What map is built using a plane. map projection

12.08.2020

map projection is a way of moving from a real, geometrically complex earth's surface.

A spherical surface cannot be deployed on a plane without deformation - compression or tension. This means that every map has certain distortions. There are distortions of the lengths of areas, angles and shapes. On large-scale maps (see), distortions can be almost imperceptible, but on small-scale they can be very large. Map projections have different properties depending on the nature and size of the distortion. Among them are distinguished:

Equangular projections. They preserve the angles and shapes of small objects without distortion, but the lengths and areas of objects are sharply deformed in them. According to maps drawn up in such a projection, it is convenient to plot the routes of ships, but it is impossible to measure areas;

Equal projections. They do not distort areas, but the angles and shapes in them are strongly distorted. Maps in equal projections are convenient for determining the size of the state, ;
Equidistant. They have a constant length scale in one direction. The distortions of angles and areas are balanced in them;

Arbitrary projections. They have distortion and angles and areas in any ratio.
Projections differ not only in the nature and size of distortions, but also in the type of surface that is used in the transition from the geoid to the map plane. Among them are distinguished:

Cylindrical when the projection from the geoid goes to the surface of the cylinder. Cylindrical projections are most often used in. They have the least distortion in the region of the equator and mid-latitudes. This projection is most often used to create maps of the world;

conical. These projections were most often chosen for creating maps. former USSR. The least amount of distortion at 47° conic projections. This is very convenient, since the main economic zones of this state were located between the indicated parallels, and here was concentrated maximum load kart. But in conic projections, regions lying in high latitudes and water areas are strongly distorted;

Azimuthal projection. This is a kind of map projection, when the projection is carried out on a plane. This type of projection is used when creating maps or any other area of the Earth.

As a result of cartographic projections, each point on the globe that has certain coordinates corresponds to one and only one point on the map.

In addition to cylindrical, conical and cartographic projections, there is a large class of conditional projections, in the construction of which they use not geometric analogues, but only mathematical equations of the desired form.

Date: 24.10.2015

map projection- a mathematical way of depicting the globe (ellipsoid) on a plane.

For projecting a spherical surface onto a plane use auxiliary surfaces.

By type auxiliary cartographic projection surface is divided into:

Cylindrical 1(auxiliary surface is the side surface of the cylinder), conical 2(lateral surface of the cone), azimuth 3(the plane, which is called the picture plane).

Also allocate polyconical

pseudocylindrical conditional

and other projections.

Orientation auxiliary figures of the projection are divided into:

normal(in which the axis of the cylinder or cone coincides with the axis of the Earth model, and the picture plane is perpendicular to it);
transverse(in which the axis of the cylinder or cone is perpendicular to the axis of the Earth model, and the picture plane is or parallel to it);
oblique, where the axis of the auxiliary figure is in an intermediate position between the pole and the equator.

Cartographic distortion is a violation geometric properties objects of the earth's surface (lengths of lines, angles, shapes and areas) when they are displayed on the map.

The smaller the scale of the map, the more significant the distortion. On large scale maps, distortion is negligible.

There are four types of distortions on the maps: lengths, areas, corners and forms objects. Each projection has its own distortions.

According to the nature of distortions, map projections are divided into:

equiangular, which store the angles and shapes of objects, but distort the lengths and areas;

equal, in which areas are stored, but the angles and shapes of objects are significantly changed;

arbitrary, in which the distortions of lengths, areas and angles, but they are evenly distributed on the map. Among them, projections are especially distinguished, in which there are no distortions of lengths either along parallels or along meridians.

Zero Distortion Lines and Points- lines along which there are also points where there are no distortions, since here, when projecting a spherical surface onto a plane, the auxiliary surface (cylinder, cone or picture plane) was tangents to the ball.

Scale indicated on the cards, persists only on lines and at zero-distortion points. It's called the main one.

In all other parts of the map, the scale differs from the main one and is called partial. To determine it, special calculations are required.

To determine the nature and magnitude of distortion on the map, you need to compare the degree grid of the map and the globe.

on the globe all parallels are at the same distance from each other, all meridians are equal and intersect with parallels at right angles. Therefore, all the cells of the degree grid between adjacent parallels have the same size and shape, and the cells between the meridians expand and increase from the poles to the equator.

