AN ALTERNATIVE APPROACH FOR MAP PROJECTION SYSTEM USED IN EGYPT
BY
Dr. Eng. Ahmed M. Youssry
Dr. Eng. Abdalla A. Saad
Dr. Eng. Ali A. Elsagheer
Shoubra Faculty of Engineering, 108 Shoubra St., Cairo, Egypt
Abstract
Mapping is one of the major aims of surveying. The obtained coordinates from the surveying operations are treated, according to the adopted technique, in the map projection section to obtain the national grid coordinates, which are suitable for mapping.
Mapping in Egypt is based on the Transverse Mercator projection. Egypt has been divided into three zones, with three numbering systems. Every zone has four degrees width of longitude, two degrees on both sides of its central meridian. At the time where the Transverse Mercator projection was chosen as a map projection system, most triangulation networks were in and around the cultivated area and the most effective part of the country was that area lying along the Nile Valley between longitudes 29ْ E and 33ْ E. This area includes most of cities, population, and activities, while the remainder areas of Egypt were neglected, which are mostly desert areas. So, the Transverse Mercator projection was a suitable projection for mapping in Egypt at that time. Recently, and after the highly precise surveying instruments and techniques are achieved, and the Egyptians started strongly to move out of the Nile Valley, the old projection system, with some defects, is not sufficient for the new needs of the current developments in Egypt. This research discusses some defects of the existed projection system in Egypt, and introduces modification for this system. The results showed that, the distortion can be reduced for the purpose of the surveying and geodetic work, and the numbering system can be modified for the purpose of the GIS work.
1- Introduction
1-1 Surfaces of the Earth
Observations and measurements are made on or near the physical surface of the earth. It would be quite impossible to perform detailed and extensive computations on such a surface whose definition requires an infinite number of parameters. The geoid is the bounding equipotential surface of the earth. It is every where normal to the direction of gravity, because of mass excesses and deficiencies within the earth. The geoid is unsuitable as a mathematical model for computations, because it is defined by an infinite number of parameters. The geometric shape of the earth is an ellipsoid of rotation with a polar diameter about one-third of one percent (1/300) shorter than the equatorial diameter. The ellipsoid chosen for any particular map series, will best describe the geoid in that part of the earth being mapped, which is named local datum. An example of local datums is Helmert 1906, which is used in Egypt. There are also international datums, which use ellipsoids fitting the whole earth, these datums are the global or international ones. The World Geodetic System 1984 (WGS84) is an example of the global datums. Nowadays, the direction towards adopting WGS84 is increasing because of increasing of using the GPS in the surveying measurements.
1-2 Distortions
Mapping, in its classical form, is representing the natural and industrial features of the earth’s surface on a sheet of paper. In surveying, the positions of these features are related to the mathematical body of the earth ( the sphere or the ellipsoid ) through a coordinate system. In map projection, the points defined on the adopted ellipsoid are shifted to a sheet of paper (the map), according to some mathematical formulations regarding some specifications. But an ellipsoidal surface can not be spread out flat without distortion, any representation of a part of the earth’s surface on the mapping plane introduces some distortions.
The two surfaces, the earth model and the map are not in coincidence and thus cannot be transformed from one into another without stretching or tearing. Accordingly, the distance among the points on the plane must be modified and this modification causes alterations. Such alterations in distances and consequently in angles and areas occur due to the process of projection, these are the distortions.
1-2-1 Types of Distortion
Four types of distortion are of particular interest in mapping [1]:
** Distortion of distances in any given direction at a certain point;
= An infinite distance in map / corresponding distance on the earth model.
** Distortion of area at a certain point;
= An infinitesimal area in the map / corresponding area on the earth.
** Distortion of angle between intersecting lines at a certain point;
= Angle on the earth - corresponding angle in the map.
** Distortion of shape; the change of shape due to the process of projection, i.e. a circle
could be projected into a closed figure and a square could be projected into a quadrilateral.
So, one can select map projection system, which retains angles without distortion at the compromise of some distortion in distance and in area. Also one can preserve areas at a cost of some distortion in shapes, and so on.
