map projection

map projection- a mathematically defined way of displaying the surface of an ellipsoid on a plane.

The essence of the projections is connected with the fact that the figure of the Earth - an ellipsoid that is not deployed in a plane, is replaced by another figure that is deployed on a plane. At the same time, a grid of parallels and meridians is transferred from the ellipsoid to another figure. The appearance of this grid is different depending on what shape the ellipsoid is replaced with.

distortion

In any projection, there are distortion They are of four types:

• length distortion
• corner distortion
• area distortion
• shape distortion

On the various maps distortions can be of various sizes: on large-scale they are almost imperceptible, but on small-scale they are very large.

Length distortion

Length distortion- basic distortion. The rest of the distortions follow logically from it. Length distortion means the inconsistency of the scale of a flat image, which manifests itself in a change in scale from point to point, and even at the same point, depending on the direction.

This means that there are 2 types of scale on the map:

• The main one, it is signed on the map, but in fact it is the scale of the original ellipsoid, by deploying which the map is obtained in the plane.

• Private scale - there are infinitely many of them on the map, it changes from point to point and even within one point.

For a visual representation of private scales, a distortion ellipse is introduced.

Area distortion

Area distortion follow logically from the distortion of lengths. The deviation of the area of ​​the distortion ellipse from the original area on the ellipsoid is taken as a characteristic of the area distortion.

Corner distortion

Corner distortion follow logically from the distortion of lengths. The angle difference between the directions on the map and the corresponding directions on the surface of the ellipsoid is taken as a characteristic of the distortion of the angles on the map.

Shape distortion

Shape distortion- graphic representation of the elongation of the ellipsoid.

Classification of projections by the nature of distortions

Equangular projections

In direct conic projections, the axes of the globe and the cone coincide. In this case, the cone is taken either tangent or secant.

After designing, the lateral surface of the cone is cut along one of the generators and unfolded into a plane. When designing using the linear perspective method, perspective conic projections are obtained that have only intermediate properties in terms of the nature of distortions.

Depending on the size of the depicted territory, one or two parallels are accepted in conic projections, along which the lengths are preserved without distortion. One parallel (tangent) is taken with a small extent in latitude; two parallels (secant) - with a large extent to reduce scale deviations from unity. In the literature they are called standard parallels.

Azimuthal projections

In azimuthal projections, parallels are depicted as concentric circles, and meridians are depicted as a bundle of straight lines emanating from the center.

The angles between the projection meridians are equal to the corresponding longitude differences. The gaps between the parallels are determined by the accepted nature of the image (equiangular or otherwise) or by the way the points of the earth's surface are projected onto the picture plane. The normal grid of azimuthal projections is orthogonal. They can be considered as a special case of conic projections.

Direct, oblique and transverse azimuth projections are used, which is determined by the latitude of the central point of the projection, the choice of which depends on the location of the territory. Meridians and parallels in oblique and transverse projections are depicted as curved lines, with the exception of the middle meridian, on which the central point of the projection is located. In transverse projections, the equator is also depicted as a straight line: it is the second axis of symmetry.

Depending on distortions, azimuth projections are subdivided into conformal, equal-area and with intermediate properties. In the projection, the length scale can be maintained at a point or along one of the parallels (along the almukantar). In the first case, a tangent picture plane is assumed, in the second - a secant one. In direct projections, formulas are given for the surface of an ellipsoid or a ball (depending on the scale of the maps), in oblique and transverse projections - only for the surface of a ball.

Azimuth equal area projection is also called stereographic projection. It is obtained by passing rays from some fixed point on the Earth's surface to a plane tangent to the Earth's surface at the opposite point.

A special kind of azimuthal projection - gnomonic. It is obtained by conducting rays from the center of the Earth to some plane tangent to the surface of the Earth. The gnomonic projection does not preserve either areas or angles, but on it the shortest path between any two points (that is, the arc of a great circle) is always represented by a straight line; respectively, the meridians and the equator on it are depicted by straight lines.

Pseudoconic projections

In pseudoconic projections, parallels are represented by arcs of concentric circles, one of the meridians, called medium- a straight line, and the rest - curves, symmetrical about the mean.

Bonn's equal-area pseudoconic projection is an example of a pseudoconic projection.

Pseudocylindrical projections

In pseudocylindrical projections, all parallels are depicted as parallel lines, middle meridian- a straight line perpendicular to the parallels, and the rest of the meridians - curves. Moreover, the middle meridian is the axis of symmetry of the projection.

