By card called a reduced generalized image of the earth's surface on a plane, made according to a certain scale and method.
Since the Earth is spherical, its surface cannot be depicted on a plane without distortion. If you cut any spherical surface into parts (along the meridians) and superimpose 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 the distortions are caused and correspond to certain mathematical laws.
Mathematically defined conditional ways of representing on the plane of all or part of the surface of a ball or an ellipsoid of revolution with small compression are called map projection, and the system of image of the network of meridians and parallels adopted for a given cartographic projection is cartographic grid.
All existing cartographic projections can be subdivided into classes according to two criteria: by the nature of distortions and by the method of constructing a cartographic grid.
By the nature of distortions, projections are divided into conformal (or conformal), equal (or equivalent) and arbitrary.
Conformal projections. On these projections, the angles are not distorted, that is, the angles on the ground between any directions are equal to the angles on the map between the same directions. Infinitesimal figures on the map, due to the property of conformity, will be similar to the same figures on Earth. If the island is round in nature, then on the map in the 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 area projections. On these projections, the proportionality of the areas of the figures is preserved, that is, if the area of \u200b\u200bany area on Earth is twice as large as another, then on the projection the image of the first area will also be twice as large as the image of the second. However, the similarity of the figures is not preserved in the equal area projection. A circular island will be projected as an ellipse of the same size.
Arbitrary projections. These projections preserve neither the similarity of figures, nor the equality of areas, but they can have some other special properties necessary for solving certain practical problems on them. Orthodromic maps, on which orthodromies (large circles of the ball) are depicted, are most widely used in navigation from maps of arbitrary projections. straight lines, and this is very important when using some radio navigation systems when sailing in a great circle.
The cartographic grid for each class of projections, in which the image of the 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, azimuthal, conditional, etc.
Conical projections. The design of the coordinate lines of the Earth is carried out according to any of the laws on the inner surface of the described or secant cone, and then, cutting the cone along the generatrix, unfold it onto a plane.
To obtain a normal straight conical mesh, make so that the axis of the cone coincides with the earth's axis PNP S (Fig, 33). In this case, meridians are depicted by straight lines emanating from one point, and parallels are represented by arcs of concentric circles. If the axis of the cone is located at an angle to the earth's axis, then such grids are called oblique conical.
Depending on the law chosen to draw parallels, conic projections can be conformal, equal, or 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 on lateral surface tangent or secant cylinder, the axis of which coincides with the axis of the Earth (Fig. 34), and subsequent sweep along the generatrix onto the 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", cc. In this case, without large distortions, the surface areas of the equatorial regions will be depicted (see, the circle K and its projection K in Fig. 34), but the sections of the polar regions in this case cannot be designed.
If you rotate the cylinder so that its axis is located in the equatorial plane, and its surface touches the poles, then a transverse cylindrical projection is obtained (for example, a transverse cylindrical Gaussian 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 drawn with curved lines.
Figure: 34
Azimuthal projections. A normal cartographic grid is obtained by projecting the coordinate lines of the Earth onto the so-called picture plane Q (Fig. 35), which is tangent to the Earth's pole. The meridians of the normal grid on the projection have the form of radial straight lines originating from. the central point of the projection P N 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 tangency is called the center point of the projection and is taken as the zenith.
The azimuthal projection depends on what radii the parallels are drawn. Subordinating the radii to one or another dependence on latitude, various azimuthal projections are obtained that satisfy the conditions of either conformal or equal-size.
Figure: 35
Perspective projections. If a cartographic grid is obtained by projecting meridians and parallels onto a plane according to the laws of linear perspective from the 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 basic diameter of the globe or on its continuation, and the plane of the sky should be perpendicular to the basic diameter.
When the main diameter passes through the pole of the Earth, the projection is called direct or polar (see Fig. 35); when the main diameter coincides with the equatorial plane, the projection is called transverse or equatorial, and at other positions of the main diameter, the projection is called oblique or horizontal.
In addition, perspective projections depend on the location of the point of view from the center of the Earth at the base 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-tereographic; when the point of view is removed to any 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 azimuthal 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 it seems appropriate to use this projection 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 while maintaining the similarity of the applied figures and areas on them. Such flat images of small areas of the earth's surface, on which distortions can be practically neglected, are called plans.
