Aircraft - Aviation modeling and navigation. Conic projections: type of cartographic grid, distribution of distortions, purpose The map is conical

Let's roll a cone from a sheet of paper in the form of a shop "pound". Let's put the cone on our wire globe so that the top of the cone is on the continuation of the axis of the globe above the "north pole". Then the cone will touch the globe along some parallel - more southern if the cone is sharp, more northern if the cone is obtuse. Let us cut the meridians along the equator and at the pole and, assuming that all the parallels, except for the parallel of contact, are elastic, we will straighten the meridians so that the meridians and parallels coincide with the surface of the cone. Cutting again the grid (together with the paper) along one of the meridians and unfolding it onto a plane, we obtain an equidistant conic projection that preserves lengths along all meridians and along the tangent parallel. The lengths of all other parallels are exaggerated, this exaggeration increases with the distance from the tangency parallel, and therefore the areas of individual cells are also exaggerated.

Like cylindrical projections, in order to obtain an equal-area conic projection, the lengths of all meridians should be shortened so that the area of each cell of the projection is equal in size to the surface of the corresponding cell on the globe. In contrast, in a conformal conic projection, the meridians lengthen to the extent that the parallels are exaggerated; the degree of elongation increases with distance from the tangency parallel.

In cartographic practice, instead of a tangent, they often take a cone that cuts the globe along two parallels. This technique improves the distribution of distortions somewhat: between the parallels of the section, the image will be underestimated against nature, outside the parallels of the section, it will be exaggerated; the main scale will be preserved along the two parallels of the section.

All conic projections have parallels in the form of concentric circles and rectilinear meridians emanating from the center of the parallels at angles proportional to the corresponding angles in nature.

It is easy to move from the equidistant conic projection to the widely used Bonn projection. To do this, we save the circular concentric parallels and the middle meridian from the conic projection. We will obtain other meridians by setting aside the distances between the meridians in kind on each parallel (of course, after transferring them to the map scale) and connecting the obtained points with smooth curves.

The Bonn projection preserves the lengths along all parallels and the middle meridian and transmits without distortion the area of each cell; she is equal. The distance between the parallels of the grid, which are concentric circles, is everywhere constant and equal to the distance between the parallels in nature. Thus, a small trapezoid on the globe and on the projection has equal bases (parallel segments) and height.

Lecture plan
1. Classification of projections according to the type of normal cartographic grid.
2. Classification of projections depending on the orientation of the auxiliary cartographic surface.
3. Choice of projections.
4. Recognition of projections.

6.1. CLASSIFICATION OF PROJECTIONS BY THE TYPE OF NORMAL GRID

In cartographic practice, the classification of projections according to the type of auxiliary geometric surface, which can be used in their construction, is common. From this point of view, projections are distinguished: cylindrical, when the lateral surface of the cylinder serves as an auxiliary surface; conical, when the auxiliary plane is the lateral surface of the cone; azimuthal, when the auxiliary surface is a plane (picture plane).
The surfaces on which the globe is projected can be tangent to it or secant to it. They can also be oriented differently.
Projections, in the construction of which the axes of the cylinder and the cone were aligned with the polar axis of the globe, and the picture plane on which the image was projected, was placed tangentially at the pole point, are called normal.
The geometric construction of these projections is very clear.

6.1.1. Cylindrical projections

For simplicity of reasoning, instead of an ellipsoid, we use a ball. We enclose the ball in a cylinder tangent to the equator (Fig. 6.1, a).

Rice. 6.1. Construction of a cartographic grid in an equal-area cylindrical projection

We continue the planes of the meridians PA, PB, PV, ... and take the intersection of these planes with the side surface of the cylinder as the image of the meridians on it. If we cut the side surface of the cylinder along the generatrix aAa 1 and deploy it on a plane, then the meridians will be depicted as parallel equally spaced straight lines aAa 1 , bBB 1 , vVv 1 ... perpendicular to the equator ABV.
The image of parallels can be obtained in various ways. One of them is the continuation of the planes of parallels until they intersect with the surface of the cylinder, which will give a second family of parallel straight lines in the development, perpendicular to the meridians.
The resulting cylindrical projection (Fig. 6.1, b) will be equal, since the lateral surface of the spherical belt AGDE, equal to 2πRh (where h is the distance between the planes AG and ED), corresponds to the area of the image of this belt in the scan. The main scale is maintained along the equator; private scales increase along the parallel, and decrease along the meridians as they move away from the equator.
Another way to determine the position of the parallels is based on the preservation of the lengths of the meridians, i.e., on the preservation of the main scale along all meridians. In this case, the cylindrical projection will be equidistant along the meridians.
For equiangular Cylindrical projection requires constant scale in all directions at any point, which requires increasing the scale along the meridians as you move away from the equator in accordance with the increase in scale along the parallels at the corresponding latitudes.
Often, instead of a tangent cylinder, a cylinder is used that cuts the sphere along two parallels (Fig. 6.2), along which the main scale is preserved during scanning. In this case, partial scales along all parallels between the parallels of the section will be smaller, and on the remaining parallels - larger than the main scale.

Rice. 6.2. Cylinder that cuts the ball along two parallels

6.1.2. Conic projections

To construct a conic projection, we enclose the ball in a cone tangent to the ball along the parallel ABCD (Fig. 6.3, a).