To determine the amount of distortion, distortion ellipses are also analyzed - ellipsoidal figures formed as a result of distortion in a certain projection of circles drawn on a globe of the same scale as the map.

Conformal projection the distortion ellipses are shaped like a circle, the size of which increases depending on the distance from the zero distortion points and lines.

In an equal area projection distortion ellipses have the shape of ellipses, the areas of which are the same (the length of one axis increases, and the second decreases).

Equidistant projection distortion ellipses have the shape of ellipses with the same length of one of the axes.

The main signs of distortion on the map

If the distances between the parallels are the same, then this indicates that the distances along the meridians are not distorted (equidistant along the meridians).
Distances are not distorted by parallels if the radii of the parallels on the map correspond to the radii of the parallels on the globe.
Areas are not distorted if the cells created by the meridians and parallels at the equator are squares, and their diagonals intersect at right angles.
The lengths along the parallels are distorted, if the lengths along the meridians are not distorted.
The lengths are distorted along the meridians, if the lengths along the parallels are not distorted.

The nature of distortions in the main groups of cartographic projections

Map projections	distortion
Equangular	Preserve angles, distort areas and lengths of lines.
isometric	They preserve areas, distort angles and shapes.
Equidistant	In one direction they have a constant length scale, the distortions of angles and areas are in equilibrium.
Arbitrary	Distort corners and squares.
Cylindrical	There are no distortions along the line of the equator, but they increase with the degree of approach to the poles.
conical	There are no distortions along the parallel of contact between the cone and the globe.
Azimuthal	There are no distortions in the central part of the map.

People have been using maps since ancient times. The first attempts to depict were made in Ancient Greece such scholars as Eratosthenes and Hipparchus. Naturally, cartography as a science has advanced far since then. Modern maps are created using satellite imagery and using computer technology, which, of course, contributes to an increase in their accuracy. And yet, on each geographical map there are some distortions relative to natural shapes, angles or distances on the earth's surface. The nature of these distortions, and, consequently, the accuracy of the map, depends on the types of cartographic projections used to create a particular map.

The concept of map projection

Let us examine in more detail what a map projection is and what types of them are used in modern cartography.

A map projection is an image on a plane. A deeper scientific definition sounds like this: a map projection is a way of displaying points on the Earth's surface on a certain plane, in which some analytical dependence is established between the coordinates of the corresponding points of the displayed and displayed surfaces.

How is a map projection built?

The construction of any types of cartographic projections occurs in two stages.

First, the geometrically irregular surface of the Earth is mapped onto some mathematical correct surface, which is called the reference surface. For the most accurate approximation in this capacity, the geoid is most often used - geometric body, limited by the water surface of all seas and oceans, interconnected (sea level) and having a single water mass. At every point on the surface of the geoid, gravity is applied normally. However, the geoid, like the physical surface of the planet, also cannot be expressed by a single mathematical law. Therefore, instead of the geoid, an ellipsoid of revolution is taken as the reference surface, giving it the maximum similarity to the geoid using the degree of compression and orientation in the Earth's body. They call this body an earth ellipsoid or a reference ellipsoid, and in different countries ah for them take different parameters.
Secondly, the accepted reference surface (reference ellipsoid) is transferred to the plane using one or another analytical dependence. As a result, we get a flat map projection

Projection distortion

Have you ever wondered why the outlines of the continents differ slightly on different maps? On some map projections, some parts of the world appear larger or smaller relative to some landmarks than on others. It's all about the distortion with which the projections of the Earth are transferred to a flat surface.

But why do map projections display in a distorted way? The answer is pretty simple. A spherical surface is not possible to deploy on a plane, avoiding folds or breaks. Therefore, the image from it cannot be displayed without distortion.

Methods for obtaining projections

When studying cartographic projections, their types and properties, it is necessary to mention the methods of their construction. So, map projections are obtained using two main methods:

geometric;
analytical.

At the core geometric method are the laws of linear perspective. Our planet is conditionally taken as a sphere of some radius and projected onto a cylindrical or conical surface, which can either touch or cut through it.