Plans are usually drawn up without any projections by direct shooting and all the details of the area being shot are applied to them.
Of the projections discussed above, the following are mainly used in navigation: conformal, cylindrical, azimuthal perspective, gnomonic and azimuthal perspective stereographic.
The scale
The scale of the map is the ratio of the infinitesimal element of the line at a given point and in a given direction on the map to the corresponding infinitesimal element of the line on the ground.This scale is called private scale, and each point of the map has its own, inherent only to it, private scale. On the maps, in addition to the private, they also distinguish main scale, which is the initial value for calculating the size of the map.
The main scale is called the scale, the value of which is preserved only along certain lines and directions, depending on the nature of the map construction. On all other parts of the same map, the magnitude of the scale is greater or less than the main one, that is, these parts of the map will correspond to their particular scales.
The ratio of the private scale of the map at a given point in a given direction to the main one is called scale up, and the difference between zoom and unit is relative length distortion. On a conformal cylindrical projection, the scale changes when moving from one parallel to another. The parallel along which the main scale is observed is called the main parallel. As you move away from the main parallel towards the pole, the values \u200b\u200bof the partial scales on the same map increase and, conversely, as you move away from the main parallel towards the equator, the values \u200b\u200bof the partial scales decrease.
If the scale is expressed as a simple fraction (or ratio), the dividend of which is one, and the divisor is a number indicating how many units of length on the horizontal projection of a given area of \u200b\u200bthe earth's surface corresponds to one unit of length on the map, then such a scale is called numerical or numeric. For example, a numerical scale of 1/100000 (1: 100000) means that 1 cm on the map corresponds to 100,000 cm on the ground.
To determine the length of the measured lines, use linear scale, showing how many units of length of the highest name on the ground are contained in one unit of length of the lowest name on the map (plan).
For example, the scale of the map is "5 miles in I cm" or 10 km in 1 cm ", etc. This means that a distance of 5 miles (or 10 km) on the ground corresponds to 1 cm on the map (plan).
A linear scale on a plan or map is placed under the frame in the form of a straight line, divided into several divisions; the starting point of the linear scale is designated by the number 0, and then numbers are placed opposite each or some of its subsequent divisions, showing the distances corresponding to these divisions on the ground.
The transition from a numerical scale to a linear one is carried out by a simple recalculation of measures of length.
For example, to go from a numerical scale of 1/100000 to a linear scale, you need to convert 100,000 cm to kilometers or miles. 100,000 cm \u003d 1 km, or approximately 0.54 miles, therefore this map is drawn on a scale of 1 km in 1 cm, or 0.54 miles in 1 cm.
If a linear scale is known, for example, 2 miles in 1 cm, then to go to a numerical scale, it is necessary to convert 2 miles to centimeters and write as a fraction with a numerator one: 2 1852 100 - \u003d 370 400 cm, therefore, the numerical scale of this map is 1 / 370400
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Conic projections
| Parameter name | Value |
| Topic of the article: | Conic projections |
| Category (thematic category) | Radio |
Classification of map projections
Maps and map projections
It is customary to call a map a reduced image of the earth's surface on a plane at a certain scale with a coordinate grid and conventional signs that represent earth objects.
The Flight Chart is the ultimate guide to flying. No flight can be performed without a card.
A map on the ground is necessary for laying and digitizing a route͵ for studying base and alternate airfields, performing the necessary measurements and calculations in preparation for flight, and in flight - for maintaining visual orientation, controlling the path, determining the location of the aircraft.
The aviation card must meet the following requirements:
1. Reliably and accurately display the state of the area:
2. Be visual, easy to read and easy to work with.
3. The map should be with minimum angular and linear distortion,
convenient for measurements and plotting.
A cartographic projection is usually called a method of representing the earth's surface on a plane. All cartographic projections differ in the following ways:
1. By the nature of the distortion;
2. By the method of constructing the coordinate grid:
By the nature of the distortion of the projection there are:
1. Conformal - equality of angles between landmarks and shape of figures is preserved.
Posted on ref.rf
Conformal maps are widely used in aviation.
2. Equal area - the constancy of the ratio of the area of \u200b\u200bthe image of the figure on the map to the area of \u200b\u200bthe same figure on the earth's surface is maintained. In this projection, there is no equality of angles and similarity of figures.