Polyconic projections

In polyconic projections, the equator is depicted as a straight line, and the remaining parallels are depicted as arcs of eccentric circles. Meridians are depicted as curves symmetrical about the central direct meridian perpendicular to the equator.

In addition to the above, there are other projections that do not belong to the indicated species.

see also

Links

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Projection A mathematically defined way of mapping the surface of a sphere or ellipsoid onto a plane, used to create a cartographic product. [GOST 21667 76] Topics cartography Generalizing terms mathematical cartography ... ...

map projection- Mathematical image method, as well as the actual image of the surface of an ellipsoid or ball on the plane of a geographical map ... Geography Dictionary

Mapping of the entire surface of the earth's ellipsoid or any part of it onto a plane, obtained mainly for the purpose of building a map. K. p. draw on a certain scale. Mentally reducing the earth's ellipsoid to Mraz, one gets its geometric. model ... ... Mathematical Encyclopedia

A mathematically defined mapping of the surface of the globe, an ellipsoid (or globe) onto a map plane. The projection establishes a correspondence between geographic coordinates point (latitude B and longitude L) and its rectangular coordinates ... ... Geographic Encyclopedia

pseudo-azimuth map projection- cartographic projection A cartographic projection in which the parallels of the normal grid are concentric circles or their arcs, and the meridians are curves emanating from the center of the parallels, symmetrical about one or two rectilinear ... ... Technical Translator's Handbook

equal area map projection- equal area projection N.d.p. Autotal projection Homologographic projection Equal area projection Equivalent projection Cartographic projection in which there are no area distortions. [GOST 21667 76] Inadmissible, not recommended ... ... Technical Translator's Handbook

conformal map projection- conformal projection Ndp. conformal projection orthomorphic projection isogonal projection autogonal projection A map projection that has no angle distortion. [GOST 21667 76] Inadmissible, not recommended autogonal ... ... Technical Translator's Handbook

azimuthal map projection- azimuthal projection Ndp. zenithal projection A map projection in which the parallels of the normal grid are concentric circles, and the meridians are their radii, the angles between which are equal to the corresponding differences in longitude. [GOST 21667 76]… … Technical Translator's Handbook

equirectangular map projection- equidistant projection Ndp. equidistant projection An arbitrary map projection in which the scale along one of the principal directions is a constant value. [GOST 21667 76] Invalid, not recommended equidistant projection ... ... Technical Translator's Handbook

conic map projection- conic projection A cartographic projection in which the parallels of the normal grid are arcs of concentric circles, and the meridians are their radii, the angles between which are proportional to the corresponding longitude differences. [GOST 21667 76] Topics… … Technical Translator's Handbook

map projection- a method of constructing an image of the Earth's surface and, above all, a grid of meridians and parallels (coordinate grid) on a plane. In every projection grid is depicted in different ways, the nature of the distortions is also different, i.e. projections have certain differences, which makes it necessary to classify them. All map projections are usually classified according to two criteria:

By the nature of the distortions;

By the form of a normal grid of meridians and parallels.

According to the nature of the projection distortions are divided into the following groups:

1. Equangular (comfortable) ) - projections in which infinitesimal figures on the maps are similar to the corresponding figures on the earth's surface. These projections are widely used in air navigation because they provide the easiest way to determine directions and angles. In addition, the configuration of small areal landmarks is transmitted without distortion, which is essential for maintaining visual orientation.

2. Equivalent (equivalent)- projections in which the ratio of areas on maps and on the earth's surface is preserved. These projections have found use in small-scale survey geographic maps.

3. Equidistant- projections in which the distance along the meridian and parallels are displayed without distortion. These projections are used to create reference maps.

4. Arbitrary are projections that do not have any of the properties listed above. These projections are widely used in air navigation, as they have practically small distortions of angles, lengths and areas, which allows them to be ignored.

By the form of the normal coordinate grid of meridians and parallels, the projections are divided into: conical, polyconical, cylindrical and azimuthal.

The construction of a cartographic grid can be represented as the result of projecting the Earth's surface onto an auxiliary geometric figure: a cone, a cylinder or a plane (Fig. 2.2).

