.
According to the nature of the distortions, the projections are divided into conformal (conformal), equal-sized (equivalent), equidistant and arbitrary.
equiangular (conformal)) are called such projections in which infinitesimal figures on the map are similar to the corresponding figures on the globe. In these projections, an infinitesimal circle taken on the globe at any of its points, when transferred to a map, will also be depicted as an infinitesimal circle, i.e., the distortion ellipse in conformal projections turns into a circle. In conformal projections in infinitesimal figures on a map and on a globe, the corresponding angles are equal to each other, and the sides are proportional. For example, in fig. 15a, b AoMoKo= AMK, a 
. The scales along the meridian and the parallel are equal to each other, i.e. T=n. The angle between the meridians and parallels on the map = 90°, and the general formulas from the theory of distortions are
= t = n = a =B, P \u003d t2, = 0.
Scale equality shows that the scale at any point on the map in conformal projections does not depend on the direction. But
Rice. 1. An infinitely small circle on the globe and on the map in a conformal projection
When moving from point to point (when the coordinates of the point change), the scale changes. This means that infinitely small circles of the same size, taken at different points of the globe, will also be depicted on the map as infinitely small circles, but of different sizes (in this case, an infinitely small circle on the globe can be understood as a circle with a diameter of about 1 cm).
equal (equivalent) such projections are called in which the scale of the area at all points of the map is equal to one. In these projections, an infinitesimal circle (Fig. 2 a),
Rice. 2. A circle on a globe and an ellipse on a map in an equal area projection
Taken on the globe, it will be depicted on the map as an infinitely small ellipse equal in area (Fig. 2 b).
Since the area of the ellipse
and the area of a circle, according to the formula
Then for these projections the equality will be true
At =1, the property of projections being equal in size is analytically expressed by the equality
P = Ab = L.
So, in equal-area projections, the product of scales in the main directions is equal to one.
If conformal projections preserve the equality of angles only in infinitesimal figures, then equal-area projections preserve the areas of any figures, regardless of their size on the map. In these projections, the angles between meridians and parallels on the map may not be equal to 90°. It should be remembered that the properties of equiangularity and equivalence in one projection are incompatible, that is, there cannot be such projections that would simultaneously maintain equality of angles and equality of areas at all points of the map.
Equidistant such projections are called in which at each point of the map the lengths in one of the main directions are preserved. In these projections, a \u003d Or b \u003d. For =1, the equidistant property is analytically expressed by the equality
A=1 Or B=1 .
Sometimes equidistant projections are also understood in which the ratio or remains constant, although not equal to unity.
In equidistant projections, a circle taken at any point on the globe (Fig. 3 a) will be depicted on the map as an ellipse (Fig. 3 b or 3 c), one of the semi-axes of which will be equal to the radius of this circle.
By the nature of the distortions, these projections occupy a middle position between conformal and equal-area projections. Without preserving either angles or areas, they distort angles less than equal-area projections, and less than conformal projections, distort areas, and therefore are used in cases where there is no need to maintain equality of angles by increasing the distortion of areas, or, conversely, due to increase the distortion of corners to maintain equality of areas.
Arbitrary projections are those that do not have the properties of equiangularity, equidistance, or equidistance. The class of arbitrary projections is the most extensive; projections that differ sharply from each other in the nature of distortions can be included here.
Arbitrary projections are mainly used for small-scale maps, in particular for hemisphere and world maps, and in some cases for large-scale maps.
Rice. 3. Circle on the globe and ellipses on the map in an equidistant projection
According to the type of meridians and parallels of the normal cartographic grid, projections are divided into conical, cylindrical, azimuthal, pseudoconical, pseudocylindrical, polyconical and others. Moreover, within each of these classes, there may be projections of different nature of distortion (equiangular, equal, etc.).
Conic projections
Conic projections are such projections in which the parallels of the normal grid are represented by arcs of concentric circles, and the meridians are their radii, the angles between which on the map are proportional to the corresponding longitude differences in nature.