3. Equidistant - the scale is maintained in one of the main directions (meridian and parallels).
4. Arbitrary - neither equality of angles nor areas is preserved.
According to the method of constructing the coordinate grid (meridians and parallels), cartographic projections are divided into cylindrical, conical, polyconical, azimuthal.
Cylindrical projections (Mercator projections)
To make maps in cylindrical projection, you need a model of the Earth made of transparent material. A light source is placed in the center of the model. The earth model is placed in the cylinder so that it touches the equator of the cylinder walls. Then backlight is produced. The light beams propagate in a straight line and all points and lines on the model are projected onto the surface of the cylinder. Next, the cylinder is cut and turned onto a plane. Meridians and parallels on the maps in this projection have the form of mutually perpendicular lines. The projection is conformal, the scale is not the same - it enlarges towards the poles. In this projection, nautical charts are produced.
In a conical projection, the surface of the Earth is projected onto the lateral surface of a cone tangent to one of the parallels. Next, the cone is cut and turned on a plane. The meridians in this projection are depicted as straight lines converging to the pole, and parallels are shown as arcs parallel to the equator. The projection is conformal, the distortion of the scale is not great. If the axis of the cone coincides with the axis of rotation of the Earth, the projection is usually called normal. In normal conic projection, side maps of scale 1 are produced : 4000000 (1cm. \u003d 40km), and 1 : 2500000 (1cm. \u003d 25km).
Conical projections - concept and views. Classification and features of the category "Conic projections" 2017, 2018.
Conical projections - the surface of a ball (ellipsoid) is projected onto the surface of a tangent or secant cone, after which it is, as it were, cut along a generatrix and turned into a plane. As in previous case, there is a normal (direct) conical projection, when the axis of the cone coincides with the axis of rotation of the Earth, transverse conical - the axis of the cone lies in the plane of the equator and oblique conical - the axis of the cone is inclined to the plane of the equator.
Conical projections are those in which the parallels of the normal grid are depicted by arcs of concentric circles, and the meridians - by 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 the meridians and parallels on the lateral surface of the cone, and then unfolding this surface into a plane.
Imagine a cone touching the globe along some parallel AoBoCo (Fig. 4). Let's extend the planes of the 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 are taken, respectively, for the images of the meridians and parallels of the globe. Let's cut the surface of the cone along the generatrix and turn it into a plane; then we get on the plane a cartographic grid in one of the conic projections (Fig. 5).
Parallels from the globe to the surface of the cone can be transferred in other ways, namely: by projecting with rays emanating from the center of the globe or from some point located on the axis of the cone, by laying on the meridians of the projection in both directions from the parallel of tangency of the straightened arcs of the meridians of the globe, enclosed between the parallels, and subsequent drawing through the deposition points 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 tangent to the Globe in parallel.
Figure: 5 Deposits of concentric circles.
The cartographic grid in a conical projection will be represented by arcs of concentric circles, and the meridians will be straight lines emanating from one point and making angles between themselves, proportional to the corresponding differences in longitudes.
Properties of the conic projections of Ptolemy, Krasovsky, Kavraisky
Krasovsky projection
There are no distortions on the map: lengths along parallels with latitudes +49.4 and +67.8 degrees; areas on parallels with latitudes + 48 °, 2 and + 68 °, 4; angles at parallels with latitudes + 50 °, 6 and + 66 °, 8. The projection is calculated under the conditions: preservation of the area of \u200b\u200bthe belt, limited by the parallels with latitudes + 39 ° 28 "42" and + 73 ° 28 "42"; equality of scales along the extreme parallels of this belt; minimum sum of squares of length distortions along parallels.
Projection should be used for maps Russian Federation, when it is essential that not only the mainland, but also the adjacent area of \u200b\u200bthe polar basin are transmitted with as little distortion as possible. The map can only be composed without including the pole in the frame, which is depicted as a polar arc.
Ptolemy's projection
The conic projection of Ptolemy is built on a straight tangent cone. Having imagined the spatial picture of the relative position of the figures, we proceed to building the projection grid.
1. The initial data for building the grid are set, namely the scale of the map, the distance in degrees between the parallels (n °) and the meridians (t °), the latitude of the parallel of tangency (ф0).