Rice. 2.2. Auxiliary location geometric figure

Depending on the location of the auxiliary geometric figure relative to the axis of rotation of the Earth, there are three types of projections (Fig. 2.2):

1. Normal- projections in which the axis of the auxiliary figure coincides with the axis of rotation of the Earth.

2. transverse- projections in which the axis of the auxiliary figure is perpendicular to the axis of rotation of the Earth, i.e. coincides with the plane of the equator.

3. oblique- projections in which the axis of the auxiliary figure makes an oblique angle with the axis of rotation of the Earth.

conical projections. To solve air navigation problems from all conic projections, a normal conformal conic projection is used, built on a tangent or secant cone.

Normal conformal conic projection on a tangent cone. On maps compiled in this projection, the meridians look like straight lines converging towards the pole (Fig. 2.3). Parallels are arcs of concentric circles, the distance between which increases with distance from the tangent parallel. In this projection, maps are published for aviation at a scale of 1: 2,000,000, 1: 2,500,000, 1: 4,000,000 and 1: 5,000,000.

Rice. 2.3. Normal conformal conic projection on a tangent cone

Normal conformal conic projection on a secant cone. On the maps compiled in this projection, the meridians are depicted by straight converging lines, and the parallels by arcs of circles (Fig. 2.4). In this projection, maps are published for aviation at a scale of 1: 2,000,000 and 1: 2,500,000.

Rice. 2.4. Normal conformal conic projection on

secant cone

Polyconic projections. Polyconic projections have no practical application in aviation, but it is the basis of the international projection, in which most aviation charts are published.

Modified polyconic (international) projection. In 1909, in London, an international committee developed a modified polyconic projection for maps at a scale of 1: 1,000,000, which was called the international one. The meridians in this projection look like straight lines converging towards the pole, and the parallels look like arcs of concentric circles (Fig. 2.5).

Rice. 2.5. Modified polyconic projection

The map sheet occupies 4° in latitude and 6° in longitude. Currently, this projection is the most common and most aviation charts are published in it at scales of 1: 1,000,000, 1: 2,000,000 and 1: 4,000,000.

Cylindrical projections. From cylindrical projections in air navigation have found application normal, transverse and oblique projection.

Normal conformal cylindrical projection. This projection was proposed in 1569 by the Dutch cartographer Mercator. On maps compiled in this projection, the meridians look like straight lines, parallel to each other and separated from each other at distances proportional to the difference in longitudes (Fig. 2.6). Parallels are straight lines perpendicular to the meridians. Distances between parallels increase with increasing latitude. Nautical charts are published in the normal conformal cylindrical projection.

Rice. 2.6. Normal conformal cylindrical projection

Equiangular transverse cylindrical projection. This projection was proposed by the German mathematician Gauss. The projection is built according to mathematical laws. To reduce the distortion of lengths, the Earth's surface is cut into 60 zones. Each such zone occupies 6° in longitude. From fig. 2.7 it can be seen that the average meridian in each zone and the equator are depicted by straight mutually perpendicular lines. All other meridians and parallels are depicted as curves of small curvature. Maps of scales 1:500,000, 1:200,000 and 1:100,000 and larger were drawn up in a conformal transverse cylindrical projection.

Rice. 2.7. Conformal transverse cylindrical projection

Oblique conformal cylindrical projection. In this projection, the inclination of the cylinder to the axis of rotation of the Earth is selected so that its side surface touches the axis of the route (Fig. 2.8). Meridians and parallels in the considered projection look like curved lines. On the maps in this projection, in the band 500–600 km from the center line of the route, the length distortion does not exceed 0.5%. In an oblique conformal cylindrical projection, maps are published at scales of 1: 1,000,000, 1: 2,000,000 and 1: 4,000,000 to ensure flights along separate long routes.

Rice. 2.8. Oblique conformal cylindrical projection

Azimuthal projections. Of all the azimuthal projections, for the purposes of air navigation, mainly central and stereographic polar projections are used.

Central polar projection. On maps compiled in this projection, the meridians look like straight lines radiating from the pole at an angle equal to the difference in longitudes (Fig. 2.9). Parallels are concentric circles, the distance between which increases with distance from the pole. In this projection, maps of the Arctic and Antarctic were previously published at scales of 1: 2,000,000 and 1: 5,000,000.

Rice. 2.10. Stereographic polar projection

In a stereographic polar projection, maps of the Arctic and Antarctic are published at scales of 1: 2,000,000 and 1: 4,000,000.

The navigator uses a map to select the most advantageous route when moving from one point to another.

card called a reduced generalized image of the earth's surface on a plane, made on a certain scale and method.