Rice. 6.3. Construction of a cartographic grid in an equidistant conic projection

Similarly to the previous construction, we continue the planes of the meridians PA, PB, PV, ... and take their intersections with the lateral surface of the cone as the image of the meridians on it. After unrolling the lateral surface of the cone on a plane (Fig. 6.3, b), the meridians will be depicted by radial straight lines TA, TB, TV, ..., emanating from the point T. Please note that the angles between them (the convergence of the meridians) will be proportional (but are not equal) to differences in longitudes. Along the tangent parallel ABV (arc of a circle with radius TA) the main scale is preserved.
The position of other parallels, represented by arcs of concentric circles, can be determined from certain conditions, one of which - the preservation of the main scale along the meridians (AE = Ae) - leads to a conic equidistant projection.

6.1.3. Azimuthal projections

To construct an azimuthal projection, we will use a plane tangent to the ball at the point of the pole P (Fig. 6.4). Intersections of meridian planes with a tangent plane give an image of the meridians Pa, Pe, Pv, ... in the form of straight lines, the angles between which are equal to the differences in longitude. Parallels, which are concentric circles, can be defined in various ways, for example, drawn with radii equal to straightened arcs of meridians from the pole to the corresponding parallel PA = Pa. Such a projection would equidistant on meridians and preserves the main scale along them.

Rice. 6.4. Construction of a cartographic grid in the azimuthal projection

A special case of azimuthal projections are promising projections built according to the laws of geometric perspective. In these projections, each point on the surface of the globe is transferred to the picture plane along the rays emerging from one point FROM called point of view. Depending on the position of the point of view relative to the center of the globe, the projections are divided into:

central - point of view coincides with the center of the globe;
stereographic - the point of view is located on the surface of the globe at a point diametrically opposite to the point of contact of the picture plane with the surface of the globe;
external - the point of view is taken out of the globe;
orthographic - the point of view is taken out to infinity, i.e. the projection is carried out by parallel rays.

Rice. 6.5. Types of perspective projections: a - central;
b - stereographic; in - external; d - orthographic.

6.1.4. Conditional projections

Conditional projections are projections for which it is impossible to find simple geometric analogues. They are built based on some given conditions, for example, the desired type of geographic grid, one or another distribution of distortions on the map, a given type of grid, etc. In particular, pseudo-cylindrical, pseudo-conical, pseudo-azimuthal and other projections obtained by converting one or several original projections.
At pseudocylindrical equator and parallel projections are straight lines parallel to each other (which makes them similar to cylindrical projections), and meridians are curves symmetrical about the average rectilinear meridian (Fig. 6.6)

Rice. 6.6. View of the cartographic grid in pseudocylindrical projection.

At pseudoconical parallel projections are arcs of concentric circles, and meridians are curves symmetrical about the average rectilinear meridian (Fig. 6.7);

Rice. 6.7. Map grid in one of the pseudoconic projections

Building a grid in polyconic projection can be represented by projecting segments of the globe's graticule onto the surface several tangent cones and subsequent development into the plane of the stripes formed on the surface of the cones. The general principle of such a design is shown in Figure 6.8.

Rice. 6.8. The principle of constructing a polyconic projection:
a - the position of the cones; b - stripes; c - sweep

in letters S the tops of the cones are indicated in the figure. For each cone, a latitudinal section of the globe surface is projected, adjacent to the parallel of the touch of the corresponding cone.
For the external appearance of cartographic grids in a polyconic projection, it is characteristic that the meridians are in the form of curved lines (except for the middle one - straight), and the parallels are arcs of eccentric circles.
In polyconic projections used to build world maps, the equatorial section is projected onto a tangent cylinder, therefore, on the resulting grid, the equator has the form of a straight line perpendicular to the middle meridian.
After scanning the cones, an image of these sections is obtained in the form of stripes on a plane (Fig. 6.8, b); the stripes touch along the middle meridian of the map. The mesh receives its final form after the elimination of gaps between the strips by stretching (Fig. 6.8, c).

Rice. 6.9. A cartographic grid in one of the polycones

Polyhedral projections - projections obtained by projecting onto the surface of a polyhedron (Fig. 6.10), tangent or secant to the ball (ellipsoid). Most often, each face is an isosceles trapezoid, although other options are possible (for example, hexagons, squares, rhombuses). A variety of polyhedral are multi-lane projections, moreover, 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. Topographic and survey-topographic are created exclusively in a multifaceted projection, and the frame of each sheet is a trapezoid composed by lines of meridians and parallels. You have to "pay" for this - a block of map sheets cannot be combined along a common frame without gaps.

Rice. 6.10. Polyhedral projection scheme and arrangement of map sheets

It should be noted that today auxiliary surfaces are not used to obtain map projections. No one puts a ball in a cylinder and puts a cone on it. These are just geometric analogies that allow us to understand the geometric essence of the projection. The search for projections is performed analytically. Computer modeling allows you to quickly calculate any projection with the given parameters, and automatic graph plotters easily draw the appropriate grid of meridians and parallels, and, if necessary, an isocol map.
There are special atlases of projections that allow you to choose the right projection for any territory. Recently, electronic projection atlases have been created, with the help of which it is easy to find a suitable grid, immediately evaluate its properties, and, if necessary, carry out certain modifications or transformations in an interactive mode.

