U.S. DEPARTMENT OF COMMERCE DANIEL C. ROPER, Secretary COAST AND GEODETIC SURVEY R. S. PATTON, Director SERIAL No. 562 PLANE COORDINATE SYSTEMS By Oscar S. Adams, Senior Mathematician U.S. Coast and Geodetic Survey INTRODUCTION What means can be employed to simplify the task of the local cadastral surveyor and the engineer making surveys for public works or for projects of a private character? For practically all such sur¬ veys the computations are made as if the region involved were a plane and the results are thus based on a local plane coordinate system. This is satisfactory when the work is limited to a very restricted area but when extensions of the surveys are made to surrounding areas trouble is apt to arise. Two adjoining local systems can be coordinated only with a great deal of extra computation. It seems advisable therefore to give study to the possibility of establishing plane coordinate systems for more extended areas so as to provide the advantages of plane coordinates and at the same time to make possible a perfect coordination of the work of different engineers throughout the whole region for which one system is adequate. DIFFICULTIES IN USING SPHERICAL COORDINATES The surface of the earth is not a plane but is very irregular. For purposes of mapping, an ideal surface is assumed that approaches very nearly to what it would be if the entire surface were covered Avith water. This assumed surface is not even a sphere but is an ellipsoid of revolution since this shape is the one that approaches most nearly the sea-level surface. This spheroidal shape introduces complications in the calculations of survey work when it is extended over a large area. Geodetic surveying is exactly such extensive work and all computations arising in it have to take into consideration this assumed spheroidal shape which fits closely the actual shape of the earth. All geodetic stations on the earth have to be located 20195—33 2 by latitude and longitude; that is, each one is referred to a definite place in the network of meridians and parallels that are assumed to cover the surface of the earth in a definite and standardized way. The latitude and longitude of a place on the curved surface of the earth thus corresponds in a general way to the x and y coordinates of a place on the usual system of plane coordinates which presup¬ poses a flat surface. But the x and y coordinates of the ordinary plane system have no relation to anything but the assumed origin and the assumed directions of the X and Y axes. Much would be gained if these coordinate systems could be made parts of a more extensive system based on a definite relationship with the meridians and parallels of the earth. UNIT OF AREA FOR PLANE COORDINATE SYSTEMS In the consideration of plans for the establishment of coordinate systems, the first question to be settled is the unit of area to be used for a single system. Since the earth’s surface is curved no part of it can be represented on a plane without distortions and discrepan¬ cies entering into the computations. At first county units were con¬ templated but it was seen that this would not give systems of suffi¬ cient extent. Some States under such an arrangement would have 50 or more separate systems which would be continually giving trouble in the matter of coordination. The ideal unit is of course a State. If the State is not too large it is possible to use a single system for it that meets the requirements in all respects. LIMITS FOR STATE SYSTEMS What can be done for a State depends upon its size, the shape of its area, and the direction of its greatest dimension. A State of greatest extent in an east-west direction and a width not greater than 158 miles in a north-south direction can have a system of plane coordinates with variations of scale limited to 1 part in 10,000. Also, a State with greatest extent in a north-south direction and with a width not more than 158 miles in the east-west direction can be fitted with a system with the same limitations of scale discrepan¬ cies. These discrepancies are well within the accuracy of general local surveys so the State system could be used as are ordinary plane coordinates in the computation of s»ch surveys. POSSIBILITY FOR GREATER EXTENSION Fortunately it is not necessary to confine a single system to even the limitations specified above. When the coordinate system is based on a definite system of projection, the scale distortion becomes known for all parts of the region. This is a definite factor that can easily be applied so as to make the computations practically as ac¬ curate as are those in geodetic work. Even if distortions of scale become as great as one part in 5,000, or more, they can be taken into consideration and a high accuracy can still be maintained. This gives the possibility of using a single system for almost any of the States. Florida, California, and Texas, however, are examples of States so irregular in shape or so large that it would be difficult to fit a suitable single system to them. In such cases it might be well 3 to divide the State into sections and then adapt a suitable system to each of the sections. The boundary between the sections could be an arbitrary straight line or if preferred a zigzag line following county boundaries. Details such as these could best be determined by the engineers of the State who would use the system. The advan¬ tage of having a plane coordinate system with very small scale factors is that the factors can be ignored except for special surveys requiring great accuracy. GEODETIC CONTROL SURVEYS As stated before, all geodetic surveys are computed directly on the ellipsoidal surface and the results are given in latitude and longi¬ tude. A vast system of such control framework has been established by the Coast and Geodetic Survey and stations of this network are now available over large areas for use as starting and ending con¬ trol for regional and local surveys. There have been completed about 40,000 miles of arcs of triangulation that cut the country up into a great checkerboard. Soon every point in the country will be within 25 miles of a control station. This distance will be reduced in the near future to a maximum of 12 or 15 miles by the cutting up of the larger sections by additional triangulation. If local surveys are made and fitted into this control, then they can serve as control for further work in the same region. All of the triangulation stations are well marked and full descriptions of them are available, so that any surveyor, after securing the necessary data, can easily locate them on the ground. As stated above, the locations of these points are given by longitude and latitude, but if a system of plane coordinates were established they could be referred to the plane of coordinates and their positions given in the x and y coordinates of the system. It would then be an easy matter to make use of them as control for any local survey. METHOD OF USING CONTROL In case a general system of plane projection is adopted by a State or a part of a State, then the Coast and Geodetic Survey would have the coordinates of all of its triangulation stations within