Gauss coordinates. The concept of cartographic projections. Classification of projections. Gaussian Conformal Transverse Cylindrical Gaussian Transverse Cylindrical Gaussian
Topic 4. ZONE SYSTEM OF RECTANGULAR COORDINATES GAUSS
4.1. EQUIANGULAR TRANSVERSE-CYLINDRICAL GAUSSIAN PROJECTION
To reduce the inevitable distortions that occur when depicting large areas on a plane, they resort to mapping areas in parts. When creating topographic maps (except for maps on a scale of 1: 1,000,000) in Ukraine and a number of other countries, Gauss-Kruger conformal transverse cylindrical projection.
Carl Friedrich (1777-1855) Gauss in 1825 developed the theory of mapping the surface of an ellipsoid of revolution onto a plane with the conservation of similarity in infinitesimal parts. In 1912, A. Krueger derived working formulas for this projection.
A projection is formed by transferring the surface of an ellipsoid onto the side surface of an elliptical cylinder, the axis of which is perpendicular to the axis of rotation of the Earth.
Rice. 4.1. Gaussian projection
Therefore, the Gaussian projection is made taking into account the compression of the Earth. A narrow strip of the earth's surface is transferred to one cylinder, occupying 6 ° in longitude
The cylinder touches the globe along the middle meridian of the zone. Each zone corresponds to a column of map sheets at a scale of 1: 1,000,000 in the international layout, i.e. each zone is bounded by meridians that are multiples of 6° of longitude. The zones are numbered from the Greenwich meridian to the east. The first zone is located between meridians 0 and 6°. There are 60 zones in total.
The surface of the globe is transferred to the side surface of the cylinder with the preservation equality of angles on the ground and on the map
. Therefore, the Gaussian projection equiangular. Length distortions will increase with distance from the equator and the meridian of contact.
In each zone, the axial meridian (as the meridian of touch) is depicted as a straight line in full size. The remaining meridians of the zone are depicted by curved lines, and their curvature increases with distance from the axial meridian.
All meridians on a globe are the same length. Consequently, all meridians in the zone, except for the middle one, are elongated compared to the corresponding meridians on the globe. The equator is depicted as a straight line, and the other parallels are curved. All parallels, including the equator, are stretched in proportion to the stretching of the meridians.
Rice. 4.2. Schematic representation of the Gaussian zone on a plane.
In the Gaussian projection, the maximum length distortions at the equator at the boundary of each zone are 0.137% (137 m per 100 km distance).
When solving many problems of geodesy, such distortions are neglected and the projection is considered not only equiangular, but also equidistant, And equal, i.e. practically no distortion of angles, distances and areas
. Maps of this projection are taken as a plan.
Each Gaussian zone along meridians and parallels is divided into separate sheets of maps . The frames of the map sheets are meridians and parallels.
In the Gaussian projection, topographic maps are compiled at a scale of 1: 500,000 and larger.
On maps at a scale of 1: 500,000, a grid of geodetic coordinates is plotted, and on this map, outputs of a kilometer grid are given.
On maps at a scale of 1:200,000 and larger, a kilometer grid of the Gaussian rectangular coordinate system is plotted.
4.2. GAUSS FLAT RECTANGULAR COORDINATE SYSTEM
On topographic maps at a scale of 1: 500,000 and larger, in addition to the geodetic grid, rectangular grid. Having accepted axial(middle) meridian in each zone per axis X(abscissa) and equator- per axle At(ordinates), and their intersection beyond the origin, we get Gaussian plane rectangular coordinate system for this zone. In topography and geodesy, orientation is carried out along the north, counting the angles in a clockwise direction. Therefore, to preserve the signs of trigonometric functions, the position of the coordinate axes in the Gaussian zone is rotated by 90° relative to the axes adopted in the Cartesian system of rectangular coordinates. For the positive direction of the axes are taken: for the axisX- north direction, for the axisY- to the East. Point position A in the coordinate zone is determined by its distance x A Andy Afrom the coordinate axes. On the territory of Ukraine, all abscissas (distances from the equator) are positive. As for the ordinates, they could be both positive and negative in each zone. For the convenience of working with maps, we agreed on the value of the ordinateYthe axial meridian of each zone is taken equal to 500 km, i.e., the origin of coordinates was, as it were, moved to the west outside the zone.