6.2. CLASSIFICATION OF PROJECTIONS DEPENDING ON THE ORIENTATION OF THE AUXILIARY CARTOGRAPHIC SURFACE

Normal projections - the projection plane touches the globe at the pole point or the axis of the cylinder (cone) coincides with the axis of rotation of the Earth (Fig. 6.11).

Rice. 6.11. Normal (direct) projections

Transverse projections - the projection plane touches the equator at some point or the axis of the cylinder (cone) coincides with the plane of the equator (Fig. 6.12).

Rice. 6.12. Transverse projections

oblique projections - the projection plane touches the globe at any given point (Fig. 6.13).

Rice. 6.13. oblique projections

Of the oblique and transverse projections, oblique and transverse cylindrical, azimuth (perspective) and pseudo-azimuth projections are most often used. Transverse azimuths are used for maps of the hemispheres, oblique - for territories that have a rounded shape. Maps of the continents are often made in transverse and oblique azimuth projections. The Gauss-Kruger transverse cylindrical projection is used for state topographic maps.

6.3. SELECTION OF PROJECTIONS

The choice of projections is influenced by many factors, which can be grouped as follows:

geographical features of the mapped territory, its position on the globe, size and configuration;
the purpose, scale and subject of the map, the intended range of consumers;
conditions and methods of using the map, tasks that will be solved using the map, requirements for the accuracy of measurement results;
features of the projection itself - the magnitude of distortions of lengths, areas, angles and their distribution over the territory, the shape of the meridians and parallels, their symmetry, the image of the poles, the curvature of the lines of the shortest distance.

The first three groups of factors are set initially, the fourth depends on them. If a map is being drawn up for navigation, the Mercator conformal cylindrical projection must be used. If Antarctica is being mapped, the normal (polar) azimuthal projection will almost certainly be adopted, and so on.
The significance of these factors can be different: in one case, visibility is put in the first place (for example, for a school wall map), in another, the features of using the map (navigation), in the third, the position of the territory on the globe (polar region). Any combinations are possible, and consequently - and different variants of projections. Moreover, the choice is very large. But still, some preferred and most traditional projections can be indicated.
World Maps usually compose in cylindrical, pseudocylindrical and polyconical projections. To reduce distortion, secant cylinders are often used, and pseudocylindrical projections are sometimes given with discontinuities on the oceans.
Hemispheric maps always built in azimuthal projections. For the western and eastern hemispheres, it is natural to take transverse (equatorial) projections, for the northern and southern hemispheres - normal (polar), and in other cases (for example, for the continental and oceanic hemispheres) - oblique azimuthal projections.
Continent maps Europe, Asia, North America, South America, Australia and Oceania are most often built in equal area oblique azimuth projections, for Africa they take transverse projections, and for Antarctica - normal azimuth projections.
Maps of selected countries , administrative regions, provinces, states are performed in oblique conformal and equal-area conic or azimuth projections, but much depends on the configuration of the territory and its position on the globe. For small areas, the problem of choosing a projection loses its relevance; different conformal projections can be used, bearing in mind that area distortions in small areas are almost imperceptible.
Topographic maps Ukraine is created in the transverse cylindrical projection of Gauss, and the United States and many other Western countries - in the universal transverse cylindrical projection of Mercator (abbreviated as UTM). Both projections are close in their properties; in fact, both are multi-cavity.
Maritime and aeronautical charts are always given exclusively in the cylindrical Mercator projection, and thematic maps of the seas and oceans are created in the most diverse, sometimes quite complex projections. For example, for the joint display of the Atlantic and Arctic oceans, special projections with oval isocols are used, and for the image of the entire World Ocean, equal projections with discontinuities on the continents are used.
In any case, when choosing a projection, especially for thematic maps, it should be borne in mind that map distortion is usually minimal in the center and increases rapidly towards the edges. In addition, the smaller the scale of the map and the wider the spatial coverage, the more attention should be paid to the "mathematical" factors of projection selection, and vice versa - for small areas and large scales, "geographical" factors become more significant.

6.4. PROJECTION RECOGNITION

To recognize the projection in which the map is drawn means to establish its name, to determine whether it belongs to one or another species, class. This is necessary in order to have an idea about the properties of the projection, the nature, distribution and magnitude of distortion - in a word, in order to know how to use the map, what can be expected from it.
Some normal projections at once recognized by the appearance of meridians and parallels. For example, normal cylindrical, pseudocylindrical, conical, azimuth projections are easily recognizable. But even an experienced cartographer does not immediately recognize many arbitrary projections; special measurements on the map will be required to reveal their equiangularity, equivalence, or equidistance in one of the directions. For this, there are special techniques: first, the shape of the frame (rectangle, circle, ellipse) is determined, how the poles are depicted, then the distances between adjacent parallels along the meridian, the area of \u200b\u200bneighboring cells of the grid, the angles of intersection of the meridians and parallels, the nature of their curvature, etc. .P.
There are special projection tables for maps of the world, hemispheres, continents and oceans. After carrying out the necessary measurements on the grid, you can find the name of the projection in such a table. This will give an idea of its properties, will allow you to evaluate the possibilities of quantitative determinations on this map, and select the appropriate map with isocoles for making corrections.