2. Calculate the radius of parallel tangency (in mm) by the formula
3. Calculate the distance between the parallels (a - a segment of the meridian - the arc of a great circle) by the formula
4. The distance between the meridians (b - parallel segment) is determined on the parallel of tangency. From the tables, the value of 1 ° of the arc of a given parallel (in km) is known, it is multiplied by the difference in longitudes between adjacent meridians (t °) and converted into millimeters, knowing the scale of this map.
After these calculations, they begin to build a projection on a sheet of paper.
1. Draw the meridian of symmetry. For Russia, the meridian of 100 ° E is considered to be such. etc.
2. With the calculated radius from the top of the cone, taken on the meridian of symmetry arbitrarily, draw a parallel of tangency. Usually latitude is chosen so that the parallel is in the middle of the map. For Russia, this may be 55 ° N. sh.
3. On both sides of the parallel of tangency on the meridian of symmetry, segments are laid - the distance between the parallels. The arcs of parallels themselves are drawn from the top of the cone.
4. On the parallel of tangency (which does not have distortions on the map), segments b are laid - the distance between the meridians.
The cartographic image of the territory of Russia or another country is limited with the inner frame, then a degree box, an outer frame is built, and the construction of the cartographic grid in the projection is completed.
Ptolemy projection properties:
1. The main scale is maintained along all meridians and parallels of touch.
2. Private scales in other parallels are greater than the main one.
3. Conformal and equal-area properties are preserved along the parallel of tangency - the line of zero distortion.
4 Distortions of contours, areas increase on both sides of the parallel of tangency. Moreover, in the 15 ° strip on both sides of it, they are small, further to the north they grow more significantly than to the south.
In 1931, V.V.Kavraisky's normal conic projection was developed for maps of the USSR. It was used for the "Atlas of the USSR" (grade 7), "The Great Soviet Atlas of the World". The projection was developed by Kavraisky with the calculation of the least distortions of lengths along the meridians and parallels for the territory of the USSR south of the Arctic Circle. To the north of it, image quality was not taken into account (Fig. 60).
The projection is built on a secant cone and has two parallels of tangency, namely 47 ° N. sh. and 62 ° N. sh., the greatest distortion of the angles is about 0.5 °. This projection has zero distortion lines of all kinds. For all meridians, the scale is the main one, for the parallels of tangency as well. When schoolchildren or students work with maps in this projection, you can use a protractor to measure angles.
Figure: 60. Grid in the Kavraisky projection
In Kavraisky's projection, published in 1949. A gypsometric map of the USSR on a scale of 1 2 500 000
Since the 50s, the normal equidistant projection FN Krasovsky has been used for maps of the USSR The principle of its construction is similar to the construction of the Kavraisky projection for calculations, the same secant cone was used, but the condition was introduced to preserve the area of \u200b\u200ba given belt and equality of length scales along its extreme parallels -39 ° 48 ′ S w and 73 ° 30 ′ s w, i.e. the strip between the parallels of tangency is widened, within which cartometric work can be performed without correcting for distortions (Fig. 61) 
The disadvantage of normal conic projections is that on a tangent cone, the main scale is retained only along the parallel
touches, there is distortion in other places. On the cutting cone, the eastern and western territories are strongly deployed, the pole is outside the image.
To maintain the scale on all parallels, it is necessary to construct a degree grid using a set of cones, namely, each parallel - on its own Then each parallel will become a parallel of tangency (its radius is calculated by the Ptolemy formula p \u003d r ctg ф0) and will be displayed without distortion. Next, find on the parallels , using the table of arc lengths in G, the points of passage of the meridians and draw them as complex curves, connecting the points of passage of the meridians on neighboring parallels. This is the principle of the structure of the cartographic grid in polyconic projections.