1-2-2 Scale Factor
The scale factor indicates the amount of distortion between the nominal scale and the actual scale at a point anywhere in the map. The scale factor will vary over the area covered by the projection and in most cases according to the direction in which the distance is measured:
Length on projection
Scale factor (k) = ------------------------------------------------------------------------
True length on the earth at the nominal projection scale
A scale factor of one is the ideal scale factor, but it may be more or less than one in different areas. For charts and small scale mapping, some distortion is acceptable to achieve other quantities. For topographical scales, one needs to measure distances and angles from the map with negligible error. The error from scale factor variations must not be allowed to exceed the plotting error, accordingly limits must be placed on scale factors.
Taking a reasonable maximum length of a measurement on a map as 50 cm, and assuming an accuracy of plotting or measuring of 0.2 mm, the scale accuracy will be 0.2/500=1:2500. Therefore a map can be considered effectively free of error if the scale factor lies between 2499/2500 and 2501/2500, i.e. between 0.9996 and 1.0004. The length distortion may be expressed in the fractional form as (k-1), e.g. if k = 1.0004 then the length distortion = 0.0004 = 1/2500 or 1:2500.
1-3 Properties of Map Projections
Because of the shape of the earth, it is impossible that the map will be in coincidence with the earth verifying the same relationships between the points on the earth. Map projection can be classified depending on the used surface as stereographic, cylindrical, conical. It can be also classified depending on used way of projection into geometrical or mathematical. And finally several kinds of map projections can be obtained when the aim of projection is considered, as follows [2]:
1- Conformal or Orthomorphic Projection: produces a map showing the correct angle between any pair of short intersecting lines, thus making small areas appear in correct shape. Because the scale varies from point to point, the shape of larger areas is incorrect.
2- Equal Area Projection: produces a map showing all areas in proper relative size, these areas may be much out of shape and the map may be having other defects.
3- Equidistant Projection: distances are correctly represented from one central point to the other points on the map.
4- Azimuthal Projection: the map shows the correct directions or azimuth of any point relative to one central point.
The conformal projection is the one adopted in the Egyptian Surveying Authority (ESA) for mapping the Egyptian terrain. Therefore this research will concentrate on the conformal projection in the next subsection.
1-3-1 Conformal Projection
Experience has shown that the best compromise projection for surveying and map applications will be achieved by using one of the projections that preserves angles of each point without distortion. Projections having this property are called Orthomorphic or True Shaped Projections. They are also known as Conformal Projections because the shape of the detail around each point conforms with its shape on the reference ellipsoid.
A Conformal Projection must verify the following three properties [3]:
1- The scale at any point is the same in all directions.
2- Angles between short intersecting lines do not change in the projection, but small corrections have to be added to the measured directions when long distances between the projected points are involved.
3- The infinitely small circle on the earth surface will always be projected as a circle on the plane.
The most commonly used Conformal Projections are:
1- The Transverse Mercator Projection
2- The Stereographic Projection
3- The Lambert Conformal Conical Projection
4- The Normal Cylindrical Mercator Projection
5- The Diagonal Mercator Projection.
1-3-2 Factors Affecting The Choice of Map Projection Systems
There are some objectives should be verified during the projection:
1- To keep distortion as small as possible
2- To keep discontinuities as few as possible, i.e. few number of zones
3- To have one Cartesian Coordinate Reference for the whole projected area if possible.
4- To regard specific areas, like the Valley and the Delta in Egypt, for their importance. In these areas, the distortion should be kept minimum as possible as it could. This is one of
the main objectives in our research.
The first two objectives are contradicting, i.e. if distortion is kept in an acceptable limits there will be many projection zones. On the other hand if a very large area is covered by one projection plane there will be unacceptable large distortions at the borders of the zone. Therefore, the choice of a projection system will be a compromise.
Here is a fourth objective where some important areas should be considered more than other areas. As example in Egypt, the Nile Valley and the Delta are still more important than the other areas. So, one of the aims of this research is to improve the projection in the middle zone which contains the Nile Valley and the Delta.
2- Transverse Mercator Projection (TM)
Transverse Mercator (TM) is the projection technique, which been adopted for producing maps in the Egyptian Surveying Authority (ESA). The following is explanation for this technique and its mathematical formulations.
