Gaussian transverse cylindrical projection. Planar Cartesian Coordinate System

Consider this projection on the ball. To do this, we introduce the system spherical coordinates, as shown in Fig. 5.9.

The point is taken as the origin of coordinates BUT, lying at the intersection of the equator with the meridian P1AP taken as initial in this system. Let's call it the axial meridian.

In the new coordinate system, the axial meridian is taken as the conditional equator, and points are taken as the conditional poles Q and Q1 lying on the equator and distant from the origin A 90° longitude. Point position M in this coordinate system is determined by the arc of the axial meridian and great circle arc .

The relationship between the coordinates , and the coordinates , is expressed by the formulas of spherical trigonometry.

(5.25)

where is the longitude of the axial meridian.

Let us now take a cylinder tangent to the ball along the axial meridian (Fig. 5.10) and transfer the conditional meridians to it Q 1 AA 1 A 2 Q, Q 1 aa 1 a 2 Q, Q 1 bb 1 b 2 Q… and conditional parallels - arcs of small circles parallel to the plane of the meridian - A 1 a 1 b 1 c 1 , A 2 a 2 b 2 c 2 since we do it in the normal conformal Mercator projection (see 5.4).

per axle x take the axial meridian. per axle at- equator Q,AQ(Fig.5.10). We obtain the equations of rectangular coordinates in this projection if in expressions (5.11) and (5.12) we replace by , and the coordinates and by and, respectively. As a result, we have

(5.26)

Since the projection is conformal, the increase along the axes will be

and distortion ellipses are circles with radius .

From (5.27) it follows that the distortion of distances and areas increases as the point moves away from the axial meridian.

In order to somehow limit these distortions, the use of this projection is limited to six-degree zones. Each zone has its own system of rectangular coordinates , . In this case, the longitude of the axial meridian in each zone is determined by the formula

where n=1,2, 3,...60 - zone number.

Within the zone, the value is quite small. Therefore, instead of the spherical ordinate, it is more convenient to use its linear value. To do this, we expand in (5.27) into a series, limited to two terms

Replacing it with a linear value , we obtain the well-known formula

. (5.28)

In Ukraine, the distortion of line lengths at the edge of the six-degree zone reaches

in the southern part or 72 cm per 1 km;

in the northern part or 52 cm per 1 km.

Distortion of areas, respectively

or 14 m 2 per 1 ha;

or 10 m 2 per 1 ha.

When compiling topographic plans on a scale of 1:5000 and larger, such distortions cannot be neglected. In this case, narrower three-degree zones are used, where the longitude of the axial meridian is calculated by the formula

As can be seen in Figure 5.10, the lines of spherical coordinates in the Gauss-Kruger projection are straight lines.

With the reverse transition from spherical to geographical coordinates, the axial meridian and the equator will be depicted as perpendicular lines (Fig. 5.11). The remaining meridians are curved lines, concavity facing the axial meridian, and parallels are curves, concavity facing the nearest pole.

So far, we have considered this projection on the ball. When passing to an ellipsoid of revolution, the general character of the projection does not change significantly.

Rectangular coordinates on the ellipsoid are calculated by formulas (2.15), and the distortion of line lengths by formula (2.18).

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MINISTRY OF AGRICULTURE OF THE RUSSIAN FEDERATION FAR EASTERN STATE AGRARIAN UNIVERSITY

FACULTY OF NATURE MANAGEMENT

DEPARTMENT OF BIOLOGY AND HUNTING

TEST

EQUIANGULAR TRANSVERSAL-CYLINDRICAL GAUSS-KRUGER PROJECTION

Completed by: 3rd year student of FZDPO gr. 918114 Shulegin T.A.

Checked: Matveeva O.A.

Blagoveshchensk

Introduction

Conclusion

Bibliography

Introduction

The conformal transverse cylindrical projection currently used for maps of scales 1:500000 and larger is named after the famous German mathematician Gauss, who developed in 1825 general theory conformal image of one surface on another.