Questions for self-control:

How are projections classified according to the type of auxiliary surface?
How are projections classified depending on the position of the axis of the auxiliary surface relative to the axis of rotation of the globe?
What is the principle of constructing a polyconic projection?
How are azimuthal projections obtained?
How to get an oblique projection on a tangent cylinder?
How to get azimuth equatorial projection?
What types of perspective projections do you know? Give them a brief description.
What projections are considered conditional?
What factors influence the choice of map projection?
In what projections are world maps, sea and aeronautical maps, topographic maps, maps of individual countries, maps of continents, maps of the hemispheres usually compiled?
How are projections identified?

In the transition from the physical surface of the Earth to its display on a plane (on a map), two operations are performed: projecting the earth's surface with its complex relief onto the surface of an earth ellipsoid, the dimensions of which are established by means of geodetic and astronomical measurements, and the image of the ellipsoid surface on a plane using one of the cartographic projections.
A map projection is a specific way of displaying the surface of an ellipsoid on a plane.
The display of the earth's surface on a plane is carried out in various ways. The simplest one is perspective . Its essence lies in projecting an image from the surface of the Earth model (globe, ellipsoid) onto the surface of a cylinder or cone, followed by a turn into a plane (cylindrical, conical) or direct projection of a spherical image onto a plane (azimuth).
One easy way to understand how map projections change spatial properties is to visualize the projection of light through the Earth onto a surface called a projection surface.
Imagine that the surface of the Earth is transparent and has a map grid on it. Wrap a piece of paper around the earth. A light source at the center of the earth will cast shadows from the grid onto the piece of paper. You can now unfold the paper and lay it flat. The shape of the coordinate grid on a flat surface of paper is very different from its shape on the surface of the Earth (Fig. 5.1).

Rice. 5.1. Geographic coordinate system grid projected onto a cylindrical surface

The map projection distorted the cartographic grid; objects near the pole are elongated.
Building in a perspective way does not require the use of the laws of mathematics. Please note that in modern cartography, cartographic grids are built analytical (mathematical) way. Its essence lies in the calculation of the position of nodal points (points of intersection of meridians and parallels) of the cartographic grid. The calculation is performed on the basis of solving a system of equations that relate the geographic latitude and geographic longitude of nodal points ( φ, λ ) with their rectangular coordinates ( x, y) on surface. This dependence can be expressed by two equations of the form:

x = f 1 (φ, λ); (5.1)
y = f 2 (φ, λ), (5.2)

called map projection equations. They allow you to calculate rectangular coordinates x, y displayed point by geographic coordinates φ and λ . The number of possible functional dependencies and, therefore, projections is unlimited. It is only necessary that each point φ , λ the ellipsoid was depicted on the plane by a uniquely corresponding point x, y and that the image is continuous.

5.2. DISTORTION

Decomposing a spheroid onto a plane is no easier than flattening a piece of watermelon peel. When going to a plane, as a rule, angles, areas, shapes and lengths of lines are distorted, so for specific purposes it is possible to create projections that will significantly reduce any one type of distortion, for example, areas. Cartographic distortion is a violation of the geometric properties of sections of the earth's surface and objects located on them when they are depicted on a plane. .
Distortions of all kinds are closely related. They are in such a relationship that a decrease in one type of distortion immediately leads to an increase in another. As area distortion decreases, angle distortion increases, and so on. Rice. Figure 5.2 shows how 3D objects are compressed to fit on a flat surface.

Rice. 5.2. Projecting a spherical surface onto a projection surface

On different maps, distortions can be of different sizes: on large-scale maps they are almost imperceptible, but on small-scale maps they can be very large.
In the middle of the 19th century, the French scientist Nicolas August Tissot gave a general theory of distortions. In his work, he proposed to use special distortion ellipses, which are infinitesimal ellipses at any point on the map, representing infinitesimal circles at the corresponding point on the surface of the earth's ellipsoid or globe. The ellipse becomes a circle at the zero distortion point. Changing the shape of the ellipse reflects the degree of distortion of angles and distances, and the size - the degree of distortion of areas.

Rice. 5.3. Ellipse on the map ( a) and the corresponding circle on the globe ( b)

The distortion ellipse on the map can take a different position relative to the meridian passing through its center. The orientation of the distortion ellipse on the map is usually determined by azimuth of its semi-major axis . The angle between the north direction of the meridian passing through the center of the distortion ellipse and its nearest semi-major axis is called the orientation angle of the distortion ellipse. On fig. 5.3, a this corner is marked with the letter BUT 0 , and the corresponding angle on the globe α 0 (Fig. 5.3, b).
Azimuths of any direction on the map and on the globe are always measured from the north direction of the meridian in a clockwise direction and can have values from 0 to 360°.
Any arbitrary direction ( OK) on a map or on a globe ( O 0 To 0 ) can be determined either by the azimuth of a given direction ( BUT- on the map, α - on the globe) or the angle between the semi-major axis closest to the northern direction of the meridian and the given direction ( v- on the map, u- on the globe).