2-1 Properties of Transverse Mercator Projection
TM is the ordinary Mercator projection turned through a 90 so that it is related to a central meridian in the same way that the ordinary Mercator is related to the equator. Because the cylinder is tangent to the spheroid at a meridian, the scale is true along that meridian, this is the central meridian and is used as the map North (N) coordinate axis. The map East (E) coordinate axis is a line parallel to the equator and tangent to latitude 30 N as it is followed in ESA. Properties of this projection are [2]:
1- Both the central meridian and the normal to it are represented by straight lines.
2- Other meridians are complex curves that are concave toward the central meridian.
3- Parallels are concave curves toward the poles.
4- The scale is true only along the central meridian.
This projection is used for areas of greater north-south extension than east-west ones.
Sometimes, the cylinder in this projection is made to cut the surface of the used spheroid along two standard lines parallel to the central meridian instead of being tangent to the spheroid. This projection is known as the secant cylindrical projection and it can slightly increases the zone width, so the number of zones becomes as few as possible. In this case, there are two lines (secant lines) on the reference spheroid with scale factor (k=1). In addition, the scale factor at the central meridian will be less than one (ko<1). Introducing, for instance, ko=0.9999 a length distortion of 1:10,000 will be obtained at the central meridian and decreases towards the secant lines and increases again to 1:10,000 at a distance 127.5 km from the central meridian. The proposed compromised projection in this research used this secant projection.
2-2 Transverse Mercator computations
The computations here is to obtain (E, N) of a point on the map from its known geographic coordinates on the used ellipsoid (geodetic datum). The formulae of obtaining (E, N) can be written as follows [4]:
E=Nr[ cos+(3 cos3/6)(1-2 + 2)+(5 cos5/120)(5-182+4+142 -582 2+134)+ ….] (2-1)
N = MA + Nr [ (2/2) sin cos + (4/24) sin cos3 (5 - 2 + 92 + 44)
(6/720) sin cos5 (61 - 582 + 4 + 270 2 - 33022 + 4454) + ….] (2-2)
where : Nr = a / (1-e2 sin2 )0.5
= tan
2 = e2 cos2 / (1 - e2)
and
MA = a (Ao - A1 sin2 + A2 sin4 - A3 sin6 +…)
in which Ao = 1- 0.25 e2 - (3/64) e4 - (5/256) e6 - ….
A1 = (3/8) e2 + (3/22) e4 + (45/256) e6 + ….
A2 = (15/256) e4 + (45/1024) e6 + ….
A3 = (35/3072) e6
The terms used in the above equations are sufficient to give E and N values that are accurate to 1 cm for zones 3 of longitude around the central meridian.
2-3 The Scale Factor
The scale factor at any point in the map can be computed as follows:
k = 1 + (2 cos2)/2) (1+2) + (4 cos4) /24)(5-42+142+134-2822+46-
4824-2426) + (6 cos6/720)(61-1482+164) +…. (2-3)
Scale factor (k) can be approximately calculated by simpler equations as follows:
k = ko( 1 + (Dist)2 / (2R2)) (2-4) or k = ko ( 1 + 2 cos2 / 2 ) (2-5)
Where ko is the scale factor along the central meridian and R=(Mr Nr)0.5 and
Mr = a (1-e2) / (1-e2 sin2)1.5
Recalling that, it is preferable to have a projection with as few zones as possible. The width of the zone can be slightly increased by introducing a secant cylinder with ko < 1, so the number of zones becomes as few as possible. In this case, there are two secant lines on the reference ellipsoid which will have scale factor k=1 (i.e. the scale error equals zero).
2-4 Zone Width in Transverse Mercator Projection
Transverse Mercator Projection can be used in different zone widths. Some famous zone widths with their specifications are given as follows:
1- Four Degree Zone: This zone width is used in Egypt. The scale factor is true along the
central meridian. A different scale factor along the central meridian can be used as the
one of the British Islands which is 0.9996012717 [5].
2- Three Degree Zone: It is one of the most applicable cases. The scale factor along the central meridian is reduced to 0.9999. Hence a smaller scale error can be expected through the zone. Accordingly, the scale error along the central meridian will be 1:10000.