41. Polyconic projections. Properties of TsNIIGAiK projections: TSB version, 1951 version
TsNIIGAiK polyconic projection (TSBB variant) was developed for world maps Large Soviet Encyclopedia... Distortions of angles and areas are of approximately the same order of magnitude, but by the nature of the distortions, it still gravitates more towards conformal projections. When displaying Europe, Africa, large parts of Asia, South and North America, Australia and even part of Antarctica, the distortion of angles does not exceed 20 degrees. The greatest distortion in the corners of the frame (more than 50 degrees). The scale of the areas varies from 0.833 (in the center of the projection) to 2 (on the northern outskirts of the continents) and up to 3 or more (in the polar regions). The length scale along the equator is 0.833. There is no distortion of lengths along the parallels + -45 degrees. There are no angular distortions on the middle meridian at two points with latitudes of + -52.7 degrees.
Projection is used for many educational, reference wall and desktop world maps.
The use of the results of topographic and geodetic works is greatly simplified if these results are attributed to the simplest - a rectangular coordinate system on a plane. In such a coordinate system, many geodetic problems in small areas of terrain and on maps are solved by applying simple formulas of analytical geometry on a plane. The law of the image of one surface on another is called projection. Cartographic projections are based on the formation of a specific mapping of the latitude parallels and longitude meridians of an ellipsoid onto some leveling or developing surface. In geometry, as you know, the most simple developable surfaces are a plane, a cylinder and a cone. This defined three families of map projections: azimuthal, cylindrical and conical ... Regardless of the type of transformation chosen, any mapping of a curved surface to a plane entails errors and distortions. For geodetic projections, projections are preferred that provide a slow increase in the distortion of the elements of geodetic constructions in them with a gradual increase in the area of \u200b\u200bthe projected territory. Especially important is the requirement that the projection provides high accuracy and ease of accounting for these distortions, and according to the simplest formulas. Errors in projection transformations arise from the accuracy of four characteristics:
conformal - the truth of the shape of any object;
equal area - equality of areas;
equidistance - the truth of distance measurements;
the truth of the directions.
None of the cartographic projections can provide the accuracy of maps on a plane for all of the above characteristics.
By the nature of distortion cartographic projections are subdivided into conformal, equal-area and arbitrary (in special cases, equidistant).
Conformal (conformal ) projections are those in which there are no distortions of the angles and azimuths of linear elements. These projections retain the angles without distortion (for example, the angle should always be right between north and east) and the shape of small objects, but their lengths and areas are sharply deformed in them. It should be noted that maintaining corners for large areas is difficult and can only be achieved in small areas.
Equal (equal area) projections are called projections in which the areas of the corresponding areas on the surface of the ellipsoids and on the plane are identically equal (proportional). In these projections, the angles and shapes of objects are distorted.
Arbitrary projections have distortions in angles, areas and lengths, but these distortions are distributed over the map in such a way that they are minimal in the central part and increase in the periphery. A special case of arbitrary projections are equidistant (equidistant)in which there are no length distortions in one of the directions: along the meridian or along the parallel.
Equidistant are called projections that preserve length in one of the main directions. As a rule, these are projections with an orthogonal cartographic grid. In these cases, the main directions are along the meridmans and parallels. Equidistant projections along one of the directions are determined accordingly. The second way to construct such projections is to preserve a single scale factor along all directions from one point, or from two. Distances measured from such points will exactly match the real ones, but this rule will not apply for any other points. When choosing this type of projection, the choice of points is very important. Usually, preference is given to the points from which the greatest number of measurements are made.
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a) conical |
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b) cylindrical |
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c) azimuth |
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Figure 11. Classes of projections by construction method
Equal-azimuth projections most often used in navigation, i.e. when preservation of directions is of greatest interest. Similar to Equal Area projection, true directions can only be maintained for one or two specific points. Straight lines drawn from only these points will correspond to true directions.
By way of construction (unfolding a surface onto a plane) there are three large classes of projections: conical (a), cylindrical (b) and azimuthal (c).
Conic projections are formed on the basis of projection of the earth's surface onto the lateral surface of a cone, oriented in a certain way relative to the ellipsoid. In direct conic projections, the axes of the globe and the cone coincide, with a secant or tangent cone selected. After design, the lateral surface of the cone is cut along one of the generatrices and unfolded into a plane. Depending on the size of the displayed area in conic projections, one or two parallels are taken, 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 deviations of scales from unity. Such parallels are called standard. A feature of conic projections is that their center lines coincide with the middle parallels. Consequently, conical projections are convenient for depicting areas located in middle latitudes and significantly elongated in longitude. That is why many maps of the former Soviet Union are drawn in these projections.