Rice. 4.3. Gaussian plane rectangular coordinate system.
Since the numerical values of the ordinates are repeated in each zone, in order to determine by the coordinates of a point which zone it belongs to, the value of the ordinate Y the zone number is assigned to the left.
For example, point coordinates X
= 6 346 650 m, at = 4 522 800 m mean that the point is located north of the equator at a distance of 6 346 650 m and east of the central meridian of the 4th zone at a distance of 22,800 m (522 800 m - 500 000 m = 22 800 m).
Another example. Coordinates X = 5 862 300 m, at = 15 323 500 m. This means that the point is located at 5,862,300 m north of the equator and at 176,500 m west of the central meridian of the 15th zone (500,000 m— 323 500 m = 176 500 m).
In order to greatly simplify the determination of rectangular coordinates, straight lines are drawn on the plane (on the map) parallel to the coordinate axes (axial meridian and equator) through an integer number of kilometers, therefore a rectangular coordinate grid is often called kilometer, and its lines are kilometric.
All lines of the kilometer grid on the maps are signed with numbers, and the lines closest to the corners of the frame of the map sheet are signed complete number of kilometers, the rest abbreviated- only the last two digits, implying the remaining digits. Thus, the signature 6081 on top of the horizontal kilometer line means that it passes at 6081 km north of the equator, and the signature 4322 near the vertical kilometer line on the right means that this line is in the 4th zone and passes in 178 km west of the axial meridian of the zone (500 km - 322 km= 178 km).
Using a kilometer grid, you can quickly find the coordinates of objects, plot points by coordinates, indicate the location of objects on the map. The rectangular coordinates of the point through which the lines of the kilometer grid pass on the map are obtained immediately by reading the digitization of the coordinate lines on the map frame.
Rice. 4.3. Digitization of lines of a rectangular coordinate grid.
The coordinates of the points lying inside the cells of the grid are determined by the coordinates of the grid lines closest to the point and the increment of the coordinates of the points relative to these lines. Coordinate increments Δ X and Δ at measured using a measuring compass and the linear scale of the map, summarized with the coordinates of kilometer lines.
y A = 7,316,000 y B = 7,313,450
Rice. 4.5. The position and digitization of the lines of a rectangular coordinate grid on a map sheet at a scale of 1: 100,000 and the determination of the rectangular coordinates of points
Coordinate increments can be measured using a coordinate meter, a small square with two perpendicular sides. Scales are marked along the inner edges of the rulers, the lengths of which are equal to the length of the side of the coordinate cells of the map of the given scale. The horizontal scale is aligned with the bottom line of the square (in which the point is located), and the vertical scale must pass through this point. The scales determine the distance from the point to the kilometer lines (Fig. 6.3).
x A = 6135 350 y A = 5577 701
R is. 4.6. Measuring rectangular coordinates of points using
coordinator
To plot a point on the map at given rectangular coordinates, proceed as follows: by the value of the abscissa x, taking into account only the integer number of kilometers, find the horizontal coordinate line, to the north of which the point will be located. By ordinate value y similarly determine the vertical coordinate line, to the east of which the desired point will be located, and find the desired square. The remaining fractions of kilometers (increments of coordinates) are set aside by the meter on a linear scale: on both horizontal sides of the square to the east - the increment of the ordinate Δ at, and along both vertical lines to the north - the increment of the abscissa Δ X. Through the points obtained, a vertical and horizontal line is drawn, at the point of intersection of which the given point is located.
For a quick location object on a given map sheet use the abbreviated coordinates of the southwest corner of the corresponding square of the kilometer grid. From the designations of both kilometer lines, they take the last two digits, printed in large print, and write them down so that the first two numbers referred to the south side, and the last two to the western side of the square . For example, in figure 4.3 Kruta is located in the square 8020, and the locality bandurka- in square 8022.