Geometrically, a cartographic grid in these projections can be obtained by projecting meridians and parallels onto the lateral surface of the cone, followed by unfolding this surface into a plane.
Imagine a cone tangent to the globe along some parallel AoBoCo (Fig. 4). Let's continue the planes of geographic meridians and parallels of the globe until they intersect with the surface of the cone. The lines of intersection of these planes with the surface of the cone will be taken as the images of the meridians and parallels of the globe, respectively. We cut the surface of the cone along the generatrix and expand it into a plane; then we will get a cartographic grid on the plane in one of the conic projections (Fig. 5).
Parallels from the globe to the surface of the cone can also be transferred in other ways, namely: by projecting rays emanating from the center of the globe or from some point located on the axis of the cone, by laying projections on the meridians in both directions from the parallel of contact of the rectified arcs of the meridians of the globe, enclosed between the parallels, and the subsequent drawing through the points of deposition of concentric circles from the point S (Fig. 5), as from the center. In the latter case, the parallels on the plane will be located at the same distance from each other as on the globe.
With the above methods of transferring the geographic grid from the globe to the surface of the cone, the parallels on the plane will be
Fig.4 Cone touching the Globe along the parallel.
Rice. 5 Deposits of concentric circles.
The cartographic grid in the conic projection will be depicted as arcs of concentric circles, and the meridians will be straight lines emanating from one point and forming angles between themselves proportional to the corresponding longitude differences.
The last dependence can be expressed by the equation
Where is the angle between adjacent meridians on the map, called the angle of convergence, or convergence, of the meridians on the plane,
The difference in longitudes of the same meridians,
The coefficient of proportionality, called the conic projection index. In conic projections Always less than one.
The radii of the Parallels on the map depend on the latitude of these parallels, i.e.
Thus, a cartographic grid can be immediately built on a plane, bypassing the projection onto the auxiliary surface of the cone, if the index AND the relationship between and is known.
When choosing conic projections for the image of a given territory, it is necessary to find such a value of a and such a dependence of p on cp in order to obtain the projection required by the nature of the distortion (equiangular, equal area, equidistant or arbitrary) with the least possible distortion in general.
The cone in relation to the globe can be located differently. The axis of the cone can coincide with the polar axis of the PP globe, form an angle of 90° with it, and finally intersect it at an arbitrary angle. In the first case, conic projections are called normal (direct), in the second - transverse and in the third - oblique. On fig. 7 shows the position of the cones for normal (a), transverse (b) and oblique (c) conic projections. Each of them, in turn, can be on a tangent or secant cone.
Obviously, in the transverse and oblique conic projections, with any methods of projection from the globe to the surface of the cone, the meridians and parallels will be displayed as complex curved lines. Converging straight lines and concentric circles on the surface of the cone in these cases, respectively, will depict the arcs of large circles passing through the points of intersection of the axis of the cone with the surface of the globe, and the arcs of small circles perpendicular to them. The indicated arcs of great circles on the sphere are called verticals, and the arcs of small circles are called almucantarates.
The cartographic grid has the simplest form in normal conic projections, in which it is called the normal or straight grid. In transverse projections, the cartographic grid is called transverse, and in oblique projections, it is called oblique.
In all normal conic projections, with the exception of conformal projections, the pole is represented by an arc. In conformal conic projections, the pole is represented by a dot.
The view of the cartographic grid in normal conic projections for the image of the northern hemisphere is shown in fig. 8 (equidistant conic).
In normal conic projections, the lines of zero distortion are the parallels of the section or the parallel of tangency, and the isocoles coincide with the parallels. Distortions grow in both directions as you move away from these parallels, and the scale along the parallels
On the map, between the parallels, the section is always less than one, on the parallels of contact and on the parallels of the section it is equal to one, and in other places it is greater than one and increases with distance from these parallels to the poles. Analytically, conic projections on a tangent cone are characterized by the expression
And on the secant cone - by the expression
Where is the minimum scale along the parallel.