Projections obtained in this way are called perspective. Depending on the position of the observation point relative to the Earth's surface, perspective projections are divided into types:

gnomonic or central (when the point of view is aligned with the center of the earth's sphere);
stereographic (in this case, the observation point is located on the reference surface);
orthographic (when the surface is observed from any point outside the sphere of the Earth; the projection is built by transferring the points of the sphere using parallel lines perpendicular to the display surface).

Analytical method construction of cartographic projections is based on mathematical expressions connecting points on the sphere of reference and the display plane. This method is more versatile and flexible, allowing you to create arbitrary projections according to a predetermined nature of the distortion.

Types of map projections in geography

To create geographical maps, many types of projections of the Earth are used. They are classified according to various criteria. In Russia, the Kavraysky classification is used, which uses four criteria that determine the main types of cartographic projections. The following are used as characteristic classifying parameters:

the nature of the distortion;
the form of displaying the coordinate lines of the normal grid;
the location of the pole point in the normal coordinate system;
mode of application.

So, what are the types of map projections according to this classification?

Projection classification

By the nature of the distortion

As mentioned above, distortion is, in fact, an inherent property of any projection of the Earth. Any characteristic of the surface can be distorted: length, area or angle. Distortion types are:

Conformal or conformal projections, in which azimuths and angles are transferred without distortion. The coordinate grid in conformal projections is orthogonal. Maps obtained in this way are recommended to be used to determine distances in any direction.
Equal area or equivalent projections, where the scale of areas is stored, which is taken equal to one, i.e. areas are displayed without distortion. Such maps are used to compare areas.
Equidistant or equidistant projections, during the construction of which the scale is preserved in one of the main directions, which is taken as unit.
Arbitrary projections, which can contain all kinds of distortions.

According to the form of displaying the coordinate lines of the normal grid

Such a classification is the most visual and, therefore, the easiest to understand. Note, however, that this criterion applies only to projections oriented normally to the observation point. So, based on this characteristic feature, the following types of cartographic projections are distinguished:

Circular, where parallels and meridians are represented by circles, and the equator and average meridian of the grid are represented by straight lines. Such projections are used to depict the surface of the Earth as a whole. Examples of circular projections are the conformal Lagrange projection, as well as the arbitrary Grinten projection.

Azimuthal. In this case, the parallels are represented as concentric circles, and the meridians as a bundle of straight lines diverging radially from the center of the parallels. A similar kind of projection is used in a direct position to display the poles of the Earth with adjacent territories, and in a transverse position as a map of the western and eastern hemispheres familiar to everyone from geography lessons.

Cylindrical, where meridians and parallels are represented by straight, normally intersecting lines. With minimal distortion, territories adjacent to the equator or stretched along some standard latitude are displayed here.

conical, representing a development of the lateral surface of the cone, where the lines of parallels are arcs of circles centered at the top of the cone, and the meridians are guides diverging from the top of the cone. Such projections most accurately depict the territories lying in the middle latitudes.

Pseudoconic projections similar to conical ones, only the meridians in this case are depicted as curved lines symmetrical with respect to the rectilinear axial meridian of the grid.

Pseudocylindrical projections resemble cylindrical, only, as well as in pseudoconical, the meridians are depicted by curved lines symmetrical to the axial rectilinear meridian. Used to depict the entire Earth (for example, the elliptical Mollweide projection, equal area sinusoidal Sanson, etc.).

Polyconic, where the parallels are depicted as circles, the centers of which are located on the middle meridian of the grid or its continuation, the meridians are in the form of curves located symmetrically to a rectilinear

By the position of the pole point in the normal coordinate system

Polar or normal- the pole of the coordinate system coincides with the geographic pole.
transverse or transversion- pole normal system aligned with the equator.
oblique or oblique- the pole of the normal coordinate grid can be located at any point between the equator and the geographic pole.

By way of application

According to the method of use, the following types of map projections are distinguished:

Solid- the projection of the entire territory onto a plane is carried out according to a single law.
Multiband- the mapped area is conditionally divided into several latitudinal zones, which are projected onto the display plane according to a single law, but with a change in the parameters for each zone. An example of such a projection is the Mufling trapezoidal projection, which was used in the USSR for large-scale maps until 1928.
multifaceted- the territory is conditionally divided into a number of zones in longitude, the projection onto the plane is carried out according to a single law, but with different parameters for each of the zones (for example, the Gauss-Kruger projection).
Composite, when some part of the territory is displayed on a plane using one pattern, and the rest of the territory on the other.