3- Two Degree Zone: A scale factor of 0.99995 can be assigned to the central meridian, so a small scale error can be expected throughout the zone. This will result a scale error of 1:20000 along the central meridian
3- Transverse Mercator as Applied in Egypt
The Transverse Mercator Projection is adopted for mapping in Egypt. The scale factor along the main central meridian, 31 E, is 1.0. The cultivated areas and most of the cities are lying around this central meridian. The N axis coincides with the central meridian of the considered zone. On the other hand, the true origins of the map coordinates are the points of intersection of the central meridians with latitude 30N, as it is applied in ESA.
3-1 Zones and Map Coordinate Systems
Egypt extends over twelve degrees of longitude (25E - 37E), it has been divided into three zones with three systems of coordinates, figures (1) and (2). Every zone is four degrees in width, two degrees at each side of the central meridian. The three zones with their three map coordinate systems are as follows [6]:
The Western Desert Zone: This zone covers the area between longitudes 25E and 29E with central meridian 27E. The zone has a false origin situated at 200 km to the south and 700 km to the west of the point of intersection of the central meridian 27E and the parallel of latitude 30N. The false origin lies therefore in Libya.
The Nile Valley Zone: This zone covers the area between longitudes 29E and 33E with central meridian 31E. Its false origin is situated at 810 km to the south and 615 km to the west of the point of true origin. The false origin lies in the southwest boundaries with Libya and Sudan.
The Eastern Desert Zone: This zone covers the area between longitudes 33E and 37E with central meridian 35E and has a point of false origin situated at 1100 km to the south and 300 km to the west of the point of true origin. The false origin lies in Sudan, figures (1) and (2).
3-2 Transverse Mercator Computations Applied in ESA
3-2-1 Computing The Cartesian Coordinates (E, N)
Equations (2-1) and (2-2) have many terms and parameters which made the computations at that time so complicated. Thus a suitable form is given in [7] is used as follows:
N = MA + (Nr 2/2) tan + (Nr 4/24) tan (5 - tan2) (3-1)
E = Nr + (Nr 3/6)((Nr/Mr) - tan2) + (Nr 5 /120)(5 - 18 tan2 + tan4) (3-2)
Where MA is the meridional distance from the equator to latitude and = sin1 cos
3-2-2 Computing The Scale Factor
The scale at a point of latitude , difference in longitude from the central meridian is given by:
k = 1 + (1/2) sin2() cos2 (3-3)
or k = 1 + (DT)2 / 2R 2 (3-4)
In the case that = 22 and = 2, so k = 1.000,524 which is the maximum scale factor can be found in mapping of Egypt.
To obtain the true distance (S) from the grid distance (s), derived from grid coordinates (E, N), or to convert a true distance to a grid distance for plotting on the map, it is necessary to calculate the scale factor and apply it in the correct way. The following connection ties between the two distances:
s = S * k or S = s / k
The scale factor changes from point to another but so slowly that for most purposes it may be taken as constant within 10 squared kilometers and taken equals to the value at the center of that square. For long lines the scale factor should be calculated at the mid point of the line. For lines up to 30 km in length the mid point value will give error within 1 or 2 PPM. When greater accuracy is needed, both ends and the mid point of the line are used as Simpson’s rule:
k = 1/6 (k1 + 4 * km + k2)
The scale factor k can be computed from the geographic coordinates (, ) as follows:
k = ko (1 + 2 (I) +4 (II))
Where
I = (cos2 /2) (1 + 2)
II = (cos4 /24) (5 - 4 tan2 + 14 2 - 28 tan2 2)
The scale factor can also be computed from the grid coordinates (E, N) as follows:
k = ko (1 + E2 (III) + E4 (V))
Where
III = 1/(2Mr1 Nr1)
V= (1 + 4 12) / (24Mr12 Nr12)
1, Mr1 and Nr1 are derived using ‘ (the geodetic latitude of the foot of the perpendicular drawn from a point on the projection to the central meridian.
3-3 Evaluation of Transverse Mercator Applied in Egypt
At the time Transverse Mercator was chosen for map projection in Egypt, most Triangulation stations were founded around the cultivated area, around the Nile Valley between longitudes 29E and 33E. This area includes the majority of cities and population. The rest of Egypt was neglected deserts. So, Transverse Mercator with the specifications explained before did not show big shortcomings and was suitable to be used for mapping in Egypt at that time.