Since the Earth has a spherical shape, its surface cannot be depicted on a plane without distortion. If we cut any spherical surface into parts (along the meridians) and impose these parts on a plane, then the image of this surface on it would turn out to be distorted and with discontinuities. There would be folds in the equatorial part, and breaks at the poles.

To solve navigation problems, distorted, flat images of the earth's surface are used - maps in which distortions are caused and correspond to certain mathematical laws.

Mathematically defined conditional ways of depicting on a plane the entire or part of the surface of a ball or an ellipsoid of revolution with low compression are called map projection, and the image system of the network of meridians and parallels adopted for this cartographic projection - cartographic grid.

All existing cartographic projections can be divided into classes according to two criteria: by the nature of distortions and by the method of constructing a cartographic grid.

According to the nature of the distortions, the projections are divided into conformal (or conformal), equal (or equivalent) and arbitrary.

Equal projections. On these projections, the angles are not distorted, i.e., the angles on the ground between any directions are equal to the angles on the map between the same directions. Infinitely small figures on the map, due to the property of equiangularity, will be similar to the same figures on the Earth. If the island round shape in nature, then on the map in a conformal projection it will be depicted as a circle of a certain radius. But the linear dimensions on the maps of this projection will be distorted.

Equal projections. On these projections, the proportionality of the areas of the figures is preserved, i.e., if the area of ​​any area on Earth is twice as large as another, then on the projection the image of the first area in terms of area will also be twice as large as the image of the second. However, in an equal area projection, the similarity of the figures is not preserved. The island of a round shape will be depicted on the projection in the form of an ellipse of equal area.

Arbitrary projections. These projections retain neither the similarity of figures nor the equality of areas, but may have some other special properties necessary for solving certain practical problems on them. From the charts of arbitrary projections, orthodromic projections have received the greatest use in navigation, on which great circles (great circles of the ball) are depicted by straight lines, and this is very important when using some radio navigation systems when navigating along a great circle arc.

The cartographic grid for each class of projections, in which the image of meridians and parallels has the simplest form, is called normal mesh.

According to the method of constructing a cartographic normal grid, all projections are divided into conical, cylindrical, azimuth, conditional, etc.

conical projections. The projection of the coordinate lines of the Earth is carried out according to one of the laws on the inner surface of the circumscribed or secant cone, and then, cutting the cone along the generatrix, it is turned onto a plane.

To obtain a normal straight conical grid, make sure that the axis of the cone coincides with the earth's axis PNP S (Fig. 33).

In this case, the meridians are depicted as straight lines emanating from one point, and parallels as arcs of concentric circles. If the axis of the cone is placed at an angle to the earth's axis, then such grids are called oblique conical.

Depending on the law chosen for constructing parallels, conic projections can be conformal, equal-area and arbitrary. Conic projections are used for geographic maps.

Cylindrical projections. A cartographic normal grid is obtained by projecting the coordinate lines of the Earth according to some law onto the side surface of a tangent or secant cylinder, the axis of which coincides with the axis of the Earth (Fig. 34), and then sweeping along the generatrix onto a plane.

In direct normal projection, the grid is obtained from mutually perpendicular straight lines of the meridians L, B, C, D, F, G and parallels aa", bb", ss. projection K in Fig. 34), but sections of the polar regions in this case cannot be projected.

If you rotate the cylinder so that its axis is located in the plane of the equator, and its surface touches the poles, then you get a transverse cylindrical projection (for example, a Gaussian transverse cylindrical projection). If the cylinder is placed at a different angle to the Earth's axis, then oblique cartographic grids are obtained. On these grids, meridians and parallels are shown as curved lines.

Azimuthal projections. A normal cartographic grid is obtained by projecting the coordinate lines of the Earth onto the so-called picture plane Q (Fig. 35) - tangent to the Earth's pole. The meridians of the normal grid on the projection have the form of radial straight lines emanating from. the central point of the projection PN at angles equal to the corresponding angles in nature, and the parallels are concentric circles centered at the pole. The picture plane can be located at any point on the earth's surface, and the point of contact is called the central point of the projection and is taken as the zenith.

The azimuth projection depends on the radii of the parallels. By subordinating the radii of one or another dependence on latitude, various azimuthal projections are obtained that satisfy the conditions of either equiangularity or equal area.

perspective projections. If a cartographic grid is obtained by projecting meridians and parallels onto a plane according to the laws of linear perspective from a constant point of view of T.Z. (see Fig. 35), then such projections are called promising. The plane can be positioned at any distance from the Earth or so that it touches it. The point of view should be on the so-called main diameter of the globe or on its continuation, and the picture plane should be perpendicular to the main diameter.