the area computed on that system and would make the results available for local use. Most of the triangulation stations have an azimuth mark that is visible from the ground at the station. The azimuth of this mark on the plane coordinate system would be furnished as a part of the data, and a traverse could be started from the station by turning off an angle from this mark to the first station of the traverse. Then, when the lengths of the traverse were measured together with the angles at the various stations, the whole could be computed directly on the plane with x and y coordinates for each of the stations. If the traverse ended on another control point, a check could be obtained on the angle measurements by means of the plane-coordinate azimuth on the azimuth mark at this end point and also a check on all measurements by seeing whether the x and y coordinates computed through the traverse agreed with the fixed x and y of the end station. The discrepancy due to the failure to check, if within allowable limits, could then be distributed through¬ out the traverse and the survey would thus be coordinated with the general control survev. o 4 KIND OF PROJECTION TO BE USED When the greatest extent of a State is in an east-and-west direc¬ tion with comparative narrow width in a north-and-south. direc¬ tion, we can make use of the Lambert conformal conic projection with two parallels held true to scale. Along these two standard parallels the scale would be exact; between them the scale would be too small and outside of them the scale would be too large. This gives a balance of scale over the projection and makes it possible to cover a greater extent of latitude. At each point the scale would be the same in all directions but only for infinitesimal distances. Along a given parallel the scale would be constant but it would vary from parallel to parallel. A table of these scale variations would be computed for every minute of latitude, so that these variations could be taken into consideration in any computations and full allowance made for them. The computations would thus be very accurate in all parts of the projection. When the greatest extent is in a north-and-south direction with limited width in an east-and-west direction, use can be made of the transverse Mercator projection. This is the well-known Mercator projection, but now related to a meridian in the same way as it is ordinarily related to the Equator. To bring about a balance of scale in this case, the scale is reduced a certain amount in the center of the projection thus providing for holding the scale along two small circles of the earth parallel to the meridian. The scale then is con¬ stant along any small circle parallel to the central meridian but varies with the distance from this meridian. A table of these scale factors, given for every 5,000 feet, is included with the table for the projection and can be used just as the one for the Lambert system, except that the variation depends upon the distance from the central meridian. These are the two most useful projections to be employed for plane coordinate systems, but others could be used in certain cases. The stereographic projection could be employed to advantage for a small circular area, but no actual State is of that shape. With this projection the scale is constant along small circles concentric with the center of the projection. The table of scale factors would thus be more troublesome to apply and hence would not be as useful as in the case of either of the other two types. All three of these projec¬ tions are conformal and hence the angles are true on the projection. This is important, since survey work makes much use of angles. It would be advisable therefore to make use of one of the three projections in all cases where plane coordinate systems are to be established. It can be seen that plane coordinate systems are special cases of general map coordinates. Any projection that is used for extensive maps could be used as the basis of a plane coordinate system. How¬ ever, the conformal projections are chosen because of their peculiar fitness for such work and because the definite scale discrepancies make it possible to carry out the computations with practically any degree of exactness. 5 SYSTEMS ALREADY ADOPTED Plane coordinate systems have already been adopted for the States of North Carolina and New Jersey and for Long Island, N.Y. Long Island was put on a separate system, since it is divided by water from adjacent land areas. It was possible with this limited area to secure a projection that has very small scale discrepancies. In North Carolina the Lambert projection was used with scale dis¬ crepancies no worse than 1 part in 6,000. The North Carolina State Highway and Public Works Commission have been using co¬ ordinates on this projection for some time, and they are very enthusi¬ astic in their praise of the method. The engineers of this depart¬ ment are making full use of the table of scale factors and are thus getting equal accuracy in all parts of the projection. The scale dis¬ crepancy of 1 part in 6,000 is no more troublesome when thus handled than one of 1 part in 20,000. In New Jersey the transverse Mercator projection was used since the greatest extent of the State is in a north-ancl-south direction. In actual practice the local 1 surveyor can use one system of coordinates as easily as the other. The computation to reduce latitude and longi¬ tude to x and y coordinates is different in the two methods, but the coordinates of the control points would be computed under the direc¬ tion of the Coast and Geodetic Survey and the local engineer would be furnished with x and y coordinates of the control stations together with grid azimuths at the stations. These data would be used in the same way in either case, except that the table of scale factors would run north and south in one case and east and west in the other. Illustrative examples of the use of the grid or coordinate systems together with the necessary tables would be furnished to any State adopting such a system. OTHER SYSTEMS IN CONTEMPLATION State systems for Maryland, Tennessee, and Delaware are under consideration and some study has been given to California and South Carolina. There is no doubt that such systems will 1 ultimately be adopted for all of the States. As soon as it becomes appreciated that such general systems can be devised there will be a general de¬ mand among engineers for a system in the region in which their work lies. USEFULNESS OF THE SYSTEMS The establishment of a plane coordinate system not only simplifies the use of control data, but it gives a permanent general grid for the whole State. County boundaries, township boundaries, property boundaries, intersections of roads and streets and any prominent features of a region can be accurately located with definite x and y coordinates. In the general system these plane coordinates can read¬ ily be transformed into latitude and longitude, and the point can thus be definitely located in the network of meridians and parallels that serve to locate points on the earth’s surface. If the marker at 3 0112 110789192