As a summary, Egypt is projected in three zones, where the cultivated area falls in the middle zone 2 of longitude at both sides of the central meridian, which is 31E. Transverse Mercator did not show any problems in that time because of:
1- Topographic maps are not needed for the desert areas in that time.
2- The computations of the desert triangulation stations were carried out in the
geographical coordinates.
3- The neighboring triangulation stations to the main (middle) zone are computed on both
their own central meridian and on the main central meridian 31E.
4- The required accuracy of the surveying applications in that time is not like the required
accuracy of today.
Nowadays, Egyptians started to move away from the narrow valley and to use every place in Egypt for their recent development. Regarding the requirements of today, the applied system has disadvantages. Some of these disadvantages are as follows:
1- The values of the scale factor are high at the outer edges of the zone. Scale factor at the central meridian of any zone equals 1.0 and it increases rapidly away from the central meridian to reach 1.000,524 at the border of the zone. The problem is felt when the area of one of the cadastral projects for developing the south of Egypt lied in two neighbor zones (shifting from zone to another).
2- Taking the false northing of the origin of the Western Desert Zone equals 200,000 meters, makes most of the northing coordinates with negative sign, figure (2).
3- Also, the choice of the false northing of the origin of the Middle Zone equals to 810,000 meters, makes some of the northing coordinates with negative sign, figure (2). So, Egypt is divided into three zones as a map projection and five zones as a numbering system.
4- The Cartesian coordinates (E, N) of some points in the three zones can have the same value and the color of the zone distinguishes it among the others, figure (2).
4- The Proposed Projection
Our concern in this research is to find a solution for the scale factor problem arising at the borders of the zones. Some limits must be placed on the magnitude of the scale factor. Two ways could be applied to limit the scale factor. The first way is to reduce the use of the projection to areas quite close to the central meridian [8]. This method reduces the distortion as minimum as possible along the area of the zone but gives a large number of zones, i. e. many discontinuities. So this way does not satisfy one of the important objectives of choosing the suitable projection system.
The second way is to apply a reduction scale factor so that the scale along the central meridian is made too small, i. e. ko1. Two lines of exact scale are established towards the outer edges of the zone. This procedure shrinks the cylinder so that it cuts the used spheroid in two curves instead of to touch in one curve. This allows the scale factor to remain within tolerable limits at all points in the projected zone.
The following will result when using such a reduction scale factor projection:
1- the distortion at the edge of the zone will be positive and equals half the value in the
case of using the tangential projection used in ESA.
2- the distortion at the middle of the zone will be negative and equals its value at the edge
of the zone.
3- the largest value for the distortion will be at the middle and at the edges of the zone.
The middle zone of Egypt is still important, so the distortion at the central meridian of this zone is preferred to stay zero.
According to the above discussion and in the light of :
1- keeping the transverse Mercator as an adopted projection in ESA
2- keeping the number of zones as it is in ESA, 3 zones
3- keeping the zero distortion value at the central meridian of the middle zone for its importance
The proposed technique through the three zones will be as follows:
1- Western Desert Zone: the projection will be secant TM. The central meridian of the zone is (27 15’ E) and the central point of the zone is (30 N and 27 15’ E) with false coordinates (1,000,000 N and 250,000 E) in meters, figures (3) and (4). The Eastern coordinates through the zone will be more than zero and less than 500,000 mt. The zone width will be 4 30’ from 25 E to 29 30’ E. The central meridian will have a negative scale factor equals 0.999,669 and at the borders the maximum scale factor will be positive 1.000,331. It should be mentioned here that the maximum scale factor at the border of the zone calculated as in ESA will be 1.000,663
2- Nile Valley Zone: which is the middle zone with tangential TM projection and central meridian 31 E as it is in ESA. The central point is (30 N and 31 E) with false coordinates (1,000,000 N and 700,000 E). The Eastern coordinates are more than 500,000 and less than 900,000. The zone width will be 3 from 29 30’ E to 32 30’ E. The scale factor at the central meridian will equal one like its value in ESA, so the scale factor at the borders of the zone will be 1.000,295. It should be mentioned here that its value in ESA is 1.000,524.