5.2.1. 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:

main scale (M);
private scale .

main scale maps call the degree of general reduction of the globe to a certain size of the globe, from which the earth's surface is transferred to the plane. It allows you to judge the decrease in the length of the segments when they are transferred from the globe to the globe. The main scale is written under the southern frame of the map, but this does not mean that the segment measured anywhere on the map will correspond to the distance on the earth's surface.
The scale at a given point on the map in a given direction is called private . It is defined as the ratio of an infinitesimal segment on a map dl To to the corresponding segment on the surface of the ellipsoid dl Z . The ratio of the private scale to the main one, denoted by μ , characterizes the distortion of lengths

(5.3)

To assess the deviation of a particular scale from the main one, use the concept zoom in (FROM) defined by the relation

(5.4)

From formula (5.4) it follows that:

at FROM= 1 the partial scale is equal to the main scale ( µ = M), i.e., there are no length distortions at a given point of the map in a given direction;
at FROM> 1 partial scale larger than the main one ( µ > M);
at FROM < 1 частный масштаб мельче главного (µ < М ).

For example, if the main scale of the map is 1: 1,000,000, zoom in FROM equals 1.2, then µ \u003d 1.2 / 1,000,000 \u003d 1/833,333, i.e. one centimeter on the map corresponds to approximately 8.3 km on the ground. The private scale is larger than the main one (the value of the fraction is larger).
When depicting the surface of a globe on a plane, the partial scales will be numerically larger or smaller than the main scale. If we take the main scale equal to one ( M= 1), then the partial scales will be numerically greater or less than unity. In this case under the private scale, numerically equal to the scale increase, one should understand the ratio of an infinitesimal segment at a given point on the map in a given direction to the corresponding infinitesimal segment on the globe:

(5.5)

Partial Scale Deviation (µ )from unity determines the length distortion at a given point on the map in a given direction ( V):

V = µ - 1 (5.6)

Often the length distortion is expressed as a percentage of unity, i.e., to the main scale, and is called relative length distortion :

q = 100(µ - 1) = V×100(5.7)

For example, when µ = 1.2 length distortion V= +0.2 or relative length distortion V= +20%. This means that a segment of length 1 cm, taken on the globe, will be displayed on the map as a segment of length 1.2 cm.
It is convenient to judge the presence of length distortion on the map by comparing the size of the meridian segments between adjacent parallels. If they are everywhere equal, then there is no distortion of the lengths along the meridians, if there is no such equality (Fig. 5.5 segments AB and CD), then there is a distortion of the line lengths.

Rice. 5.4. Part of a map of the Eastern Hemisphere showing cartographic distortions

If a map depicts such a large area that it shows both the equator 0º and the parallel 60° of latitude, then it is not difficult to determine from it whether there is a distortion of lengths along the parallels. To do this, it is enough to compare the length of the segments of the equator and parallels with a latitude of 60 ° between adjacent meridians. It is known that the parallel of 60° latitude is two times shorter than the equator. If the ratio of the indicated segments on the map is the same, then there is no distortion of the lengths along the parallels; otherwise, it exists.
The largest indicator of length distortion at a given point (the major semi-axis of the distortion ellipse) is denoted by the Latin letter a, and the smallest one (semi-minor axis of the distortion ellipse) - b. Mutually perpendicular directions in which the largest and smallest indicators of length distortion act, called the main directions .
To assess various distortions on maps, of all partial scales, partial scales in two directions are of greatest importance: along meridians and along parallels. private scale along the meridian usually denoted by the letter m , and the private scale parallel - letter n.
Within the limits of small-scale maps of relatively small territories (for example, Ukraine), the deviations of the length scales from the scale indicated on the map are small. Errors in measuring lengths in this case do not exceed 2 - 2.5% of the measured length, and they can be neglected when working with school maps. Some maps for approximate measurements are accompanied by a measuring scale, accompanied by explanatory text.
On the nautical charts , built in the Mercator projection and on which the loxodrome is depicted by a straight line, no special linear scale is given. Its role is played by the eastern and western frames of the map, which are meridians divided into divisions through 1′ in latitude.
In maritime navigation, distances are measured in nautical miles. Nautical mile is the average length of the meridian arc of 1′ in latitude. It contains 1852 m. Thus, the frames of the sea chart are actually divided into segments equal to one nautical mile. By determining in a straight line the distance between two points on the map in minutes of the meridian, the actual distance in nautical miles along the loxodrome is obtained.

Figure 5.5. Measuring distances on a sea chart.

5.2.2. Corner distortion

Angular distortions follow logically from length distortions. 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.
For angle distortion between the lines of the cartographic grid, they take the value of their deviation from 90 ° and designate it with a Greek letter ε (epsilon).
ε = Ө - 90°, (5.8)
where in Ө (theta) - the angle measured on the map between the meridian and the parallel.