Conic projections have found wide application for depicting territories stretched out in a narrow or wide strip along the parallels. In the first case, it is more advantageous to use conic projections on a tangent cone, in the second - on a secant cone. In particular, conic projections on a secant cone are widely used for maps of Ukraine.
It is advantageous to use transverse and oblique conic projections, respectively, for maps of countries stretched along arcs of small circles parallel to the axial meridian and arcs of small circles of an arbitrary direction, but these projections, due to the complexity of their calculation, have not found practical application.
Cylindrical projections
Cylindrical projections are such projections in which the parallels of the normal grid are depicted as parallel lines, and the meridians are equidistant lines perpendicular to the lines of parallels.
Geometrically, a cartographic grid in these projections can be obtained by projecting the meridians and parallels of the globe onto the side surface of the cylinder, followed by unfolding this surface into a plane.
Fig.8. Cartographic grid in equidistant conic projection.
Imagine a cylinder touching the globe along the equator (Fig. 9). Let us continue the planes of geographic meridians and parallels until they intersect with the side surface of the cylinder. Let us take, respectively, for the images of meridians and parallels on the surface of the cylinder, the lines of intersection of the indicated planes with the surface of the cylinder. We cut the surface of the cylinder along the generatrix and unfold it into a plane. Then on this plane a cartographic grid will be obtained in one of the cylindrical projections, as well as in conical projections, the parallels of the normal cartographic grid can be transferred to the surface of the cylinder in other ways, namely: by projecting rays emanating from the center of the globe or from some point located on the axis cylinder by laying on the meridians of the projection in both directions from the equator of the rectified arcs of the meridians of the globe, enclosed between the parallels, and then drawing straight lines parallel to the equator through the deposition points. In the latter case, the parallels on the map will be located at the same distance from each other.
The considered cylindrical projection (Fig. 9) is a projection on a tangent cylinder. In the same way, one can construct a projection on a secant cylinder.
Figure 10 shows a cylinder traversing the globe along the parallels AFB and CKD. Obviously, in the first case on the equator (Fig. 9), and in the second case on the parallels of the section AFB and CKD (Fig. 10), the scale on the map will be equal to the main one, i.e. the equator
Rice. 9. A cylinder touching the globe along the equator, and a part of the surface of the cylinder turned into a plane and the indicated parallels of the section will retain their length on the map. The cylinder in relation to the globe can be located differently.
Rice. 10. Cylinder cutting the globe along parallels
Depending on the position of the axis of the cylinder relative to the axis of the globe, cylindrical projections, like conic projections, can be normal, transverse, and oblique. In accordance with this, the cartographic grid in these projections will have the name normal, transverse and oblique. Transverse and oblique cartographic grids in cylindrical projections look like complex curved lines.
As in the case of conic projections, to construct normal grids of cylindrical projections, there is no need to project the surface of the globe first onto a cylinder, and then unfold the latter into a plane. To do this, it is enough to know the rectangular coordinates x and y of the intersection points of parallels and meridians on the plane. Moreover, in cylindrical projections, the abscissas x express the removal of parallels from the equator, and the ordinates y - the removal of meridians from the middle (axial) meridian.
Based on this, the general equations of all normal cylindrical projections can be represented as:
Where C is a constant factor, which is the radius of the equator (for projections on a tangent cylinder) or the radius of the parallel section of the globe (for projections on a secant cylinder),
I - latitude and longitude of the given point, expressed in radian measure,
X, y - rectangular coordinates of the same point on the map. Depending on the choice of the function, Cylindrical projections can be conformal, equal-area, equidistant or arbitrary by the nature of the distortion. The dependence of x on the mean also determines the distances between the parallels on the map. The distances between the meridians depend on the factor C. Thus, choosing one or another dependence of x on and one or another value of C, you can get the required projection both in terms of the nature of the distortions and their distribution relative to the equator or the middle parallel of the map (parallel of the section).