Additional kilometer grid applied at the border of adjacent zones. Since the vertical kilometer grid lines are parallel to their axial meridian of the zone, and the axial meridians of neighboring zones are not parallel to each other, when gluing two map sheets located at the junction of two zones, the vertical kilometer lines of both grids will be located at some angle to each other. When determining the coordinates of points located in two adjacent zones, it is necessary to recalculate the coordinates of objects in one zone into another zone. This is a labor-intensive work that requires special tables and computer technology.
Rice. 4.7. Mutual arrangement of kilometer lines of the network of adjacent zones (a) and additional coordinate grid (b)
To eliminate this inconvenience, in each zone, on all sheets of maps located within 2 ° to the east and west of the zone boundary, in addition to the kilometer grid of their zone, the conclusions of the kilometer grid of the neighboring (western or eastern) zone are also plotted in the form of dashes outside the outer frame . Signatures of the additional network are made on the outside of the outer frame.
The presence of an additional grid on the map allows you to graphically recalculate the coordinates of objects (targets) from one zone to another zone. To build an additional grid on the map, it is necessary to connect the outputs of the additional coordinate grid with the same values along the eastern and western frames, as well as along the southern and northern frames, with a straight line.
4.3. DETERMINATION OF GEOGRAPHICAL COORDINATES OF POINTS,
SET ON THE MAP
Each sheet of maps at a scale of 1: 1,000,000 and larger is limited by meridians and parallels. Values geographical coordinates are signed at the corners of the frame of the map sheet. In addition, along the sides of the frame are shown (on a map scale) images of arcs of meridians and parallels corresponding to a certain number of minutes of latitude and longitude.
Rice. 4.8. Making the frame of the topographic map sheet.
Geographical coordinates of the corners of the inner frame of the sheet (northeast, southeast, southwest, northwest) signed on the map
.
Frame has divisions into segments corresponding to one minute of latitude (on the western and eastern frames) and one minute of longitude (on the northern and southern frames). Minute segments are represented on the map as long dotted lines.
To determine geographic coordinates on a map
points
draw the parallel closest to it from the south and the meridian closest from the west. The desired latitude will be the sum of the latitude of the drawn parallel and the increment of the latitude of the point relative to this parallel. Similarly, you can get the longitude of a point. Latitude and longitude increments are usually determined by second marks next to minute divisions
or by interpolation.
To define coordinate increments interpolation method
it is necessary to measure on the map the length of one minute of latitude and longitude, as well as the distance from the point to the nearest parallel from the south and from the point to the nearest meridian from the west. Based on these data, proportions are compiled and coordinate increments are determined.
For example:
On a map of scale 1: 25,000, the length of the minute line in longitude is 42 mm. The distance from the point to the nearest western meridian is 20 mm. Find the increment of longitude in seconds.
We make a proportion:
60 sec corresponds 43 mm
x sec corresponds 20 mm
x \u003d (60 × 20): 43 \u003d 27,9 ≈ 28 sec
When determining the geographical coordinates of points on maps of scales 1: 500,000 and 1: 1,000,000, apply special palette . It is a system of straight lines drawn on transparent paper, the distances between which correspond to 5 "latitude and longitude. Such a palette is placed on a map sheet so that its lines, multiples of whole degrees of latitude and longitude, coincide with the corresponding lines of the cartographic grid. After that, they evaluate the position of the point being determined relative to the nearest western and southern lines of the palette.
Tasks and questions for self-control
- What projections are used to create topographic maps in Ukraine?
- What is the essence of creating a Gaussian projection?
- Why is the Gaussian projection called: "Conformal transverse cylindrical"
- How are meridians and parallels depicted in the Gaussian projection?
- In what areas of the Gaussian projection map is the distortion maximum?
- What is the purpose of the Gaussian map sheet borders?