The Gaussian projection is the projection of an ellipsoid onto a plane, and it is determined by the following conditions:

Image equiangularity;

The image of the axial (middle) meridian in the form of a straight line, in relation to which all meridians and parallels are located symmetrically;

Preservation of the length of the axial meridian.

In this paper, in addition to the disclosure of the concept of the Gaussian projection, methods for determining points and the corresponding formulas for translating coordinates depending on their location in a particular projection or plane are reflected.

1. Gauss-krueger conformal transverse cylindrical projection

The working formulas for the conformal projection of an ellipsoid without an intermediate transition to a sphere were given by L Krueger in 1912, as a result of which this projection is also called the Gauss Kruger projection in the literature.

In the transverse cylindrical projection of Gauss, in contrast to the conformal cylindrical projection of Mercator, the projection is made on the surface of a cylinder touching the surface earth ellipsoid(and not a ball) not along the equator, but along the meridian (Fig. 1). Therefore, the scale is preserved not along the equator of HOH1, but along the meridian of contact POP1. When designing, the cylinder is taken with an elliptical cross section.

Figure 1 - Cylinder touching the earth's ellipsoid along the meridian

Distortions in the Gaussian projection increase with distance from the axial meridian to the west and east, and the isocoles look like straight lines parallel to the touch meridian (axial meridian).

Mutually perpendicular lines in the Gaussian projection depict not meridians and parallels, but arcs of small circles ABC and DEP (almucantarata) and arcs of great circles HQ, NK, HO, HL, perpendicular to the axial meridian (verticals). If the almucantars ABC, DEF are drawn on the ellipsoid at regular intervals, and the verticals divide the axial meridian into equal segments LO=OK=KQ, then they, by analogy with the Mercator projection, form a grid of rectangles on the map, as shown in Fig. 2. The abscissa lines here are the images of almucantarates, and the ordinate lines are the images of verticals.

Also, by analogy with the Mercator projection with a known tolerance, it can be argued that the scale ( m) in the Gaussian conformal transverse cylindrical projection at any point of the map in any direction is expressed by the formula

where c"- the central angle measuring the almucantar of a given point.

Figure 2 - Coordinates of a point in the Gaussian projection

Injection c", expressed in radian measure, is equal to the length of the vertical arc that subtends it, divided by the radius of the ball (in this case, the ellipsoid can be equated to a ball). If the contracting arc of the angle c" denote by y0, then

where R is the radius of the globe. Expanding in a row, we get

This formula, as well as the formula, shows that in the Gaussian projection, distortions increase with distance from the axial meridian, i.e., with an increase in the ordinate y on the map.

Meridians and parallels, with some exceptions, have the form of complex curves in the Gaussian projection (Fig. 3). The equator, middle axial) meridian and the meridians 90° from the mean longitude are straight lines.

Figure 3 - Cartographic grid in the Gaussian projection

The Gaussian projection with a continuous image of large areas elongated in longitude gives large distortions (points that are 90 ° away from the axial meridian along the equator go to infinity). Therefore, in order to reduce distortion, it is applied in areas limited by meridian lines. Each zone is depicted on a plane separately, and the image of the middle (axial) meridian of each zone is taken as the X-axis, and the image of the equator is taken as the Y-axis. The length of the zones in longitude is taken such that the distortions at their edges are negligibly small.

When moving west or east of the axial meridian by 3 °, the relative distortion of lengths reaches 1/750 at the equator, and 1/1500 at a latitude of 45 °. Such distortion is acceptable for maps of scales 1:25,000 and smaller. However, with the distance from the axial meridian of the zone by more than 3°, linear distortions begin, grow rapidly, and become unacceptable. On this basis, in the CIS, the length of zones in longitude is set at 6°.

Figure 4 - Image of zones in the Gaussian projection

The image of the zones on the plane is shown in fig. 4. Knowing the zone number, you can determine the longitude of its axial (middle) meridian by the formula

L 0 \u003d 6N - 3,(4)

where N is the zone number,

L 0 - longitude of the axial meridian.