Fig 11 Cartographic grid in a square cylindrical projection.
The view of the cartographic grid in normal cylindrical projections for the image of the entire earth's surface is shown in fig. 11 (square cylindrical projection).
In cylindrical projections, as well as in conical ones, the lines of zero distortion in normal cartographic grids are the parallels of the section or the tangency parallel, and the isocoles coincide with the parallels. Distortions increase with distance from the tangent parallel (parallels of the section) in both directions.
Normal cylindrical projections are mainly used to depict territories elongated along the equator, and relatively rarely to depict territories elongated along an arbitrary parallel, since in the latter case they give greater distortions than conical projections.
In transverse and oblique cylindrical projections, the line of zero distortion is the arc of the great circle along which the cylinder touches the ball or ellipsoid. Isocoles are depicted as straight lines parallel to the zero distortion line, and distortion increases on both sides of the zero distortion line.
Transverse cylindrical projections are used to depict territories stretched along the meridian, and oblique projections are used to depict territories stretched in an arbitrary direction along a great circle arc.
Azimuthal projections
Azimuthal (zenithal) projections are those in which the parallels of the normal grid are depicted by concentric circles, and the meridians are their radii, the angles between which are equal to the corresponding longitude differences in nature. Geometrically, the cartographic grid in these projections can be obtained as follows. If planes are drawn through the axis of the globe and meridians until they intersect with a plane tangent to the globe at one of the poles, then meridians are formed on the latter in the azimuthal projection. In this case, the angles between the meridians on the plane will be equal to the corresponding dihedral angles on the globe, i.e., the differences in the longitudes of the meridians. To obtain parallels in the azimuth projection from the point of intersection of the meridians of the projection, as from the center, one should draw concentric circles with radii equal, for example, to straightened arcs of the meridians from the pole to the corresponding parallels. With such radii of parallels, an equidistant azimuthal projection will be obtained
The plane can not only touch, but also cut the surface of the globe in some small circle, from this the essence of the azimuthal projection does not change. Just as in conic projections, depending on the location of the plane relative to the polar axis of the globe, the cartographic grid in azimuth projections can be normal (straight), transverse and oblique. With a normal cartographic grid, the plane touches the globe at one of the poles, with a transverse grid, at a point lying on the equator, and with an oblique one, at some arbitrary point with a latitude greater than 0° and less than 90°. Normal azimuth projections are also called polar, transverse - equatorial and oblique - horizontal azimuthal projections.
Based on the definition of normal azimuthal projections, their general equations can be expressed as follows
Depending on the nature of the relationship between the radius of the parallel on the map and its latitude, the azimuthal projections, by the nature of the distortions, can be equiangular, equal in area, equidistant and arbitrary.
Figure 12 Cartographic grid and isocols of angles in oblique azimuth projection.
In azimuth projections on the tangent plane, the point of contact of the ball or ellipsoid is the point of zero distortion, and in projections on the cutting plane, the section circle serves as the line of zero distortion. In both cases, the isocoles look like concentric circles coinciding with the parallels of the normal grid. Distortion increases as you move away from the zero distortion point (from the zero distortion line).
Normal, transverse and oblique azimuth projections are widely used to depict areas that have a rounded shape. In particular, for the image of the northern and southern hemispheres, only normal projections are used, and for the western and eastern hemispheres, only transverse azimuth projections. Oblique azimuth projections are used for maps of individual continents. The view of the cartographic grid and isocol angles in one of the oblique azimuthal projections is shown in Fig. 12. A special case of azimuthal projections are perspective projections.
Perspective projections are those in which parallels and meridians from a ball or ellipsoid are transferred to a plane according to the laws of linear perspective, that is, with the help of direct rays emanating from the so-called point of view. In this case, a mandatory condition is accepted that the point of view is on the main beam, i.e., on a line passing through the center of the ball or ellipsoid, and the projection plane (picture plane) is perpendicular to this beam.