- What is taken as the coordinate axes (abscissa and ordinate) in the Gaussian plane rectangular coordinate system?
- What does writing coordinate values mean: X = 6 346 650, at = 4 522 800?
- What is the purpose of a kilometer grid on topographic maps?
- How to determine the flat rectangular coordinates of a given point using a topographic map?
- What is the purpose of abbreviated coordinates?
- What is the solution of the direct geodesic problem?
- What is the solution of the inverse geodesic problem?
- What is the procedure for determining geographic coordinates on a topographic map?
The solution of geodesic problems in this system is carried out according to simple formulas of analytical geometry, for which it is necessary to first project the elements of the surface of the ellipsoid in one way or another onto a plane. Such a projection will be accompanied by inevitable distortions, and their magnitude and nature depend on the type of the selected surface.
For large-scale mapping and engineering geodesy, projections that provide best preservation a similar image of figures in the transition from an ellipsoid to a plane. This will become possible if the earth's surface is divided into parts (zones), and then depicted in its entirety on a plane. The resulting distortions will be small and easily taken into account.
These requirements are met by the Gauss-Kruger transverse cylindrical projection adopted in the USSR since 1928. This projection was proposed by Gauss in 1825-1830, in 1912 Kruger worked out the details of the application and gave working formulas for the calculation in this projection.
The essence of the Gauss-Kruger projection is as follows.
The surface of the earth's ellipsoid is divided by meridians into spherical bicagons - zones through 60 (Fig. 6).
Mentally position such an ellipsoid in the cylinder so that the axial meridian of the first zone touches the side surface of the cylinder (Fig. 7).
The "y" axis of the cylinder is located across the minor axis of the ellipsoid (which is why the projection is called transverse cylindrical), and the "x" axis, in contrast to mathematics, is located north in geodesy. From the center of the ellipsoid, all points (T, E, Z, M, K ...) of the six-degree zone are projected along plumb lines onto the side surface of the cylinder (t, e, Z, m, k ...). Then, the ellipsoid is rotated in the cylinder in such a way that the axial meridian of the second zone touches the side surface of the cylinder, and all points on the boundary meridians of this zone are designed similarly. And so on, all 60 zones of the ellipsoid are designed.
By cutting the cylinder along generatrices AA1 and BB1 and deploying half of its lateral surface, an image of the earth's surface on a plane is obtained in the form of separate zones that are in contact with one another only at points along the equator. (Fig. 8).
The axis of the cylinder and the line of the equator lie in the same plane and, after a sweep, will be depicted as a straight line - the “y” axis. The "x" axis is located perpendicular to the "y" axis, coincides with the axial meridian of the zone and the minor axis of the ellipsoid. Thus, the axial meridian and the equator are depicted as mutually perpendicular straight lines. After unfolding the surface of the ellipsoid projected onto the lateral surface of the cylinder, the earth's surface breaks are obtained, noticeable in the northern and southern hemispheres at a latitude of more than 580 (Fig. 8). Six-degree zones are numbered in Arabic numerals from west to east, starting from the Greenwich meridian. The longitude of the axial meridian (0) of any zone can be determined by the formula:
λ 0 = 6 0 N - 3 0 (4)
where N is the zone number.
The intersection of the images of the axial meridian - the abscissa axis "x" and the equator "y" is taken as the beginning of the counting of coordinates in each zone. Shown in fig. 8 lines parallel to the image of the axial meridian (in the desired zone) and the equator form a rectangular coordinate system. Coordinates (x and y) can have "+" and "-" signs. Since the territory of Russia is located north of the equator, then all "x" will be positive; "y" - west of the axial meridian will have negative values. In order to avoid confusion in signs, that is, for the ordinates to be positive, the points of the axial meridian are conditionally assigned a value of 500 km, or the axial meridian of each zone is conditionally transferred 500 km to the west (Fig. 9).
Then the ordinates get conditional values and are called transformed y A, y D. The number of the zone in which the given point is located is written in front of the changed ordinate.