Cylindrical projectionsare formed on the basis of the projection of the earth's surface onto the lateral surface of the cylinder, oriented in a certain way relative to the earth's ellipsoid. In straight cylindrical projections, parallels and meridians are depicted by two families of straight parallel lines perpendicular to each other. Thus, a rectangular grid of cylindrical projections is defined. Cylindrical projections can be considered as a special case of conical projections, when the vertex of the cone is referred to infinity ( \u003d 0). There are different ways to form cylindrical projections. The cylinder can be tangent to the ellipsoid or intersecting it. In the case of using a tangent cylinder, the length measurement accuracy is maintained along the equator. If a secant cylinder is used - along two standard parallels, symmetrical about the equator. Direct, oblique and transverse cylindrical projections are used, depending on the location of the imaged area. Cylindrical projections are used for small and large scale maps.
Azimuth projections are formed by projecting the earth's surface onto a certain plane oriented in a certain way relative to the ellipsoid. In them, the parallels are depicted as concentric circles, and the meridians are represented by a bundle of straight lines emanating from the center of the circle. The angles between the meridians of the projections are equal to the corresponding differences in longitudes. The intervals between parallels are determined by the adopted character of the image (conformal or otherwise). The normal projection grid is orthogonal. Azimuthal projections can be viewed as a special case of conic projections in which \u003d 1.
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, in turn, depends on the location of the territory. Depending on the distortion, azimuthal projections are classified as conformal, equal, and intermediate.
There is a wide variety of projections: pseudocylindrical, polyconic, pseudoazimuth and others. The possibility of conditions for the optimal solution of the assigned tasks depends on the correct choice of cartographic projection. The choice of projections is due to many factors that can be conditionally combined into three groups.
The first group of factors characterizes the object of mapping from the point of view of the geographic location of the study area, its size, configuration, and the significance of its individual parts.
The second group includes the factors that characterize the created map. This group includes the content and purpose of the map as a whole, methods and conditions for its use in solving GIS problems, requirements for the accuracy of their solution.
The third group includes factors that characterize the resulting cartographic projection. This is a condition for ensuring a minimum of distortion, the maximum permissible values \u200b\u200bof distortion, the nature of their distribution, the curvature of the image of the meridians and parallels.
The choice of cartographic projections is proposed to be carried out in two stages.
At the first stage, a set of projections is established taking into account the factors of the first and second groups. In this case, it is necessary that the central lines or projection points, near which the scales change little, are in the center of the study area, and the central lines coincide, if possible, with the direction of the greatest distribution of these territories. At the second stage, the desired projection is determined.
Consider the choice of different projections depending on the location of the study area. Azimuth projections are chosen, as a rule, to represent the territories of the polar regions. Cylindrical projections are preferred for areas located near and symmetrically to the equator and extended in longitude. Conic projections should be used for the same area, but not symmetrical about the equator or located in mid-latitudes.
For all projections of the selected population, the partial scales and distortions are calculated using the formulas of mathematical cartography. Preference should be given, of course, to the projection that has the least distortion, a simpler type of cartographic grid, and, under equal conditions, a simpler mathematical projection apparatus. When considering using equal area projections, consider the size of the area of \u200b\u200binterest and the magnitude and distribution of angular distortion. Small areas are displayed with much less angular distortion when using equal area projections, which can be useful when the area and shapes of objects are important. In the case when the problem of determining the shortest distances is being solved, it is better to use projections that do not distort directions. The choice of projection is one of the main processes of creating a GIS.
When solving the problems of mapping in subsoil use on the territory of Russia, two projections are most often used, described below.
Modified simple polyconic projection it is applied as multifaceted, i.e. each sheet is defined in its own version of the projection.