Classification of map projections - 4.2 out of 5 based on 6 votes
Map projections maps of the entire surface of the earth's ellipsoid (see Earth's ellipsoid) or any part of it onto a plane, obtained mainly for the purpose of constructing a map. Scale. K. items are built on a certain scale. Mentally reducing the earth's ellipsoid into M times, for example, 10,000,000 times, they get its geometric model - Globe, the image of which is already life-size on a plane gives a map of the surface of this ellipsoid. Value 1: M(in example 1: 10,000,000) defines the main, or general, scale of the map. Since the surfaces of an ellipsoid and a sphere cannot be unfolded onto a plane without ruptures and folds (they do not belong to the class of developable surfaces (see Developable surface)), distortions in the lengths of lines, angles, and so on are inherent in any C.P. characteristic of any map. The main characteristic of a C.P. at any point is the partial scale μ. This is the reciprocal of the ratio of the infinitesimal segment ds on the earth's ellipsoid to its image dσ on the plane: μ min ≤ μ ≤ μ max , and equality here is possible only at certain points or along some lines on the map. Thus, the main scale of the map characterizes it only in general terms, in some average form. Attitude μ/M called the relative scale, or increase in length, the difference M = 1. General information. Theory of K. p. - Mathematical cartography -
aims to study all types of distortions of mappings of the surface of the earth's ellipsoid onto a plane and to develop methods for constructing such projections in which the distortions would have either the smallest (in some sense) values or a predetermined distribution. Proceeding from the needs of cartography (see Cartography), in the theory of cartography, maps of the surface of the earth's ellipsoid onto a plane are considered. Since the earth's ellipsoid has a small compression, and its surface slightly recedes from the sphere, and also due to the fact that K. n. are necessary for compiling maps on medium and small scales ( M> 1,000,000), we often confine ourselves to mapping onto the plane of a sphere of some radius R, whose deviations from the ellipsoid can be neglected or taken into account in some way. Therefore, in what follows we mean maps onto the plane hoy sphere referred to the geographic coordinates φ (latitude) and λ (longitude). The equations of any K. p. have the form x = f 1 (φ, λ), y = f 2 (φ, λ), (1) where f 1 and f 2 - functions that satisfy some general conditions. Images of meridians λ = const and parallels φ = const in a given map they form a cartographic grid. The K. p. can also be determined by two equations in which non-rectangular coordinates appear X,at planes, and any others. Some projections [for example, Perspective projections (in particular, orthographic, rice. 2
) perspective-cylindrical ( rice. 7
) and others] can be determined by geometric constructions. A map grid is also determined by the rule for constructing a cartographic grid corresponding to it, or by such characteristic properties of it, from which equations of the form (1) can be obtained, which completely determine the projection. Brief historical information. The development of the theory of cartography, as well as of all cartography, is closely connected with the development of geodesy, astronomy, geography, and mathematics. The scientific foundations of cartography were laid in Ancient Greece (6th-1st centuries BC). The most ancient projection is considered to be the Gnomonic projection, which was used by Thales of Miletus to map the starry sky. After the establishment in the 3rd century. BC e. the sphericity of the Earth K. p. began to be invented and used in the preparation of geographical maps (Hipparchus,
Ptolemy and others). A significant upsurge in cartography in the 16th century, caused by the Great Geographical Discoveries, led to the creation of a number of new projections; one of them, proposed by G. Mercator,
is still used today (see Mercator projection). In the 17th and 18th centuries, when the extensive organization of topographic surveys began to supply reliable material for compiling maps over large areas, maps were developed as the basis for topographic maps (French cartographer R. Bonn and J. D. Cassini).
and studies were also carried out on some of the most important groups of C. p. (I. Lambert, L. Euler, J. Lagrange
and etc.). The development of military cartography and a further increase in the volume of topographic work in the 19th century. They demanded that a mathematical basis be provided for large-scale maps and that a system of rectangular coordinates be introduced on a basis more suitable to the map. This led K. Gauss to develop the fundamental geodetic projection. Finally, in the middle of the 19th century. A. Tissot (France) gave a general theory of distortions of the C.P.