§ 9. Relationship between spherical rectangular coordinates of an ellipsoid and plane rectangular coordinates in the Gaussian projection
Principal radii of curvature at a given point on the ellipsoid.
If the Earth had the shape of a ball of radius R, then the curvature of its surface at all points would be constant and the same, equal, and the coordinates of any point on its surface would depend on B and L (Fig. 5).
In fact, as noted above, the Earth is close in shape to an ellipsoid of revolution with different sizes semiaxes (§6). Thus, the curvature of the earth's surface varies from point to point, which means that the coordinates of any point will depend not only on B and L, but also on the curvature of the earth's surface, that is, on the main ellipsoid curvature radii.
An infinite number of planes can be drawn through the normal to the surface of the ellipsoid (Fig. 5). These planes, perpendicular to the tangent plane to the surface of the ellipsoid at a given point, are called normal. Curves formed from the intersection of normal planes drawn at a given point with the surface of an ellipsoid are called normal sections. At each point of the ellipsoid, there are two mutually perpendicular normal sections, the curvature of which has a maximum and minimum value; these normal sections are called principal normal sections.
At some point M (Fig. 10) of the surface of the earth's ellipsoid, the main normal sections, as is known from differential geometry, are:
The meridional section passing through the given point M and both poles of the ellipsoid P and P1 (in Fig. 10 the meridional section at the point M is represented by the ellipse PME1P1E);
Section of the first vertical passing through point M and perpendicular to the meridian section of point M.
The cross section of the first vertical is shown in Fig. 9 of the WME curve, which is also an ellipse. Denote by M and N the radius of curvature of the meridian and the first vertical and write, respectively:
(5)
(6)
where a is the major semiaxis of the ellipsoid;
e 2 is the square of the first eccentricity of the meridional section;
B - geodetic latitude of point M.
Relationship between spheroid and planar rectangular coordinates.
Point A (Fig. 11) on the earth's ellipsoid has spherical coordinates ХА, YА and is located in a 6-degree zone at some distance (YА) from the axial meridian.
Since, according to the projection condition, the axial meridian of the zone touches the side surface of the cylinder, then after designing and developing the side surface of the cylinder, the abscissa XA will be displayed on the plane with the value xA without distortion (Fig. 12), that is
x A \u003d X A (7)
The YA ordinate is distorted and after projection it is calculated by the formula.
To compile topographic maps on the territory b. In the USSR, since 1928, the Gauss-Kruger transverse cylindrical conformal projection has been adopted.
Using the Gauss-Kruger projection, the entire earth's surface is divided by meridians into six- or three-degree zones (Fig. 4, a). This is due to the fact that at a large distance, the points of the axial meridian receive large distortions at this point on the map. The choice of a zone with a width of 3 or 6 ° of longitude depends on the scale of the map being compiled. When drawing up a map at a scale of 1:10,000 or smaller, a six-degree zone is used, and when drawing up a map at a scale of 1:5,000 or larger, a three-degree zone is used.
Figure 4. View of the zone in the Gauss-Kruger projection on the ball and on the plane
Six-degree zones are numbered with Arabic numerals, starting from the Greenwich meridian, from west to east. Since the western boundary of the first zone coincides with the Greenwich (initial) meridian, the longitudes of the axial meridians of the zones will be: 3, 9, 15, 21 °, .... The longitude of the axial meridian can be determined by the formula:
L 0 \u003d 6 ° N-3,
where N is the number of this zone.
Three-degree zones are located on the earth's surface in such a way that all the axial and boundary meridians of the six-degree zones are the axial meridians of the three-degree zones.
The coordinate systems in each zone of the Gauss-Kruger projection are exactly the same: the flat rectangular coordinates x and y, calculated from the geodetic (geographical) coordinates B and L in any coordinate zone, have the same values. In the Gauss-Kruger projection, the axial meridian, representing the abscissa axis (x), and the equator - the ordinate axis (y), are depicted by mutually straight perpendicular lines, and the remaining meridians are curved, converging at the poles (Fig: b). All abscissas of points in the northern parts of the zones (north of the equator) are positive. In order for all ordinates to be positive, 500 km are added to all ordinates (negative and positive). In addition, to fully determine the position of a point on the earth's surface, the zone number is written ahead of the changed ordinate. For example, in zone 7 point A has a real ordinate:
U A \u003d +14 837.4 m.