3- Eastern Desert Zone: the projection will be secant TM with central meridian 34 45’ E and central point (30 N and 34 45’ E). The false coordinates of the central point are (1,000,000 N and 1,150,000 E). The Eastern coordinates through the zone will be more than 900,000 mt. As in the Western zone, the scale factor at the central meridian of the Eastern zone will equals 0.999,669 and at the borders of the zone will be 1.000,331. Figure (3) and (4) are illustrating these projections.
The largest value of the distortion at the edge of every zone is calculated using the equation:
k = ko( 1 + (Dist)2 / (2R2))
Where ko is the scale factor at the central meridian. In the middle zone case, ko equals 1 and the distance from the central meridian to the edge of the zone equals half the width of the zone which is (3/2 =1 30’ 155 km). Substituting in the above equation, the scale factor at the edge of the middle zone is 1.000,295 where the distortion is 0.000,295. This value of distortion corresponds 0.000,524 at the border of the zone in the existing projection in ESA.
5- Conclusion
Based on applying the proposed technique, The following will be verified:
1- Exceeding the limits of the middle zone to the Eastern and Western zones, when it is
necessary, for 30’ ( 55 km) distance without increasing the already existed distortion in
ESA current projection. ESA current distortion at the edge of the zone is 0.000,524.
2- Exceeding the limits of the Eastern and the Western zones towards the middle zone for
18’ 20” (33 km) distance without increasing the already existed distortion in ESA
current projection which equals 0.000,524.
3- With the proposed system of the false coordinates, eighboring maps can be collected in realistic style instead of the style in ESA which exchanges the positions of the Eastern and the Western Deserts. This could be not important in the geodetic applications but it is important in the GIS applications.
4-In Egypt, and for the sake of avoiding the defects of the already existed projection, two solutions are sought. The first solution is to adopt WGS84 as a datum for Egypt, so, UTM projection will be used directly. This will change all the maps and numbering system in the hand of people, owners, authorities, …etc. The second solution is to change the projection into more suitable technique like Lambert, the one improved the projection in Egypt more than any other system as studied in [8]. This also will change all the maps and numbering system in the hand of people, owners, authorities, … etc.
5- To change the maps and the numbering system of almost one century of work and ownership system is a real revolution in that field. It is difficult matter in any country. The proposed solution in this research will keep the mapping and the numbering system as they are in ESA in most of the middle zone. The middle zone contains the majority of the produced maps and the surveying activities in Egypt. Therefore most of the existing mapping work will not be lost at the contrary of the first two solutions. Eastern and Western Deserts anyhow need new maps, so, the proposed system will offer less distortion projection.
6- In ESA, the Eastern and the middle zones have unsuitable false northings, so, negative grid values are obtained at the south of the two zones. Accordingly, 1000,000 meters are added as a constant to change these negative values to positive ones, and the numbering system five zones not three. The proposed projection introduces three zones in the projection and in the numbering system.
6- REFERENCES
[1] Porter W. and Jr. McDonnell (1979). “Introduction to Map Projections”. Marcel Dekker, INC., New York and Basel.
[2] Davis, and Mikhail (1998). “Surveying, Theory and Practice”. McGraw-Hill Book Company.
[3] Habib M.I. (1997). “Assessment of Some Conformal Projection Properties and Their Uses in Mapping”. Master Thesis Submitted to the Transportation Engineering Department, Alexandria University, Egypt.
[4] Thomas P.D. (1952). “Conformal Projections in Geodesy and Cartography”. U.S. Coast and Geodetic Survey, Special Publication No. 251.
[5] Anonyme (1983). “Transverse Mercator Projection”. Ordnance Survey Information.
[6] Youssry A.M. (1984). “New Consideration for the Use of T. M. P. System in Egypt. Master Thesis, Cairo University, Egypt.
[7] Craig J. I. (1933). “The Theory of Map Projections”. National Printing Department, Cairo
.
[8] Youssry A.M. (1997). “A Proposed System for Map Projection in Egypt”. Ph.D. Thesis, Shoubra Faculty of Engineering, Zagazig University, Benha Branch.
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