P. L. Chebyshev, D. A. Grave and others). The works of the Soviet cartographers V. V. Kavrayskii, N. A. Urmaev, and others developed new groups of maps, some of their variants (up to the stage of practical use), and important questions in the general theory of maps. , their classification, etc. The theory of distortions. Distortions in an infinitely small area near any projection point obey some general laws. At any point on the map in a projection that is not conformal (see below), there are two such mutually perpendicular directions, which also correspond to mutually perpendicular directions on the displayed surface, these are the so-called main display directions. The scales in these directions (principal scales) have extreme values: μ max = a and μ min = b. If in any projection the meridians and parallels on the map intersect at a right angle, then their directions are the main ones for this projection. The length distortion at a given point in the projection visually represents an ellipse of distortion, similar and similarly located to the image of an infinitesimal circle circumscribed around the corresponding point on the displayed surface. The half-diameters of this ellipse are numerically equal to the partial scales at a given point in the corresponding directions, the semi-axes of the ellipse are equal to the extreme scales, and their directions are the main ones. The connection between the elements of the distortion ellipse, the distortions of the C.P., and the partial derivatives of functions (1) is established by the basic formulas of the theory of distortions. Classification of cartographic projections according to the position of the pole of the used spherical coordinates. The poles of the sphere are special points of geographical coordination, although the sphere at these points does not have any features. This means that when mapping areas containing geographic poles, it is sometimes desirable to use not geographic coordinates, but others in which the poles turn out to be ordinary points of coordination. Therefore, spherical coordinates are used on the sphere, the coordinate lines of which are the so-called verticals (conditional longitude on them a = const) and almucantarates (where the polar distances z = const), are similar to geographic meridians and parallels, but their pole Z0 does not coincide with the geographic pole P0 (rice. one
). Transition from geographic coordinates φ
, λ
any point on the sphere to its spherical coordinates z, a at a given pole position Z 0 (φ 0 , λ 0) carried out according to the formulas of spherical trigonometry. Any C. p. given by equations (1) is called normal or direct ( φ 0 \u003d π / 2). If the same projection of the sphere is calculated by the same formulas (1), in which instead of φ
, λ
appear z, a, then this projection is called transverse when φ 0 = 0, λ 0
and oblique if 0 . The use of oblique and transverse projections leads to a reduction in distortion. On the rice. 2
normal (a), transverse (b) and oblique (c) orthographic projections (See. Orthographic projection) of a sphere (surface of a ball) are shown. Classification of cartographic projections according to the nature of distortions. In equiangular (conformal) K. p. the scale depends only on the position of the point and does not depend on the direction. The distortion ellipses degenerate into circles. Examples are Mercator projection, Stereographic projection.