The advantage of both multi-lane and multi-faceted projections is the high display accuracy within each zone. However, a significant disadvantage in this case is the impossibility of obtaining a continuous image.

Of course, each map projection can be classified using each of the above criteria. So, the famous projection of the Earth Mercator is conformal (equiangular) and transverse (transversion); Gauss-Kruger projection - conformal transverse cylindrical, etc.

All cartographic projections are classified according to a number of characteristics, including the nature of distortions, the type of meridians and parallels of the normal cartographic grid, and the position of the pole of the normal coordinate system.

1. Classification of map projections

according to the nature of the distortion:

a) equiangular, or conformal leave without distortion the corners and the shape of the contours, but have significant distortion of the areas. An elementary circle in such projections always remains a circle, but its dimensions vary greatly. Such projections are especially useful for determining directions and plotting routes along a given azimuth, so they are always used on navigation charts.,

These projections can be described by equations in the characteristics of the form:

m=n=a=b=m

q=90 0 w=0 m=n

Rice. Distortions in conformal projection. World map in Mercator projection

b) equal or equivalent- keep the area without distortion, however, the angles and shapes are significantly disturbed on them, which is especially noticeable in large areas. For example, on a world map, the polar regions look very flattened. These projections can be described by equations of the form R = 1.

Rice. Distortions in equal area projection. World map in Mercator projection

c) equidistant (equidistant).

In these projections, the linear scale in one of the main directions is constant and is usually equal to the main scale of the map, i.e., there is

or a= 1, or b= 1;

d) arbitrary.

They do not preserve angles or areas.

2. Classification of cartographic projections according to the method of construction

Auxiliary surfaces in the transition from an ellipsoid or a ball to a map can be a plane, a cylinder, a cone, a series of cones, and some other geometric shapes.

1) Cylindrical projections— the projection of a ball (ellipsoid) is carried out on the surface of a tangent or secant cylinder, and then its lateral surface unfolds into a plane.

In these projections, the parallels of normal grids have straight parallel lines, the meridians are also straight lines orthogonal to the parallels. The distances between the meridians are equal and always proportional to the difference in longitudes

Rice. View of the cartographic grid of a cylindrical projection

Conditional projections - projections for which it is impossible to find simple geometric analogues. They are built on the basis of 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., obtained by converting one or more similar projections.

Pseudocylindrical projections: parallels are depicted by straight parallel lines, meridians - by curved lines, symmetrical about the average rectilinear meridian, which is always orthogonal to the parallels (used for maps of the world and the Pacific Ocean).

Rice. View of the cartographic grid of the pseudocylindrical projection

We assume that the geographic pole coincides with the pole of the normal coordinate system

a) Normal (straight) cylindrical - if the axis of the cylinder coincides with the axis of rotation of the Earth, and its surface touches the ball along the equator (or cuts it along parallels) . Then the meridians of the normal grid appear as equally spaced parallel lines, and the parallels appear as lines perpendicular to them. In such projections, the least distortion is in the tropical and equatorial regions.

b) transverse cylindrical projection - the axis of the cylinder is located in the plane of the equator. The cylinder touches the ball along the meridian, there are no distortions along it, and therefore, in such a projection, it is most advantageous to depict territories stretched from north to south.

c) oblique cylindrical - the axis of the auxiliary cylinder is located at an angle to the plane of the equator . It is convenient for elongated territories oriented to the northwest or northeast.

2) Conic 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 the generatrix and unfolded into a plane.

Distinguish:

· normal (straight) conical projection when the axis of the cone coincides with the axis of rotation of the Earth. Meridians are straight lines radiating from the pole point, and parallels are arcs of concentric circles. An imaginary cone touches the globe or cuts it in the mid-latitude region, therefore, in such a projection, it is most convenient to map the territories of Russia, Canada, and the United States, elongated from west to east in mid-latitudes.

· transverse conical - the axis of the cone does not live in the plane of the equator

· oblique conical- the axis of the cone is inclined to the plane of the equator.