The transformed ordinate will be 7,500,000 m more, i.e., U A = 7,514,837.4 m. The abscissas of points throughout Russia are positive, they are left unchanged.
To obtain a cartographic grid in the transverse-cylindrical Gauss-Kruger projection, the Earth is placed in a transverse cylinder. The projection center is located in the center of the ball and the surface of the ball is projected onto the generatrix of the cylinder with direct rays. Design each zone in turn. In this case, the earth is rotated inside the cylinder so that the generatrix of the cylinder coincides (touches) with the axial meridian of the zone, Fig. 5.
As a result of the design, the cartographic grid had the following form, Fig. 6.
The Gauss-Kruger projection is conformal because it does not distort horizontal angles. geometric shapes earth's surface. Therefore, infinitesimal figures in these projections are similar to the corresponding figures on an ellipsoid.
Figure 5. Transverse cylindrical projection
In the Gauss-Kruger projection, in addition, the lengths of the arcs of the axial meridians are not distorted. The lengths of other lines and the areas of figures in this projection are distorted.
Figure 6. View of the cartographic grid in the Gauss-Kruger projection
Gauss-Kruger projection
As follows from the previous paragraphs, here we mean that the projection of the Earth's surface is made on a cylindrical surface, the axis of which coincides with the plane of the equator. In the projections of Mercator and Lambert, the axis of the auxiliary cylinder coincides with the axis of rotation of the Earth. In addition, as the title of the paragraph implies, the projection is conformal, which means that the directions that are in the horizontal plane are preserved on the image and on the earth's surface, and the horizontal angles between the directions are also preserved, respectively. But, as M.V. Lomonosov roughly said, if somewhere a certain amount is added, then somewhere the same amount will decrease. In this projection, the angles are preserved, but everything else is distorted: the scales of lengths and areas, the shapes of objects. The only undistorted image of the Earth's surface can only be a globe, i.e. volumetric image. But just imagine, in order to get an image of the Earth at a scale of at least 1:1,000,000, it will be necessary to make a globe with a diameter of more than 12.5 meters. You can't bring it into the audience. But, really, you can't take it. What if the scale is larger? So we have to make concessions: to depict one thing without distortion, while not paying close attention to the other.
The transverse cylindrical projection for depicting the surface of the earth's ellipsoid on a plane was developed by a German surveyor Zoldner and French surveyor Cassini. Subsequently, K. Gauss applied the principle of equiangularity to this projection, and the image scales in the new projection at each point in any direction were the same. Information about the new projection was published by K. Gauss in 1825, and almost 90 years later, in 1912, the scientist L.I. Kruger(1857 - 1923) published working formulas for this projection. Now this projection is named after Gauss and Kruger.
Suppose that the figure of the Earth is a ball of radius R. Let's build a cylindrical surface around the Earth, touching the surface of the ball along the meridian (Fig. 2.14). The axis of the cylindrical surface in this case must coincide with the plane of the equator. In the Gaussian projection on the cylinder, only a part of the surface of the ball (or ellipsoid) is projected, limited in longitude by meridians 3 o to the sides of the meridian tangent to the cylinder, the so-called 6 o (six-degree) zone. In total, there are 60 such zones for the entire Earth.
Fig.2.14. Gauss-Kruger conformal transverse cylindrical projection.