Areas are preserved in equal-sized (equivalent) squares; more precisely, the areas of figures on maps compiled in such projections are proportional to the areas of the corresponding figures in nature, and the coefficient of proportionality is the reciprocal of the square of the main scale of the map. Distortion ellipses always have the same area, differing in shape and orientation. Arbitrary squares are neither equal-angled nor equal-sized. Of these, equidistant ones are distinguished, in which one of the main scales is equal to one, and orthodromic, in which the great circles of the ball (orthodromes) are depicted as straight lines. When a sphere is depicted on a plane, the properties of equiangularity, equal area, equidistance, and orthodromy are incompatible. To show distortions in different places of the depicted area, the following are used: a) distortion ellipses built in different places of the grid or map sketch ( rice. 3
); b) isocoles, i.e. lines of equal distortion (on rice. 8c
see isocoles of the greatest distortion of angles ω and isocoles of the area scale R); c) images in some places of the map of some spherical lines, usually orthodromes (O) and loxodromies (L), see fig. rice. 3a
,3b
and etc. Classification of normal map projections according to the type of images of meridians and parallels, which is the result of the historical development of the theory of quantum projections, encompasses most of the known projections. It retained the names associated with the geometric method of obtaining projections, however, their groups under consideration are now determined analytically. Cylindrical projections ( rice. 3
) - projections in which the meridians are depicted as equally spaced parallel lines, and parallels - as straight lines perpendicular to the images of the meridians. Beneficial for depicting territories stretched along the equator or any parallels. Navigation uses the Mercator projection, a conformal cylindrical projection. The Gauss-Kruger projection is an equiangular transverse-cylindrical K. p. - used in the preparation of topographic maps and the processing of triangulations.
Azimuthal projections ( rice. 5
) - projections in which the parallels are concentric circles, the meridians are their radii, while the angles between the latter are equal to the corresponding longitude differences. A special case of azimuth projections are perspective projections. Pseudoconic projections ( rice. 6
) - projections in which the parallels are depicted by concentric circles, the middle meridian - by a straight line, the rest of the meridians - by curves. Bonn's equal area pseudoconic projection is often used; since 1847, a three-verst (1:126,000) map of the European part of Russia has been drawn up in it. Pseudocylindrical projections ( rice. eight
) - projections in which the parallels are depicted by parallel lines, the middle meridian - by a straight line perpendicular to these lines and which is the axis of symmetry of the projections, the remaining meridians - by curves. Polyconic projections ( rice. 9
) - projections in which parallels are depicted by circles with centers located on the same straight line, depicting the middle meridian. When constructing specific polyconic projections, additional conditions are imposed. One of the polyconic projections is recommended for the international (1:1,000,000) map. There are many projections that do not belong to these types. Cylindrical, conic and azimuthal projections, called the simplest ones, are often referred to as circular projections in the broad sense, distinguishing from them circular projections in the narrow sense - projections in which all meridians and parallels are represented by circles, for example, Lagrange conformal projections, Grinten projection, etc. Using and choosing map projections depend mainly on the purpose of the map and its scale, which often determine the nature of the allowable distortions in the chosen c. determining the ratio of the areas of any territories - in equal areas. In this case, some violation of the defining conditions of these projections is possible ( ω ≡ 0
or p ≡ 1), which does not lead to tangible errors, i.e., we allow the choice of arbitrary projections, of which projections that are equidistant along the meridians are more often used. The latter are also resorted to when the purpose of the map does not at all provide for the preservation of angles or areas. When choosing a projection, one starts with the simplest, then moves on to more complex projections, even possibly modifying them. If none of the known C.P. satisfies the requirements for the map being compiled on the part of its purpose, then a new, most suitable C.P. is sought, trying (as far as possible) to reduce distortions in it. The problem of constructing the most advantageous C.P., in which distortions are in any sense reduced to a minimum, has not yet been completely solved. K. the item are also used in navigation, astronomy, crystallography, etc.; they are sought for the purposes of mapping the moon, planets, and other celestial bodies. Projection transformation. Considering two K. p., given by the corresponding systems of equations: x = f 1 (φ, λ), y = f 2 (φ, λ) and X = g 1 (φ, λ), Y = g 2 (φ, λ), it is possible, by excluding φ and λ from these equations, to establish the transition from one of them to another: X \u003d F 1 (x, y), Y \u003d F 2 (x, y). These formulas, when concretizing the type of functions F 1 ,F 2 , firstly, they give a general method for obtaining the so-called derived projections; secondly, they form the theoretical basis for all sorts of methods of technical methods for compiling maps (see Geographical maps). For example, affine and fractional-linear transformations are carried out with the help of mapping transformers (See Cartographic transformer).