Pseudoconic projections- those in which all parallels are represented by arcs of concentric circles (as in normal conic circles), the middle meridian is a straight line, and the remaining meridians are curves, and their curvature increases with distance from the middle meridian. They are used for maps of Russia, Eurasia, and other continents.

Polyconic projections- projections obtained as a result of projecting a ball (ellipsoid) onto a set of cones. In normal polyconic projections, the parallels are represented by arcs of eccentric circles, and the meridians are curves symmetrical about the direct median meridian. Most often, these projections are used for world maps.

3) Azimuthal projections — the surface of the globe (ellipsoid) is transferred to the tangent or cutting plane. If the plane is perpendicular to the Earth's axis of rotation, then normal (polar) azimuth projection . In these projections, parallels are depicted as single-center circles, meridians - as a bunch of straight lines with a vanishing point coinciding with the center of the parallels. In this projection, the polar regions of our and other planets are always mapped.

a - normal or polar projection onto a plane; v - grid in transverse (equatorial) projection;

G - grid in oblique azimuthal projection.

Rice. Map grid view of the azimuth projection

If the projection plane is perpendicular to the equatorial plane, then it turns out transverse (equatorial) azimuthal projection. It is always used for maps of the hemispheres. And if the projection is made on a tangent or secant auxiliary plane located at any angle to the equatorial plane, then it turns out oblique azimuth projection.

Among the azimuth projections, there are several of their varieties, which differ in the position of the point from which the ball is projected onto the plane.

Pseudo-azimuth projections - modified azimuthal projections. In polar pseudo-azimuthal projections, parallels are concentric circles, and meridians are curved lines symmetrical about one or two straight meridians. Transverse and oblique pseudo-azimuth projections have a common oval shape and are usually used for maps Atlantic Ocean or the Atlantic Ocean together with the Arctic.

4) Polyhedral projections — projections obtained by projecting a ball (ellipsoid) onto the surface of a tangent or secant polyhedron. Most often, each face is an isosceles trapezoid.

3) Classification of map projections according to the position of the pole of the normal coordinate system

Depending on the position of the pole of the normal system R o, all projections are divided into the following:

a) straight or normal- pole of the normal system R o coincides with the geographic pole ( φ o= 90°);

b) transverse or equatorial- pole of the normal system R o lies on the surface in the plane of the equator ( φ o = 0°);

c) oblique or horizontal- pole of the normal system R o located between the geographic pole and the equator (0°< φ o<90°).

In direct projections, the main and normal grids coincide. There is no such coincidence in oblique and transverse projections.

Rice. 7. The position of the pole of the normal system (P o) in an oblique map projection

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

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

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

/Li= a = b = m = n; a>= 0°; 0 = 90°; k = 1 and a0=0°(or ±90°).

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

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

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

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

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

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

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

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

Fig.6. Normal cylindrical projection

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

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

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

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

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

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

Rice. 7. Normal azimuthal projection

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

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

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

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

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

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

Rice. 8. Normal conic projection

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

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

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

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

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

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

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

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

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

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

Topic 3. Distortions on maps. Types of distortion

Goals and objectives of studying the topic:

To give an idea of the distortions on the maps and the types of distortions:

- to form an idea of distortions in lengths;

— form an idea of distortions in areas;

— to form an idea of distortions in the corners;

— form an idea of distortions in forms;

The result of mastering the topic:

The surface of an ellipsoid (or sphere) cannot be turned into a plane while maintaining the similarity of all outlines.

If the surface of the globe (a model of the earth's ellipsoid), cut into strips along meridians (or parallels), is turned into a plane, gaps or overlaps will occur in the cartographic image, and with distance from the equator (or from the middle meridian) they will increase.

As a result, it is necessary to stretch or compress the strips in order to fill the gaps along the meridians or parallels.

As a result of stretching or compression in the cartographic image, distortions occur in lengthsm (mu) , areas p, cornersw and forms k.

In this regard, the scale of the map, which characterizes the degree of reduction of objects in the transition from nature to the image, does not remain constant: it changes from point to point and even at one point in different directions. Therefore, one should distinguish main scale ds , equal to the given scale in which the earth ellipsoid decreases.