When the main diameter passes through the Earth's pole, the projection is called direct or polar (see Fig. 35); when the main diameter coincides with the plane of the equator, the projection is called transverse or equatorial, and at other positions of the main diameter, the projections are called oblique or horizontal.

In addition, perspective projections depend on the location of the point of view from the center of the Earth on the main diameter. When the point of view coincides with the center of the Earth, the projections are called central or gnomonic; when the point of view is on the surface of the Earth stereographic; when the point of view is removed at some known distance from the Earth, the projections are called external, and when the point of view is removed to infinity - orthographic.

On polar perspective projections, the meridians and parallels are depicted similarly to the polar azimuth projection, but the distances between the parallels are different and are due to the position of the point of view on the line of the main diameter.

On transverse and oblique perspective projections, meridians and parallels are depicted as ellipses, hyperbolas, circles, parabolas, or straight lines.

Of the features inherent in perspective projections, it should be noted that on a stereographic projection, any circle drawn on the earth's surface is depicted as a circle; on the central projection, any large circle drawn on the earth's surface is depicted as a straight line, and therefore, in some special cases, this projection seems appropriate to use in navigation.

Conditional projections. This category includes all projections that, according to the method of construction, cannot be attributed to any of the above types of projections. They usually satisfy some pre-set conditions, depending on the purposes for which the card is required. The number of conditional projections is not limited.

Small areas of the earth's surface up to 85 km can be depicted on a plane with the similarity of the applied figures and areas preserved on them. Such flat images of small areas of the earth's surface, on which distortions can practically be neglected, are called plans.

Plans are usually drawn up without any projections by direct shooting and all the details of the area being filmed are applied to them.

Classifications of map projections

By the nature of the distortion projections are divided into conformal, equal-area and arbitrary.

Equangular(or conformal) projections preserve the angles and shapes of infinitesimal figures. The length scale at each point is constant in all directions (which is ensured by a regular increase in the distances between adjacent parallels along the meridian) and depends only on the position of the point. Distortion ellipses are expressed as circles of various radii.

For each point in conformal projections, the dependencies are valid:

/ L i= a = b = m = n; a>= 0°; 0 = 90°; k = 1 and and 0 \u003d 0 °(or ±90°).

Such projections especially useful for determining directions and laying routes along a given azimuth (for example, when solving navigation problems).

isometric(or equivalent) projections do not distort the area. In these projections the areas of the distortion ellipses are. An increase in the scale of lengths along one axis of the distortion ellipse is compensated by a decrease in the scale of lengths along the other axis, which causes a regular decrease in the distances between adjacent parallels along the meridian and, as a result, a strong distortion of shapes.

Such projections are convenient for measuring areas objects (which, for example, is essential for some economic or morphometric maps).

In the theory of mathematical cartography, it is proved that no, and there cannot be a projection that would be both conformal and equal-area at the same time. In general, the more distortion of angles, the less distortion of areas and vice versa.

Arbitrary projections distort both angles and areas. When constructing them, they try to find the most favorable distribution of distortions for each specific case, reaching, as it were, a certain compromise. This group of projections used in cases where excessive distortion of angles and areas is equally undesirable. Arbitrary projections by their properties lie between equiangular and equal areas. Among them are equidistant(or equidistant) projections, at all points of which the scale along one of the main directions is constant and equal to the main one.

Classification of cartographic projections according to the type of auxiliary geometric surface .

According to the type of auxiliary geometric surface, projections are distinguished: cylindrical, azimuthal and conical.

Cylindrical are called projections in which the network of meridians and parallels from the surface of the ellipsoid is transferred to the side surface of the tangent (or secant) cylinder, and then the cylinder is cut along the generatrix and unfolded into a plane (Fig. 6).

Fig.6. Normal cylindrical projection

Distortions are absent on the line of contact and are minimal near it. If the cylinder is secant, then there are two lines of contact, which means 2 LNI. Between LNI distortion is minimal.