However, more general transformations require the use of new, in particular electronic, technology. The task of creating perfect transformers for K.p. is an urgent problem of modern cartography. Lit.: Vitkovsky V., Cartography. (Theory of cartographic projections), St. Petersburg. 1907; Kavraysky V.V., Mathematical cartography, M. - L., 1934; his own, Fav. works, vol. 2, c. 1-3, [M.], 1958-60; Urmaev N. A., Mathematical cartography, M., 1941; his, Methods for finding new cartographic projections, M., 1947; Graur A. V., Mathematical cartography, 2nd ed., Leningrad, 1956; Ginzburg G. A., Cartographic projections, M., 1951; Meshcheryakov G. A., Theoretical Foundations of Mathematical Cartography, Moscow, 1968. G. A. Meshcheryakov. 2. The ball and its orthographic projections. 3a. Cylindrical projections. Equangular Mercator. 3b. Cylindrical projections. Equidistant (rectangular). 3c. Cylindrical projections. Equivalent (isocylindrical). 4a. conical projections. Equangular. 4b. conical projections. Equidistant. 4c. conical projections. Equal. Rice. 5a. Azimuthal projections. Equiangular (stereographic) on the left - transverse, on the right - oblique. Rice. 5 B. Azimuthal projections. Equidistant (left - transverse, right - oblique). Rice. 5th century Azimuthal projections. Equal-sized (on the left - transverse, on the right - oblique). Rice. 8a. Pseudocylindrical projections. Mollweide Equal Area Projection. Rice. 8b. Pseudocylindrical projections. Equal area sinusoidal projection of VV Kavraysky. Rice. 8c. Pseudocylindrical projections. Arbitrary projection TSNIIGAiK. Rice. 8y. Pseudocylindrical projections. BSAM projection. Rice. 9a. Polyconic projections. Simple. Rice. 9b. Polyconic projections. Arbitrary projection of G. A. Ginzburg.
Great Soviet Encyclopedia. - M.: Soviet Encyclopedia. 1969-1978 .
See what "Map projections" are in other dictionaries:
Mathematical methods of image on the plane of the surface of the earth's ellipsoid or ball. Map projections determine the relationship between the coordinates of points on the surface of the earth's ellipsoid and on the plane. Due to the inability to deploy ... ... Big Encyclopedic Dictionary
CARTOGRAPHIC PROJECTIONS, system methods of plotting the meridians and parallels of the Earth on a flat surface. Only on a globe can one reliably represent territories and forms. On flat maps of large areas, distortions are inevitable. Projections are... Scientific and technical encyclopedic dictionary
3. And finally, the final stage of creating a map is displaying the reduced surface of the ellipsoid on a 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 that maintain a constant 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. As you can easily see, this corresponds to dividing the 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 a section of the earth's surface, the 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 of 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 (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 of Remizov's Atlas of Russia.
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 in 1728 presented a project for compiling an atlas of the 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. By order of the Minister of Defense of the Russian Federation of November 9, 2003, the date of the annual holiday of the VTU of the 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 a 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 indicated ruins belong to archaeological monuments, their identification is a matter worthy of consideration.
The coordinates of the 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-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 activities, 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 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 values of the Earth's 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.
Map projections- these are mathematical methods of depicting the surface of the globe (ellipsoid) on a plane.
The globe most accurately conveys the shape of the Earth, because it is as spherical as our planet. But globes take up a lot of space, they are difficult to take on the road, they cannot be put into 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 - azimuth, cylindrical, conical. Depending on the type of map projection, the greatest distortion may be in one or another place on the map, and the degree network may look different.
Which projection to choose depends on the purpose of the map, on the size of the depicted territory and the latitude at which it is located. For example, for countries elongated in middle latitudes, such as Russia, it is convenient to use a conical projection, for polar regions an azimuth projection, and for maps of the world, individual continents, and oceans, a cylindrical projection is often used.