The main scale shows the overall reduction rate adopted for this map.

The main scale is always signed on maps.

In all other places map scales will differ from the main one, they will be larger or smaller than the main one, these scales are called private and denoted by the letter ds1.

The scale in cartography is understood as the ratio of an infinitely small segment taken on a map to the corresponding segment on the earth's ellipsoid (globe). It all depends on what is taken as the basis for constructing the projection - the globe or the ellipsoid.

The smaller the change in scale within a given area, the more perfect the map projection will be.

To perform cartographic work, you need to know distribution on a map of partial scales so that corrections can be made to the measurement results.

Private scales are calculated using special formulas.

Analysis calculation of particular scales shows that among them there is one direction with largest scale , and the other with least.

largest the scale, expressed in fractions of the main scale, is denoted by the letter " a", a least - letter « v" .

The directions of the largest and smallest scales are called main directions .

The main directions only coincide with the meridians and parallels when the meridians and parallels intersect under right angles.

In such cases scale by meridians denoted by the letter « m" , and by parallels - letter « n" .

The ratio of the private scale to the main one characterizes the distortion of lengths m (mu).

In other words, the value m (mu) is the ratio of the length of an infinitesimal segment on the map to the length of the corresponding infinitesimal segment on the surface of an ellipsoid or ball.

m(mu) = ds1

Area distortion.

Area distortion p defined as the ratio of infinitesimal areas on a map to infinitesimal areas on an ellipsoid or ball:

p= dp1

Projections in which there are no area distortions are called equal.

While creating physical and geographical and socio-economic cards, it may be necessary to save correct area ratio. In such cases, it is advantageous to use equal-area and arbitrary (equidistant) projections.

In equidistant projections, the area distortion is 2-3 times less than in conformal projections.

For political maps world, it is desirable to maintain the correct ratio of the areas of individual states without distorting the external contour of the state.

In this case, it is advantageous to use an equidistant projection.

The Mercator projection is not suitable for such maps, since areas are greatly distorted in it.

Corner distortion. Let's take the angle u on the surface of the globe (Fig. 5), which on the map is represented by the angle u ′ .

Each side of the angle on the globe forms an angle α with the meridian, which is called the azimuth. On the map, this azimuth will be represented by the angle α ′.

In cartography, two types of angular distortions are accepted: direction distortions and angle distortions.

A A ′

α α ′

0 u 0 ′ u ′

B B ′

Corner distortion

The difference between the azimuth of the side of the corner on the map α ′ and the azimuth of the side of the angle on the globe is called direction distortion , i.e.

ω = α′ — α

The difference between the angle u ′ on the map and the value u on the globe is called angle distortion, those.

2ω = u′ — u

The distortion of the angle is expressed by the value 2ω because the angle consists of two directions, each of which has a distortion ω .

Projections in which there are no angle distortions are called equiangular.

The distortion of shapes is directly related to the distortion of angles (specific values w match certain values k ) and characterizes the deformation of the figures on the map in relation to the corresponding figures on the ground.

Form distortion will be the greater, the more the scales differ in the main directions.

As shape distortion measures accept coefficient k .

k = a / b

where a and v are the largest and smallest scales at a given point.

Distortions on geographical maps are the greater, the larger the depicted territory, and within the same map, distortions increase with distance from the center to the edges of the map, and the slew rate changes in different directions.

In order to visualize the nature of distortions in different parts of the map, they often use the so-called ellipse of distortion.

If we take an infinitely small circle on the globe, then when moving to the map, due to stretching or contraction, this circle will be distorted like outlines geographical objects and takes the form of an ellipse.

This ellipse is called ellipse distortion or Tissot's indicatrix.

The dimensions and degree of elongation of this ellipse compared to the circle reflect all kinds of distortions inherent in the map in this place. Type and dimensions ellipse are not the same in different projections and even in different points the same projection.

The largest scale in the distortion ellipse coincides with the direction of the major axis of the ellipse, and the smallest scale coincides with the direction of the minor axis.

These directions are called main directions .

The distortion ellipse is not displayed on the maps.

It is used in mathematical cartography to determine the magnitude and nature of distortions at some projection point.

The directions of the axes of the ellipse may coincide with the meridians and parallels, and in some cases the axes of the ellipse may occupy an arbitrary position relative to the meridians and parallels.