Depending on the orientation of the cylinder relative to the axis of the earth's ellipsoid, projections are distinguished:

- normal, when the axis of the cylinder coincides with the minor axis of the earth's ellipsoid; the meridians in this case are equidistant parallel lines, and the parallels are straight lines perpendicular to them;

- transverse, when the axis of the cylinder lies in the plane of the equator; type of grid: the middle meridian and the equator are mutually perpendicular straight lines, the remaining meridians and parallels are curved lines (Fig. c).

- oblique, when the axis of the cylinder is with the axis of the ellipsoid sharp corner; in oblique cylindrical projections, the meridians and parallels are curved lines.

Azimuthal are called projections in which the network of meridians and parallels is transferred from the surface of the ellipsoid to the tangent (or secant) plane (Fig. 7).

Rice. 7. Normal azimuthal projection

The image near the point of contact (or section line) of the plane of the earth's ellipsoid is almost not distorted at all. The touch point is the zero distortion point.

Depending on the position of the point of contact of the plane on the surface of the earth's ellipsoid, among the azimuthal projections, there are:

- normal, or polar, when the plane touches the Earth at one of the poles; type of grid: meridians - straight lines, radially diverging from the pole, parallels - concentric circles with centers at the pole (Fig. 7);

- transverse, or equatorial, when the plane touches the ellipsoid at one of the points of the equator; type of grid: the middle meridian and the equator are mutually perpendicular straight lines, the remaining meridians and parallels are curved lines (in some cases, parallels are depicted as straight lines;

– oblique, or horizontal, when the plane touches the ellipsoid at some point lying between the pole and the equator. In oblique projections, only the middle meridian, on which the point of contact is located, is a straight line, the other meridians and parallels are curved lines.

conical projections are called in which the network of meridians and parallels from the surface of the ellipsoid is transferred to the side surface of the tangent (or secant) cone (Fig. 8).

Rice. 8. Normal conic projection

Distortions are little perceptible along the line of contact or two lines of section of the cone of the earth's ellipsoid, which are the line (s) of zero distortion LNI. Like cylindrical conic projections are divided into:

- normal, when the axis of the cone coincides with the minor axis of the earth's ellipsoid; the meridians in these projections are represented by straight lines radiating from the top of the cone, and the parallels are represented by arcs of concentric circles.

- transverse, when the axis of the cone lies in the plane of the equator; type of grid: the middle meridian and the parallel of contact are mutually perpendicular straight lines, the remaining meridians and parallels are curved lines;

- oblique, when the axis of the cone makes an acute angle with the axis of the ellipsoid; in oblique conic projections, the meridians and parallels are curved lines.

In normal cylindrical, azimuth and conic projections, the cartographic grid is orthogonal - the meridians and parallels intersect at right angles, which is one of the important diagnostic features of these projections.

If, when obtaining cylindrical, azimuthal and conical projections, a geometric method is used (linear projection of an auxiliary surface onto a plane), then such projections are called perspective-cylindrical, perspective-azimuth (ordinary perspective) and perspective-conical, respectively.

polyconical called projections, in which the network of meridians and parallels from the surface of the ellipsoid is transferred to the side surfaces of several cones, each of which is cut along the generatrix and unfolds into a plane. In polyconic projections, parallels are represented by arcs of eccentric circles, the central meridian is a straight line, all other meridians are curved lines symmetrical with respect to the central one.

conditional called projections, the construction of which does not resort to the use of auxiliary geometric surfaces. A network of meridians and parallels is built according to some predetermined condition. Conditional projections include pseudocylindrical, pseudo-azimuth and pseudoconical projections that preserve the appearance of parallels from the original cylindrical, azimuthal and conical projections. In these projections the middle meridian is a straight line, the other meridians are curved lines.

To conditional projections are also polyhedral projections , which are obtained by projecting onto the surface of a polyhedron that touches or cuts the earth's ellipsoid. Each face is an isosceles trapezoid (less often - hexagons, squares, rhombuses). A variety of polyhedral projections are multi-lane projections , and the strips can be cut both along the meridians and along the parallels. Such projections are advantageous in that the distortion within each facet or band is very small, so they are always used for multi-sheet maps. The main disadvantage of polyhedral projections is in the impossibility of combining a block of map sheets along a common frame without gaps.

Practically valuable is the division by territorial coverage. By territorial coverage map projections for maps of the world, hemispheres, continents and oceans, maps of individual states and their parts. According to this principle tables-determinants of cartographic projections were built. Besides, recently attempts are being made to develop genetic classifications of cartographic projections based on the form of differential equations describing them. These classifications cover the entire possible set of projections, but are extremely obscure, because are not related to the type of grid of meridians and parallels.