Figure 5.4 indicates that the angle Ө is equal to 115°, therefore, ε = 25°.
At a point where the angle of intersection of the meridian and the parallel remains right on the map, the angles between other directions can be changed on the map, since at any given point the amount of angle distortion can change with direction.
For the general indicator of the distortion of angles ω (omega), the greatest distortion of the angle at a given point is taken, equal to the difference between its magnitude on the map and on the surface of the earth's ellipsoid (ball). When known x indicators a and b value ω determined by the formula:

(5.9)

5.2.3. Area distortion

Area distortions follow logically from length distortions. 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.
A simple way to identify the distortion of this type is to compare the areas of the cells of the cartographic grid, limited by parallels of the same name: if the areas of the cells are equal, there is no distortion. This takes place, in particular, on the map of the hemisphere (Fig. 4.4), on which the shaded cells differ in shape, but have the same area.
Area Distortion Index (R) is calculated as the product of the largest and smallest indicators of length distortion at a given location on the map
p = a×b (5.10)
The main directions at a given point on the map may coincide with the lines of the cartographic grid, but may not coincide with them. Then the indicators a and b according to famous m and n calculated according to the formulas:

(5.11)
(5.12)

The distortion factor included in the equations R recognize in this case by the product:

p = m×n×cos ε, (5.13)

Where ε (epsilon) - the deviation of the angle of intersection of the cartographic grid from 9 0°.

5.2.4. Form distortion

Shape distortion consists in the fact that the shape of the site or the territory occupied by the object on the map is different from their shape on the level surface of the Earth. The presence of this type of distortion on the map can be established by comparing the shape of the cartographic grid cells located at the same latitude: if they are the same, then there is no distortion. In figure 5.4, two shaded cells with a difference in shape indicate the presence of a distortion of this type. It is also possible to identify the distortion of the shape of a certain object (continent, island, sea) by the ratio of its width and length on the analyzed map and on the globe.
Shape Distortion Index (k) depends on the difference of the largest ( a) and least ( b) indicators of length distortion in a given location of the map and is expressed by the formula:

(5.14)

When researching and choosing a map projection, use isocoles - lines of equal distortion. They can be plotted on the map as dotted lines to show the amount of distortion.

Rice. 5.6. Isocoles of the greatest distortion of angles

5.3. CLASSIFICATION OF PROJECTIONS BY THE NATURE OF DISTORTIONS

For various purposes, projections of various types of distortion are created. The nature of the projection distortion is determined by the absence of certain distortions in it. (angles, lengths, areas). Depending on this, all cartographic projections are divided into four groups according to the nature of distortions:
- equiangular (conformal);
- equidistant (equidistant);
— equal (equivalent);
- arbitrary.

5.3.1. Equangular projections

Equangular such projections are called in which directions and angles are depicted without distortion. The angles measured on the conformal projection maps are equal to the corresponding angles on the earth's surface. An infinitely small circle in these projections always remains a circle.
In conformal projections, the scales of lengths at any point in all directions are the same, therefore they have no distortion of the shape of infinitesimal figures and no distortion of angles (Fig. 5.7, B). This general property of conformal projections is expressed by the formula ω = 0°. But the forms of real (final) geographical objects occupying entire sections on the map are distorted (Fig. 5.8, a). Conformal projections have especially large area distortions (which is clearly demonstrated by distortion ellipses).

Rice. 5.7. View of distortion ellipses in equal-area projections — BUT, equiangular - B, arbitrary - AT, including equidistant along the meridian - G and equidistant along the parallel - D. The diagrams show 45° angle distortion.

These projections are used to determine directions and plot routes along a given azimuth, so they are always used on topographic and navigational maps. The disadvantage of conformal projections is that areas are greatly distorted in them (Fig. 5.7, a).

Rice. 5.8. Distortions in cylindrical projection:
a - equiangular; b - equidistant; c - equal

5.6.2. Equidistant projections

Equidistant projections are projections in which the scale of the lengths of one of the main directions is preserved (remains unchanged) (Fig. 5.7, D. Fig. 5.7, E.) They are used mainly to create small-scale reference maps and star charts.

5.6.3. Equal Area Projections

Equal-sized projections are called in which there are no area distortions, that is, the area of \u200b\u200bthe figure measured on the map is equal to the area of \u200b\u200bthe same figure on the surface of the Earth. In equal area map projections, the scale of the area has the same value everywhere. This property of equal-area projections can be expressed by the formula:

P = a × b = Const = 1 (5.15)

An inevitable consequence of the equal area of these projections is a strong distortion of their angles and shapes, which is well explained by the distortion ellipses (Fig. 5.7, A).

5.6.4. Arbitrary projections

to arbitrary include projections in which there are distortions of lengths, angles and areas. The need to use arbitrary projections is explained by the fact that when solving some problems, it becomes necessary to measure angles, lengths and areas on one map. But no projection can be at the same time conformal, equidistant, and equal area. It has already been said earlier that with a decrease in the imaged area of the Earth's surface on a plane, image distortions also decrease. When depicting small areas of the earth's surface in an arbitrary projection, the distortions of angles, lengths and areas are insignificant, and in solving many problems they can be ignored.

5.4. CLASSIFICATION OF PROJECTIONS BY THE TYPE OF NORMAL GRID

In cartographic practice, the classification of projections according to the type of auxiliary geometric surface, which can be used in their construction, is common. From this point of view, projections are distinguished: cylindrical when the side surface of the cylinder serves as the auxiliary surface; conical when the auxiliary plane is the lateral surface of the cone; azimuthal when the auxiliary surface is a plane (picture plane).
The surfaces on which the globe is projected can be tangent to it or secant to it. They can also be oriented differently.
Projections, in the construction of which the axes of the cylinder and the cone were aligned with the polar axis of the globe, and the picture plane on which the image was projected, was placed tangentially at the pole point, are called normal.
The geometric construction of these projections is very clear.