On the contrary, knowing the longitude of the axial meridian, it is easy to determine the zone number using the formula

The abscissas x in each zone are counted from the equator to the north with a plus sign, and to the south - with a minus sign. For the entire territory of Ukraine, the abscissas x are positive, so the plus sign is not placed in front of them. The ordinates y are measured from the axial meridian of each zone with a plus sign to the east and a minus sign to the west. To avoid negative values of the ordinates, they are conditionally increased by algebraic addition by 500,000 m. In addition, the zone number is placed in front of the resulting sum in order to know in which zone the given point is located. For example, some point is in zone 7 and has ordinate

Y \u003d - 243 435.15 m.

According to the specified rule, the transformed, conditional value of the ordinate will be

Y \u003d 7 256 564.85 m.

Thus, to calculate the conditional ordinate of any point, the number of the zone in which the point is located must be known.

Figure 5 Basic symbols on the ellipsoid and plane in the Gaussian projection.

The zone number can be determined by knowing the longitude of a given point or the nomenclature of a sheet of some topographic or survey topographic map on which it is located.

For the Gaussian projection, the following main notations are accepted (Fig. 5a - on an ellipsoid and Fig. 5b - on a plane):

B is the geodetic latitude of an arbitrary point M on the ellipsoid;

L -- geodesic longitude from Greenwich of the same point on the ellipsoid;

L 0 -- longitude from Greenwich of the axial meridian;

L \u003d L - L 0 - the difference between the longitudes of the meridian of a given point and the axial meridian;

A is the azimuth of the geodesic line on the ellipsoid;

X in -- the length of the meridian from the equator to the parallel with the latitude of the given point;

x and y are rectangular Gaussian coordinates of the corresponding

Points M1 on the plane;

g - Gaussian convergence of meridians;

b - directional angle of the chord of the geodesic line M1N1ґ on the plane;

e - correction for the curvature of the image of the geodesic line M1N1ґ (curve) on the plane;

N is the radius of curvature of the first vertical at a point with latitude B.

The rectangular Gaussian coordinates of any point on the earth's ellipsoid are the flat rectangular coordinates of the image of the corresponding point on the plane in the Gaussian projection.

The Gaussian convergence of the meridians at a given point is the angle formed on the plane by the meridian passing through the given point and the line parallel to the axial meridian.

A geodesic line between two points on an ellipsoid is the line of shortest distance on the surface of the ellipsoid between these points. The geodesic line in the Gaussian projection is depicted as a curve that forms an angle 5 with its chord, called the correction for the curvature of the curve. Angle 3 is small and is taken into account only when processing triangulation.

The directional angle of any direction on the plane is the angle between the positive direction of the x-axis and this direction. This angle varies from 0 to 360° and is measured from the positive direction of the X-axis in a clockwise direction. The connection between the azimuth, directional angle and Gaussian convergence of the meridians of an arbitrary point M1 on the plane is easily determined from Fig. 7.

Figure 7 - Relationship between azimuth, directional angle and convergence of meridians in the Gaussian projection.

When point M1 is located east of the central meridian

When point M1 is located west of the central meridian

In contrast to the polyhedral projection, in the Gaussian projection, due to the increase in distortions in both directions from the axial meridian, the trapezium of a topographic or survey-topographic map, the sides of which are segments of meridians and parallels, does not represent a geometrically correct figure. The concavity of the meridians in it is directed towards the axial meridian (Fig. 8). However, the deviation of the meridians from the straight line is much less than the graphical accuracy required when constructing trapezoid maps at scales of 1:500,000 and larger. Therefore, the sides of the trapezoids of these maps in the Gaussian projection are depicted as straight lines. projection Gauss plane ordinate

Figure 8 - Trapezoid in Gaussian projection

The deviation of the parallels from the straight line begins to be practically felt on the trapezoids of maps of scales 1:100000 and smaller (with a difference in longitudes of the extreme meridians of 30 "and more). Based on this, each parallel (north or south side) of the trapezoid is plotted: for a map of a scale of 1:100 000 by coordinates of three points, for a map of scale 1:200,000 by coordinates of five points, and for a map of scale 1:500,000 by coordinates of seven points.Accordingly, to build trapezoid maps of scales 1:100,000, 1:200,000 and 1: 500 000, you need to know the coordinates of six, ten and fourteen points, respectively.Trapezoid maps of scales 1: 50 000 and larger are built according to the coordinates of four points (corner vertices).