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, maintaining the constancy of the scale in one of the main directions.

4. Arbitrary projections - projections that do not belong to any of the considered groups, but have some other properties that are important for practice, are called arbitrary.

Rice. Projection of an ellipsoid onto a figure unfolded into a plane.

Depending on which figure the ellipsoid surface is projected onto (cylinder, cone or plane), projections are divided into three main types: cylindrical, conical and azimuthal. The type of figure on which the ellipsoid is projected determines the type of parallels and meridians on the map.

Rice. The difference in projections according to the type of figures on which the surface of the ellipsoid is projected and the type of development of these figures on the plane.

In turn, depending on the orientation of the cylinder or cone relative to the ellipsoid, cylindrical and conical projections can be: straight - the axis of the cylinder or cone coincides with the axis of the Earth, transverse - the axis of the cylinder or cone is perpendicular to the axis of the Earth and oblique - the axis of the cylinder or cone is inclined to the axis of the Earth at an angle other than 0° and 90°.

Rice. The difference in projections is the orientation of the figure onto which the ellipsoid is projected relative to the Earth's axis.

The cone and cylinder can either touch the surface of the ellipsoid or intersect it. Depending on this, the projection will be tangent or secant. Rice.

Rice. Tangent and secant projections.

It is easy to see (Fig) that the length of the line on the ellipsoid and the length of the line on the figure that it is projected will be the same along the equator, tangent to the cone for the tangent projection and along the secant lines of the cone and cylinder for the secant projection.

Those. for these lines, the map scale will exactly match the scale of the ellipsoid. For other points on the map, the scale will be slightly larger or smaller. This must be taken into account when cutting map sheets.

The tangent to the cone for the tangent projection and the secant of the cone and cylinder for the secant projection are called standard parallels.

For the azimuthal projection, there are also several varieties.

Depending on the orientation of the plane tangent to the ellipsoid, the azumuthal projection can be polar, equatorial or oblique (Fig)

Rice. Views of the Azimuthal projection by the position of the tangent plane.

Depending on the position of an imaginary light source that projects the ellipsoid onto a plane - in the center of the ellipsoid, at the pole, or at an infinite distance, there are gnomonic (central-perspective), stereographic and orthographic projections.

Rice. Types of azimuthal projection by the position of an imaginary light source.

The geographical coordinates of any point on the ellipsoid remain unchanged for any choice of map projection (determined only by the selected system of "geographical" coordinates). However, along with geographical projections of an ellipsoid on a plane, so-called projected coordinate systems are used. These are rectangular coordinate systems - with the origin at a certain point, most often having coordinates 0,0. Coordinates in such systems are measured in units of length (meters). More about it we will talk below for 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 the two map projections that are 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. Like a truly great mathematician, he solved this particular problem in general view 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 given 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 common system layouts 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 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 latitudes of the northern and southern 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 are indicated full values coordinates, for intermediate ones - abbreviated (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 the 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 compass arrow shows a direction slightly different from the direction to geographic belt. The magnitude of this deviation is called the magnetic declination. It can be east or west. By adding to the value of the magnetic declination the annual change in the magnetic declination, multiplied by the number of years that have passed since the creation of the map until the current moment, determine the magnetic declination at the current moment.

In concluding the topic on the basics of cartography, let us briefly dwell on the history of cartography in Russia.

The first maps with a displayed geographical coordinate system (maps of Russia by F. Godunov (published in 1613), G. Gerits, I. Massa, N. Witsen) appeared in the 17th century.

In accordance with the legislative act of the Russian government (the boyar "verdict") of January 10, 1696 "On the removal of a drawing of Siberia on canvas with an indication of cities, villages, peoples and distances between tracts" S.U. Remizov (1642-1720) created a huge (217x277 cm) cartographic work "Drawing of all Siberian cities and lands", which is now in the permanent exhibition of the State Hermitage. 1701 - January 1 - the date on the first title page Atlas of Russia Remizov.