Determination of distortions for a number of map points and subsequent drawing on them isocol - lines connecting points with the same distortion values gives a clear picture of the distribution of distortions and allows you to take into account distortions when using the map.

To determine the distortions within the map, you can use special tables or diagrams isokol. Isocols can be for angles, areas, lengths, or shapes.

No matter how one deploys the earth's surface onto a plane, gaps and overlaps will inevitably occur, which in turn leads to tensions and compressions.

But on the map, at the same time, there will be places where there will be no compressions and tensions.

Lines or points on a geographical map that are not distorted and the main scale of the map is preserved, called lines or zero-distortion points (LNI and TNI) .

As you move away from them, the distortion increases.

Questions for repetition and consolidation of the material

What causes cartographic distortions?

Types of map projections and their characteristics

What types of distortions occur during the transition from the surface
ellipsoid to plane?

3. Explain what is the point and line of zero distortion?

4. On which maps does the scale remain constant?

5. How to determine the presence and magnitude of distortion in certain areas of the map?

What is Tissot's indicatrix?

7. What is the purpose of the distortion ellipse?

8. What are isocoles and what is their purpose?

map projection- this is a way to move from a real, geometrically complex earth's surface to the plane of the map.

A spherical surface cannot be deployed on a plane without deformation - compression or tension.

This means that every map has certain distortions. There are distortions of the lengths of areas, angles and shapes. On large scale maps (cf.

Scale) distortions can be almost imperceptible, but on small scales they can be very large. Map projections have different properties depending on the nature and size of the distortion.

Topic 5. CARTOGRAPHIC PROJECTIONS AND DISTORTIONS

Among them are distinguished:

Equal projections. They do not distort areas, but the angles and shapes in them are strongly distorted.

Maps in equal projections are convenient for determining the size of the state, land;
Equidistant. They have a constant length scale in one direction. The distortions of angles and areas are balanced in them;

Among them are distinguished:

Cylindrical when the projection from the geoid goes to the surface of the cylinder.

Cylindrical projections are most often used in cartography. They have the least distortion in the region of the equator and mid-latitudes. This projection is most often used to create maps of the world;

conical. These projections were most often chosen for creating maps of the former USSR. The least amount of distortion in conic projections was at the parallels of 47° north latitude and 62° north longitude.

This is very convenient, since the main economic zones of this state were located between the indicated parallels, and the maximum load of maps was concentrated here. On the other hand, in conic projections, regions lying in high latitudes and in the waters of the Arctic Ocean are strongly distorted;

Azimuthal projection. This is a kind of map projection, when the projection is carried out on a plane.

This type of projection is used when creating maps of Antarctica or the Arctic or any other region of the Earth.

As a result of cartographic projections, each point on the globe, which has certain geographic coordinates, corresponds to one and only one point on the map.

In addition to cylindrical, conic and azimuth map projections, there is a large class of conditional projections, in the construction of which they use not geometric analogues, but only mathematical equations of the desired type.

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

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

Since the Earth has a spherical shape, its surface cannot be depicted on a plane without distortion.

If we cut any spherical surface into parts (along the meridians) and impose these parts on a plane, then the image of this surface on it would turn out to be distorted and with discontinuities. There would be folds in the equatorial part, and breaks at the poles.

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

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

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

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

Equal projections. On these projections, the angles are not distorted, i.e.

e. the angles on the ground between any directions are equal to the angles on the map between the same directions. Infinitely small figures on the map, due to the property of equiangularity, will be similar to the same figures on the Earth.

If the island is round in nature, then on the map in a conformal projection it will be depicted as a circle of a certain radius. But the linear dimensions on the maps of this projection will be distorted.

Equal projections. On these projections, the proportionality of the areas of the figures is preserved, i.e.

That is, if the area of any area on Earth is twice as large as another, then on the projection the image of the first area in terms of area will also be twice as large as the image of the second. However, in an equal area projection, the similarity of the figures is not preserved. The island of a round shape will be depicted on the projection in the form of an ellipse of equal area.

Arbitrary projections. These projections retain neither the similarity of figures nor the equality of areas, but may have some other special properties necessary for solving certain practical problems on them.