Figure 12. Nomenclature trapezoids of sheets of scale 1: 200000 in polyconic projection
The features of the modified simple polyconic projection and the distribution of distortions within individual sheets of a million scale are as follows:
all meridians are depicted by straight lines, there are no length distortions at the extreme parallels and on the meridians that are ± 2º from the mean,
the extreme parallels of each sheet (north and south) are circular arcs, the centers of these parallels are located on the middle meridian, their length is not distorted, the middle parallels are determined by proportional division in latitude along the rectilinear meridians,
The earth's surface, taken as the surface of an ellipsoid, is divided by lines of meridians and parallels on a trapezoid. Trapeziums are depicted on separate sheets in the same projection (for a map with a scale of 1: 1,000,000 in a modified simple polyconic). The sheets of the International Map of the World at a scale of 1: 1,000,000 have certain trapezoid sizes - along the meridians of 4 degrees, along the parallels of 6 degrees; at a latitude of 60 to 76 degrees, the sheets are doubled, they have dimensions along the parallels 12; above 76 degrees, the four sheets are united and their parallel size is 24 degrees.
The use of a projection as a multifaceted one is inevitably associated with the introduction of the nomenclature, i.e. single sheet designation systems. For a map of the millionth scale, the designation of trapeziums along latitudinal zones is adopted, where in the direction from the equator to the poles, the designation is carried out by letters of the Latin alphabet (A, B, C, etc.) and by columns in Arabic numerals, which are counted from the meridian with a longitude of 180 (by Greenwich) counterclockwise. The sheet on which Yekaterinburg is located, for example, has the O-41 nomenclature.
Figure 13. Nomenclature division of the territory of Russia
The advantage of the modified simple polyconic projection, applied as a multifaceted one, is a small amount of distortion. Analysis within the map sheet showed that the distortions of lengths do not exceed 0.10%, area 0.15%, angles 5´ and are practically not perceptible. The disadvantage of this projection is the appearance of breaks when connecting sheets along the meridians and parallels.
Conformal (conformal) pseudocylindrical Gauss-Kruger projection. To apply such a projection, the surface of the earth's ellipsoid is divided into zones enclosed between two meridians with a difference in longitudes of 6 or 3 degrees. Meridians and parallels are depicted by curves symmetrical about the axial meridian of the zone and the equator. The axial meridians of six-degree zones coincide with the central meridians of the map sheets at a scale of 1: 1,000,000. The ordinal number is determined by the formula
where N is the column number of the map sheet at a scale of 1: 1 000 000.
D 
the lengths of the axial meridians of the six-degree zones are determined by the formula
L 0 \u003d 6n - 3, where n is the zone number.
The rectangular coordinates x and y within the zone are calculated relative to the equator and the axial meridian, which are depicted by straight lines
Figure 14. Conformal pseudocylindrical Gauss-Kruger projection
Within the territory of the former USSR, the abscissas of the Gauss-Kruger coordinates are positive; ordinates are positive to the east, negative to the west of the axial meridian. To avoid negative ordinates, the points of the axial meridian are conventionally assigned the value y \u003d 500,000 m, with the obligatory indication of the number of the corresponding zone in front. For example, if a point is located in zone 11 at 25,075m east of the axial meridian, then the value of its ordinate is written as follows: y \u003d 11,525,075 m: if the point is located west of the axial meridian of this zone at the same distance, then y \u003d 11 474 925 m.
In conformal projection, the angles of triangulation triangles are not distorted, i.e. remain the same as on the surface of the earth's ellipsoid. The scale of the image of linear elements on the plane is constant at a given point and does not depend on the azimuth of these elements: linear distortions on the axial meridian are equal to zero and gradually increase with distance from it: at the edge of the six-degree zone they reach their maximum value.
In the countries of the Western Hemisphere, the Universal Transverse Cylindrical Mercator (UTM) projection in six-degree zones is used for compiling topographic maps. This projection is close in its properties and distortion distribution to the Gauss-Kruger projection, but on the axial meridian of each zone the scale is m \u003d 0.9996, and not one. The UTM projection is obtained by double projection - an ellipsoid onto a ball, and then a ball onto a plane in the Mercator projection.
Figure 15. Coordinate transformation in geographic information systems
The presence in the GIS of software that performs projection transformations makes it easy to transfer data from one projection to another. This is sometimes necessary if the received initial data exists in a projection that does not coincide with the one selected in your project or you need to change the projection of the project data to solve a specific problem. The transition from one projection to another is called projection transformations. It is possible to translate the coordinates of digital data, originally entered in the conditional coordinates of the digitizer or raster substrate using plane transformations.
Each spatial object, in addition to a spatial reference, has some meaningful essence, and in the next chapter we will consider the possibilities of describing it.