The meridian of the zone tangent to the cylindrical surface is called central or axial meridian of the zone. The zones are counted east of the Greenwich meridian and are designated by Arabic numerals (1, 2, ..., 60). The axial meridian of the 1st zone has an east longitude of 3 o. The longitude of the axial meridian of any zone with a number n in the Eastern Hemisphere can be determined by the formula
, (2.7)
and in the Western Hemisphere, for zones whose numbers are greater than 30, according to the formula
But not in all cases, the 6 o projection of the zone can be used. It is obvious that the line of zero distortion in this projection is its axial meridian in each zone. For all other points of the earth's surface (meaning its geometric spherical shape) there is a "gap" with an auxiliary cylindrical surface. And this means that the distortions gradually increase when moving from the axial meridian to the west or east and reach their maximum value at the edges of the zones. As seen in fig. 2.14, dot A, located on the extreme (eastern) meridian of the 1st zone on the plane (in the image) will move away from itself to another place and will be on the extreme (western) meridian of the 2nd zone.
In mine surveying, such distortions of 6 o zones are too large, so mine surveyors also use the Gauss-Kruger projection to compile cartographic materials, but using 3 o (three-degree) zones. Distortions in images constructed in 3 o zones are four times less than distortions obtained in 6 o zones. The axial meridian of the 1st 3rd zone coincides with the axial meridian of the 1st 6th zone. The axial meridian of the 2nd 3rd zone coincides with the extreme meridians of the 1st and 2nd 6th zones, etc. In total, 120 3 o zones are obtained.
The main properties of the Gauss-Kruger projection are as follows:
The axial meridian of the zone is depicted without distortion and is a straight line on the plane. All other meridians of this zone are represented by complex curves;
The equator in the projection is a straight line perpendicular to the projection of the axial meridian. All other parallels of this zone are compound curves;
Directions on the ground in the projection are transmitted almost without distortion;
The scale of the image (private scale) of small areas of the Earth's surface is preserved.
Pay attention to the following. Above, we talked about the fact that projection is performed on a cylindrical surface from the surface of a ball. But it is not so. Or rather, not at all. The geometric auxiliary surface of the Earth, the reference surface, is, as you already know, not the surface of a ball, but the surface of a reference ellipsoid. Therefore, the auxiliary cylindrical surface, on which the earth's surface is projected, must correspond to this, i.e. conjugate with the meridian, which is not a circle, but an ellipse. Thus, an elliptical cylinder must necessarily be used for design.
§ 11. Layout and nomenclature of topographic maps and plans
The concept of nomenclature in cartography is completely different from its other meanings in our everyday non-geodetic life (lat. - nomenklatura). This is a collection or list of names, terms used in any branch of science, technology, art, etc., this is also a circle of officials appointed by a higher authority. The semantic concept of nomenclature in geodesy comes from the fact that the adopted provisions should provide an unambiguous designation of topographic sheets or any other maps of various scales. It cannot be said that the nomenclature adopted by the cartographers in their work is convenient. Any other notation system will not be convenient either, since there are so many consecutive divisions from the primary sheet of the map that one can only rely on one's memory or use a reference book, which is much more reliable in these cases.
Rice. 2.15. Layout and nomenclature of topographic maps at a scale of 1:1000000.
Nomenclature is a system for designating sheets of maps of different scales.
The system of dividing maps into separate sheets using cartographic grid lines (lines of meridians and parallels) or a rectangular coordinate grid (coordinate lines) is called delineation.
The basis for dividing maps into sheets in our country is the international layout of maps at a scale of 1: 1,000,000 (Fig. 2.15). Breakdown into ranks parallels are made from the equator every 4 o latitude. Rows are labeled Latin alphabet: A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, V, W. columns within their boundaries they coincide with 6 o zones of the Gauss-Kruger projection, but they are numbered from the meridian ± 180 o to the east. Thus, the column number differs from the zone number by 30 units in one direction or another. Columns are designated (by numbers) with Arabic numerals.
Assume that the column number is 47. Then the corresponding zone number will be 47 - 30 = 17. If the column number is less than 30, then to determine the zone number, add 30 to the column number.
The nomenclature of the first sheet of a topographic map at a scale of 1: 1,000,000 is made up of the letter of the row and the number of the column. For example, H - 47.