In 1726-34. the first Atlas of the All-Russian Empire is published, the head of the work on the creation of which was the chief secretary of the Senate I.K. Kirillov. The atlas was published in Latin, and consisted of 14 special and one general maps under the title "Atlas Imperii Russici". In 1745 the All-Russian Atlas was published. Initially, the work on compiling the atlas was led by academician, astronomer I. N. Delil, who presented in 1728 a project for compiling the atlas Russian Empire. Starting from 1739, the work on compiling the atlas was carried out by the Geographical Department of the Academy of Sciences, established on the initiative of Delisle, whose task was to compile maps of Russia. Delisle's atlas includes comments on maps, a table with the geographical coordinates of 62 Russian cities, a map legend and the maps themselves: European Russia on 13 sheets at a scale of 34 versts per inch (1:1428000), Asian Russia on 6 sheets on a smaller scale and a map of all of Russia on 2 sheets on a scale of about 206 versts per inch (1:8700000) The Atlas was published in the form of a book in parallel editions in Russian and Latin with the application of the General Map.

When creating the Delisle atlas, much attention was paid to the mathematical basis of the maps. For the first time in Russia, an astronomical determination of the coordinates of strong points was carried out. The table with coordinates indicates the way they were determined - "for reliable reasons" or "when compiling a map" During the 18th century, a total of 67 complete astronomical determinations of coordinates were made relating to the most important cities of Russia, and 118 determinations of points in latitude were also made . On the territory of Crimea, 3 points were identified.

From the second half of XVIII in. the role of the main cartographic and geodetic institution of Russia gradually began to be performed by the Military Department

In 1763 a Special General Staff was created. Several dozen officers were selected there, who officers were sent to remove the areas where the troops were located, the routes of their possible following, the roads along which messages passed by military units. In fact, these officers were the first Russian military topographers who completed the initial scope of work on mapping the country.

In 1797, the Card Depot was established. In December 1798, the Depot received the right to control all topographic and cartographic work in the empire, and in 1800 the Geographical Department was attached to it. All this made the Map Depot the central cartographic institution of the country. In 1810 the Kart Depot was taken over by the Ministry of War.

February 8 (January 27, old style) 1812, when the highest approved "Regulations for the Military Topographic Depot" (hereinafter VTD), which included the Map Depot as a special department - the archive of the military topographic depot. Order of the Minister of Defense Russian Federation dated November 9, 2003, the date of the annual holiday of the VTU General Staff of the Armed Forces of the Russian Federation was set - February 8.

In May 1816, the VTD was included in the General Staff, while the head of the General Staff was appointed director of the VTD. Since this year, the VTD (regardless of renaming) has been permanently part of the Main or General Staff. VTD led the Corps of Topographers, created in 1822 (after 1866, the Corps of Military Topographers)

The most important results of the work of the VTD for almost a whole century after its creation are three large maps. First - special card 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 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 was formed armed forces Russia, 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..

Cards large scale give the most full information about the territory. They can be directly used to search for objects marked on them and preserved to this day. The ruins of buildings are one of the elements included in the legend of topographic maps, and although only a few of the ruins indicated are archaeological monuments, their identification is a matter worthy of consideration.

The coordinates of the preserved objects, determined from the topographic maps of the USSR, or by direct measurements using the global space positioning system (GPS), can be used to link old maps to modern systems coordinates. However, even maps of the early-mid 19th century can contain significant distortions in the proportions of the terrain in certain areas of the territory, and the procedure for linking maps consists not only of correlating the origins of coordinates, but also requires uneven stretching or compression of individual sections of the map, which is carried out on the basis of knowing the coordinates of a large number of reference points. points (the so-called map image transformation).

After the binding, it is possible to compare the signs on the map with the objects present on the ground at the present time, or that existed in the periods preceding or following the time of its creation. To do this, it is necessary to compare the available maps of different periods and scales.

Large-scale topographic maps of the 19th century seem to be very useful when working with boundary plans of the 18th-19th centuries, as a link between these plans and large-scale maps of the USSR. Boundary plans were drawn up in many cases without substantiation at strong points, with an orientation along the magnetic meridian. Due to changes in the nature of the terrain caused by natural factors and human activity, a direct comparison of boundary and other detailed plans of the last century and maps of the XX century is not always possible, however, a comparison of detailed plans of the last century with a modern topographic map seems to be simpler.

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 allows you to overlay each other and link them to modern maps, combine layers of large-scale topographic maps of 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 meanings Earth parameters: Equatorial radius, 6378 km. Polar radius, 6357 km. The average radius of the Earth, 6371 km. Equator length, 40076 km. Meridian length, 40008 km...

Here, of course, it must be taken into account that the value of the “stage” itself is a debatable issue.

A diopter is a device used to direct (sight) a known part of a goniometric tool to given subject. 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.