5.4.1. Cylindrical projections

For simplicity of reasoning, instead of an ellipsoid, we use a ball. We enclose the ball in a cylinder tangent to the equator (Fig. 5.9, a).

Rice. 5.9. Construction of a cartographic grid in an equal-area cylindrical projection

We continue the planes of the meridians PA, PB, PV, ... and take the intersection of these planes with the side surface of the cylinder as the image of the meridians on it. If we cut the side surface of the cylinder along the generatrix aAa 1 and deploy it on a plane, then the meridians will be depicted as parallel equally spaced straight lines aAa 1 , bBB 1 , vVv 1 ... perpendicular to the equator ABV.
The image of parallels can be obtained in various ways. One of them is the continuation of the planes of parallels until they intersect with the surface of the cylinder, which will give a second family of parallel straight lines in the development, perpendicular to the meridians.
The resulting cylindrical projection (Fig. 5.9, b) will be equal, since the lateral surface of the spherical belt AGED, equal to 2πRh (where h is the distance between the planes AG and ED), corresponds to the area of the image of this belt in the scan. The main scale is maintained along the equator; private scales increase along the parallel, and decrease along the meridians as they move away from the equator.
Another way to determine the position of the parallels is based on the preservation of the lengths of the meridians, i.e., on the preservation of the main scale along all meridians. In this case, the cylindrical projection will be equidistant along the meridians(Fig. 5.8, b).
For equiangular A cylindrical projection requires constancy of scale in all directions at any point, which requires an increase in scale along the meridians as you move away from the equator in accordance with an increase in scale along the parallels at the corresponding latitudes (see Fig. 5.8, a).
Often, instead of a tangent cylinder, a cylinder is used that cuts the sphere along two parallels (Fig. 5.10), along which the main scale is preserved during sweeping. In this case, partial scales along all parallels between the parallels of the section will be smaller, and on the remaining parallels - larger than the main scale.

Rice. 5.10. Cylinder that cuts the ball along two parallels

5.4.2. Conic projections

To construct a conic projection, we enclose the ball in a cone tangent to the ball along the parallel ABCD (Fig. 5.11, a).

Rice. 5.11. Construction of a cartographic grid in an equidistant conic projection

Similarly to the previous construction, we continue the planes of the meridians PA, PB, PV, ... and take their intersections with the lateral surface of the cone as the image of the meridians on it. After unfolding the side surface of the cone on a plane (Fig. 5.11, b), the meridians will be depicted by radial straight lines TA, TB, TV, ..., emanating from the point T. Please note that the angles between them (the convergence of the meridians) will be proportional (but are not equal) to differences in longitudes. Along the tangent parallel ABV (arc of a circle with radius TA) the main scale is preserved.
The position of other parallels, represented by arcs of concentric circles, can be determined from certain conditions, one of which - the preservation of the main scale along the meridians (AE = Ae) - leads to a conic equidistant projection.

5.4.3. Azimuthal projections

To construct an azimuthal projection, we will use a plane tangent to the ball at the point of the pole P (Fig. 5.12). Intersections of meridian planes with a tangent plane give an image of the meridians Pa, Pe, Pv, ... in the form of straight lines, the angles between which are equal to the differences in longitude. Parallels, which are concentric circles, can be defined in various ways, for example, drawn with radii equal to straightened arcs of meridians from the pole to the corresponding parallel PA = Pa. Such a projection would equidistant on meridians and preserves the main scale along them.

Rice. 5.12. Construction of a cartographic grid in the azimuthal projection

central - point of view coincides with the center of the globe;
stereographic - the point of view is located on the surface of the globe at a point diametrically opposite to the point of contact of the picture plane with the surface of the globe;
external - the point of view is taken out of the globe;
orthographic - the point of view is taken out to infinity, i.e. the projection is carried out by parallel rays.

Rice. 5.13. Types of perspective projections: a - central;
b - stereographic; in - external; d - orthographic.

5.4.4. Conditional projections

Rice. 5.14. View of the cartographic grid in pseudocylindrical projection.

At pseudoconical parallel projections are arcs of concentric circles, and meridians are curves symmetrical about the average rectilinear meridian (Fig. 5.15);

Rice. 5.15. Map grid in one of the pseudoconic projections

Rice. 5.16. The principle of constructing a polyconic projection:
a - the position of the cones; b - stripes; c - sweep

in letters S the tops of the cones are indicated in the figure. For each cone, a latitudinal section of the globe surface is projected, adjacent to the parallel of the touch of the corresponding cone.
For the external appearance of cartographic grids in a polyconic projection, it is characteristic that the meridians are in the form of curved lines (except for the middle one - straight), and the parallels are arcs of eccentric circles.
In polyconic projections used to build world maps, the equatorial section is projected onto a tangent cylinder, therefore, on the resulting grid, the equator has the form of a straight line perpendicular to the middle meridian.
After scanning the cones, these sections are imaged as stripes on a plane; the stripes touch along the middle meridian of the map. The mesh receives its final form after the elimination of gaps between the strips by stretching (Fig. 5.17).