There are no other sheets of the map at a scale of 1:1000000 with this designation. But you can ask a perfectly reasonable question: “The cartographers used all the letters of the Latin alphabet for the northern hemisphere to designate the rows. But what about the sheets of maps for the southern hemisphere? Should I repeat it?" Why repeat. Nomenclature is a system of notation. So let's slightly change this system for the southern hemisphere. For example, in the southern hemisphere, a similar sheet is designated as 47 - N. And no problems or questions. And you can come up with something else, for example, take in brackets (H) for the southern hemisphere. And again - no problem. And again, a reasonable question from your side: “Well, how do they actually designate it?” It’s easier than you and I thought: after the nomenclature, they indicate in brackets (Yu.P.). Our methods are still more interesting with you.
The layout of larger scales from the 1:1000000 sheet can be traced according to the diagram below, Table. 2.2 and figures 2.16 - 2.21 related to the example below.
Also known as the Transverse Mercator, this projection is similar to the Mercator projection, but in this case, the cylinder does not rotate around the equator, but around one of the meridians. The result is a conformal projection that does not preserve the correct directions. The central meridian is in the region that can be selected. On the central meridian, the distortions of all properties of objects in the region are minimal. This projection is most suitable for mapping areas stretching from north to south. The Gauss-Kruger coordinate system is based on the Gauss-Kruger projection.
Projection method
Transverse cylindrical projection with the central meridian located in a particular region. In the Gauss-Kruger coordinate system, the Earth's surface is divided into 60 zones six degrees wide, and the central meridian of the first zone is 177° W. Projection is carried out in each zone separately onto a cylinder, the axis of which rotates in the plane of the equator by 6 degrees from zone to zone. The scale factor is 1.000, not 0.9996, unlike UTM. In some countries, a number is added to the 500,000 meter Y offset, which is equal to the zone number. Zone 5 of the Gauss-Kruger coordinate system can have X-shift values of 500,000 meters or 5,500,000 meters.
Lines of contact
Any meridian for the tangent projection. (Gauss-Kruger). Two lines at the same distance from the central meridian for a secant projection (Transverse Mercator).
Linear elements of the cartographic grid
Equator and central meridian of the zone.
Properties
shape
Equangular. Small forms are preserved. The distortion of the shape of large areas increases with distance from the central meridian.
Region
The distortion increases with distance from the central meridian.
Direction
Local angles are exact everywhere.
Distance
Accurate scale along the central meridian if the scale factor is 1.0. If it is less than 1.0, then the exact scale is maintained on straight lines located at equal distances on both sides of the central meridian.
Restrictions
Spheroid or ellipsoid objects cannot be projected beyond 90° from the central meridian. In fact, the extent of the spheroid or ellipsoid should be within 10-12 degrees on either side of the central meridian. Outside this range, the projected data may not be projected to the same position when the operation is reversed. For data on a sphere, these restrictions do not exist.
A new projection called the Transverse Mercator complex (Transverse_Mercator_complex) has been added to the projection engine in ArcGIS. This provides accurate forward and backward UTM conversion up to 80 degrees from the central meridian. Involvement of a complex mathematical method makes this transformation preferable.
Areas of use
Gauss-Kruger coordinate system. Topographic mapping in the USSR and Russia on a scale from 1:10,000 to 1:500,000 of the entire surface of the Earth. In this system, the globe is divided into zones six degrees wide. The scale factor is 1, the X offset is 500,000 meters, and the southern hemisphere also has a Y offset of 10,000,000 meters. The central meridian of zone 1 is 177° W. Some countries add the zone number to the easting offset of 500,000 meters. In the fifth zone in the Civil Code, the shift along the X axis is 500,000 or 5,500,000 meters. There are also 3-degree Gauss-Kruger zones used for surveys at a scale of 1:5,000 and larger.
The UTM system is similar to the Gauss-Kruger system. The scale factor is 0.9996 and the central meridian of the first UTM zone is 177 degrees WH. The X offset is 500,000 meters and the southern hemisphere also has a Y offset of 10,000,000 meters.