On fig. 9 shows schematic representations of trapezoid maps in scales 1:10000--1:500000. For map trapezoids at scales of 1:100000, 1:200000 and 1:500000, intermediate points are indicated, along whose coordinates parallels are plotted, and the dimensions of the trapezoid are given in degree measure (DV and DL are the dimensions of the trapezoid, respectively, in latitude and longitude).

The Gaussian rectangular coordinates of the vertex angles of the trapezoids and intermediate points are selected from special tables (Tables of coordinates of the Gauss-Kruger edition of 1947). The construction of a trapezoid is done by drawing these points in the usual way.

Figure 9 - Schematic representations of trapezoids indicating intermediate points, the coordinates of which are plotted parallels on maps in the Gaussian projection.

On a coordinator or with a caliper and scale bar. In the latter case, a square or rectangle is first constructed, and then the vertices of the corners of the trapezoid and intermediate points, if necessary, are plotted from its sides along the coordinates.

For the convenience of processing geodetic measurements made at the junction of two adjacent zones, mutual overlapping of coordinate zones in longitude is established. At the same time, the western zone overlaps the eastern one by 30", and the eastern one overlaps the western one by 7". In accordance with this, in the catalogs of geodetic points for all points located in the overlap band, rectangular coordinates are given for both zones. In some cases, it may be necessary to obtain the coordinates of the adjacent zone for points located outside the zone overlap zone. In these cases, the rectangular coordinates of the points are converted from one six-degree zone to another, adjacent six-degree zone. Usually performed using special tables (Tables for recalculating rectangular Gauss-Kruger coordinates from one six-degree zone to another six-degree zone of the 1947 edition). Tables for recomputing rectangular Gauss-Krüger coordinates from one six-degree zone to an adjacent six-degree zone of the 1946 edition), etc. In the introductory parts of these tables, explanations are given for their use and examples of coordinate recalculation are given.

To solve a number of practical problems, in particular military ones, a grid of rectangular Gaussian coordinates, or a coordinate grid, is applied on topographic maps. It is a network of squares formed by lines parallel to the axial meridian of the zone, and lines perpendicular to it. In each zone, the coordinate grid is applied from the equator and the axial meridian of this zone. The presence of a coordinate grid greatly facilitates the determination of the coordinates of points on the map and the drawing of points on the map by coordinates.

Conclusion

The Gaussian projection used for maps of scales 1:10000 -- 1:500000 has a number of advantages in comparison with the polyhedral projection used earlier. The first advantage of this projection is its connection on maps with a coordinate grid and rectangular coordinates of geodetic points. Drawing the vertices of the corners of the trapezium and geodetic points in the Gaussian projection is preceded by the construction of a coordinate grid. When applying a polyhedral projection, a trapezoid is first built, and then a grid of Gaussian rectangular coordinates is applied from the vertices of its corners. This reduces the graphic accuracy of plotting geodetic points.

The second advantage of the Gaussian projection is the theoretical possibility of gluing any a large number map sheets within the six-degree zone.

Finally, the third advantage of the Gaussian projection is its equiangularity. In comparison with other projections used for topographic and survey topographic maps, the Gaussian projection has the advantage that distortions are taken into account in it using fairly simple formulas.

Bibliography

1. Guidelines for the implementation of laboratory work on the discipline "Geodesy and topography" for full-time students of the direction 130201 "Geophysical methods of prospecting and Exploration of mineral deposits" and 130202 "Geophysical methods of well research". - Tomsk: ed. TPU, 2006 - 82 p.