Rice. 5.17. A cartographic grid in one of the polycones

Polyhedral projections - projections obtained by projecting onto the surface of a polyhedron (Fig. 5.18), tangent or secant to the ball (ellipsoid). Most often, each face is an isosceles trapezoid, although other options are possible (for example, hexagons, squares, rhombuses). A variety of polyhedral are multi-lane projections, moreover, 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. Topographic and survey-topographic are created exclusively in a multifaceted projection, and the frame of each sheet is a trapezoid composed by lines of meridians and parallels. You have to "pay" for this - a block of map sheets cannot be combined along a common frame without gaps.

Rice. 5.18. Polyhedral projection scheme and arrangement of map sheets

5.5. CLASSIFICATION OF PROJECTIONS DEPENDING ON THE ORIENTATION OF THE AUXILIARY CARTOGRAPHIC SURFACE

Normal projections - the projection plane touches the globe at the pole point or the axis of the cylinder (cone) coincides with the axis of rotation of the Earth (Fig. 5.19).

Rice. 5.19. Normal (direct) projections

Transverse projections - the projection plane touches the equator at some point or the axis of the cylinder (cone) coincides with the plane of the equator (Fig. 5.20).

Rice. 5.20. Transverse projections

oblique projections - the projection plane touches the globe at any given point (Fig. 5.21).

Rice. 5.21. oblique projections

5.6. SELECTION OF PROJECTIONS

The choice of projections is influenced by many factors, which can be grouped as follows:

geographical features of the mapped territory, its position on the globe, size and configuration;
the purpose, scale and subject of the map, the intended range of consumers;
conditions and methods of using the map, tasks that will be solved using the map, requirements for the accuracy of measurement results;
features of the projection itself - the magnitude of distortions of lengths, areas, angles and their distribution over the territory, the shape of the meridians and parallels, their symmetry, the image of the poles, the curvature of the lines of the shortest distance.

5.7. PROJECTION RECOGNITION

Video
Types of projections by the nature of distortions

Questions for self-control:

What elements make up the mathematical basis of the map?
What is the scale of a geographic map?
What is the main scale of a map?
What is the private scale of a map?
What is the reason for the deviation of the private scale from the main one on the geographical map?
How to measure the distance between points on a sea chart?
What is a distortion ellipse and what is it used for?
How can you determine the largest and smallest scales from the distortion ellipse?
What are the methods of transferring the surface of the earth's ellipsoid to a plane, what is their essence?
What is a map projection?
How are projections classified according to the nature of distortion?
What projections are called conformal, how to depict an ellipse of distortion on these projections?
What projections are called equidistant, how to depict an ellipse of distortions on these projections?
What projections are called equal areas, how to depict an ellipse of distortions on these projections?
What projections are called arbitrary?

Classification of map projections

Maps and map projections

A map is a reduced image of the earth's surface on a plane on a certain scale with the application of a coordinate grid and conventional signs that display earth objects.

The flight chart is the main aid for aircraft navigation. No flight can be carried out without a card.

The map on the ground is necessary for laying and digitizing the route, studying the main and alternate airfields, performing the necessary measurements and calculations in preparation for the flight, and in flight - for visual orientation, path control, and determining the position of the aircraft.

The aviation card must meet the following requirements:

1. Reliably and accurately display the state of the terrain:

2. Be visual, well-read and easy to work with.

3. The card must be with minimal angular and linear distortion,

convenient for measurements and graphic constructions.

A map projection is a way of depicting the earth's surface on a plane. All map projections differ in the following ways:

1. By the nature of the distortion;

2. According to the method of constructing the coordinate grid:

By the nature of projection distortion can be:

1. Equangular- the equality of the angles between the landmarks and the shape of the figures are preserved. Maps in conformal projection are widely used in aviation.

2. Equal- the ratio of the area of the image of a figure on the map to the area of the same figure on the earth's surface remains constant. In this projection there is no equality of angles and similarity of figures.

3. Equidistant– the scale is kept in one of the main directions (meridian and parallels).

4. Arbitrary- Neither the 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, polyconic, azimuthal.

Cylindrical projections (Mercator projections)

To make maps in a cylindrical projection, you need a model of the Earth made of a transparent material. A light source is placed in the center of the model. The model of the earth is placed in the cylinder so that it touches the walls of the cylinder with the equator. Then light is produced. Rays of light propagate in a straight line and all points and lines present on the model are projected onto the surface of the cylinder. Then the cylinder is cut, unfolded on a plane. Meridians and parallels on the maps in this projection look like mutually perpendicular lines. The projection is conformal, the scale is not the same - it is enlarged towards the poles. In this projection, nautical charts are produced.

In a conic projection, the Earth's surface is projected onto the lateral surface of a cone tangent to one of the parallels. Then the cone is cut and unfolded on the plane. Meridians in this projection are depicted as straight lines converging towards the pole, and parallels as arcs parallel to the equator. The projection is conformal, scale distortion is not large. If the axis of the cone coincides with the axis of rotation of the Earth, the projection is called normal. In normal conic projection, onboard maps of scale 1 are produced. : 4000000 (1cm = 40km), and 1 : 2500000 (1cm = 25km).

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