From the charts of arbitrary projections, orthodromic projections have received the greatest use in navigation, on which great circles (great circles of the ball) are depicted by straight lines, and this is very important when using some radio navigation systems when navigating along a great circle arc.

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

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

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

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

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

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

Conic projections are used for geographic maps.

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

In direct normal projection, the grid is obtained from mutually perpendicular straight lines of the meridians L, B, C, D, F, G and parallels aa ', bb', ss. projection K in fig.

34), but sections of the polar regions cannot be designed in this case.

If you rotate the cylinder so that its axis is located in the plane of the equator, and its surface touches the poles, then you get a transverse cylindrical projection (for example, a Gaussian transverse cylindrical projection).

If the cylinder is placed at a different angle to the Earth's axis, then oblique cartographic grids are obtained.

Lecture: Types of map projections

On these grids, meridians and parallels are shown as curved lines.

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

The picture plane can be located at any point on the earth's surface, and the point of contact is called the central point of the projection and is taken as the zenith.

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

perspective projections. If a cartographic grid is obtained by projecting meridians and parallels onto a plane according to the laws of linear perspective from a constant point of view of T.Z.

(see Fig. 35), then such projections are called promising. The plane can be positioned at any distance from the Earth or so that it touches it. The point of view should be on the so-called main diameter of the globe or on its continuation, and the picture plane should be perpendicular to the main diameter.

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

In addition, perspective projections depend on the location of the point of view from the center of the Earth on the main diameter.

When the point of view coincides with the center of the Earth, the projections are called central or gnomonic; when the point of view is on the surface of the Earth stereographic; when removing the point of view on any known distance projections from the Earth are called external, and when the point of view is removed to infinity - orthographic.

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

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

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

Conditional projections. This category includes all projections that, according to the method of construction, cannot be attributed to any of the above types of projections.

They usually satisfy some pre-set conditions, depending on the purposes for which the card is required. The number of conditional projections is not limited.

Small areas of the earth's surface up to 85 km can be depicted on a plane with the similarity of the applied figures and areas preserved on them.

Such flat images of small areas of the earth's surface, on which distortions can practically be neglected, are called plans.

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

Along with the usual radio amateurs use maps with an azimuthal projection, in which the surface of the mainland is projected onto a plane. The point of zero distortion is the point of contact of the plane with the earth's surface, the maximum distortion is in the peripheral parts of the map.

Parallels in direct azimuth projections (point of contact - poles) are depicted by concentric circles, and meridians - by straight lines (rays). In the transverse-azimuth projection (the point of contact is on the equator), a map of the hemispheres has been compiled, in which curves correspond to the meridians and parallels, with the exception of the equator and the middle meridians of the hemispheres.

To depict individual continents, the points of contact are chosen in their center (maps of Africa, Australia and America).

§ 17. CARTOGRAPHIC PROJECTIONS

In modern conditions, cartographic projections are also built using mathematical calculations without auxiliary surfaces; they are called conditional projections

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It turns out that the familiar map of the world is just one of the few projections of the globe onto the plane of the map.

Mercator projection- one of the main map projections.

Developed by Gerard Mercator for use in his Atlas 450 years ago.

The Mercator projection hangs on the wall in the geography classroom and is used by Google Maps and other mapping services. People look at the map in the Mercator projection and lose touch with reality. They believe that tiny Greenland is a real ice continent, the size of Australia, and North America more Africa, etc.

Classifications of map projections

Ask all your friends, what is the second largest continent in size? Almost always, it turns out to be North America.

Such a projection flatters the circumpolar states, because the size of the countries in this projection is amazing - Africa is on the outskirts of the territory.

In a geographical sense real world looks different. Not in the popular and largely illusory Mercator projection, but in real proportions.

Gall-Peters projection

In the Gall-Peters projection, the areas of all countries are shown on the same scale.

It is on this map that the areas of states or continents should be compared with each other. On this map, Russia does not look like the usual giant, but occupies a strip along the north pole.

Now everything has fallen into place Greenland is 3 times smaller than Australia.

Art. Lebedev's studio made a website and a poster as part of the Susha project. On the site you can compare the areas of different countries. I advise you to look at the process of creating infographics.