2. Basics of geodesy and topography: tutorial/ V.M. Perederin, N.V. Chukharev, N.A. Antropova. - Tomsk: Publishing House of Tomsk Polytechnic University, 2008. -123 p.

3. Maslov A.V., Gordeev A.V., Batrakov Yu.G., Geodesy. - M.: KolosS, 2008. -598s.

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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 Earth divided into zones 6 degrees wide. The scale factor is 1, the x-offset is 500,000 meters, and Southern Hemisphere also has a shift along the Y axis - 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.

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 geodesic (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

Modern ideas about the shape and size of the Earth.

In geodesy, the shape of the Earth is defined as a body bounded by a level surface. A level surface is a surface that intersects sheer lines at a right angle. An ideal figure bounded by a level surface is called a geoid and is taken as the general figure of the Earth. Due to the special complexity of the geometric orientation of the geoid, it is replaced by another figure - an ellipsoid, which is obtained by rotating the ellipse around its minor axis PP1. (a=6378245m; b=6356863m; compression a=(a-b)/a=1/298.3; R=6371.11km).

Flat images of areas of the earth's surface.

A reduced image on paper of a horizontal projection of a small area of area is called a plan. On a plan, the area is depicted without noticeable distortion, since a small area of \u200b\u200bthe surface can be taken as a plane. A map is a reduced image on paper of a horizontal projection of a section of the earth's surface in the accepted map projection, that is, taking into account the curvature of the reference surface. When designing small areas of the earth's surface, a small part of the level surface can be replaced by a plane. In this case, the plumb lines are parallel to each other and the horizontal projection of the earth's surface is converted into an orthogonal projection. The projection of the terrain line onto a horizontal plane is called the horizontal distance. The formula for the horizon is (s=S*cosv). In geodesy, central and cartographic projections are also used.

Geographic coordinate system.

The position of a point on the surface of the Earth is determined by two coordinates - latitude and longitude. The geodesic coord system refers to the surface of an ellipsoid of revolution. Geodes latitude (B) - the angle between the normal and the plane of the equator. 0º≤В≤90º Geodes longitude (L) – angle between the plane of the prime meridian (Greenwich Mean Time) and the meridian plane of the given point. Longitudes vary from 0º to 180º, west of Greenwich - west and east - east. All points of the same meridian have the same longitude. Astronomical CS refers to the surface of a sphere. Astronomer latitude (φ) - the angle between the plumb line and the plane of the equator. Astronomer longitude (λ) - the angle between the meridian plane of a given point and the plane of the prime meridian. 0º≤φ≤90º 0º≤λ≤180º

convergence of meridians.

The angle between the midday lines of two points lying on the same parallel is called the convergence of the meridians of these points.

The concept of the Gauss-Kruger conformal transverse-cylindrical projection.

The essence of this projection is as follows.

1. The earth ellipsoid is divided by meridians into six and three degree zones. The middle meridian is called the axial meridian. Zones are numbered eastward. The axial meridians lie on the inner surface of the cylinder, in which the spherical surface is divided into separate sections (60 in total).

2. Each zone separately is projected confermally onto a plane in such a way that the axial meridian is depicted as a straight line without distortion (ie, with exact preservation of lengths along the axial meridian). The equator will also be drawn as a straight line. The intersection of the image of the axial meridian - the x-axis of the abscissa and the equator - the y-axis of the coordinates is taken as the origin of the coordinate count in each zone. Lines parallel to the axial meridian and the equator form a rectangular coordinate grid.

3. Line length distortions in the Gauss-Kruger projection increase with distance from the axial meridian in proportion to the square of the ordinate. These distortions at the edges of the six-degree zone can reach values of the order of 1/1500 of the line length, and in the three-degree zone 1/6000. For a segment with the coordinates of the end points x1y1 and x2y2, the correction formula for line length distortion on the plane has the form, where and R is the average radius of curvature. In large-scale surveys, such distortions cannot be neglected. In this case, when the site is located at the edge of the zone, one should either take into account distortions, or use a particular coordinate system with an axial meridian passing approximately through the middle of the work site.