GEOLOGICAL SURVEY CffiCULAR 57 

May 1949 

STATE COORDINATES AND POLYCONIC MAPS 

SOME NOTES ON THE STATE PLANE-COORDINATE SYSTEMS AND 

THEIR RELATION TO POLYCONIC QUADRANGLE MAPS 

Prepared by 

Topographic Division 

Research and Technical Control Branch 
UNITED STATES DEPARTMENT OF THE INTERIOR 

J. A. Krug, Secretary 

GEOLOGICAL SURVEY 

W. E. Wrather, Director 

WASHINGTON, D. C. 

Free on application to the Director, Geological Survey, Washington 25, D. C. 
STATE COORDINATES AND POLYCONIC MAPS 

SOME NOTES ON THE STATE PLANE-COORDINATE SYSTEMS AND 

THEm RELATION TO POLYCONIC QUADRANGLE MAPS 

Prepared by 

Topographic Division 

Research and Technical Control Branch 

Since the introduction of the State plane-coordinate systems on the Lambert and 

transverse Mercator projections in 1935, numerous questions have arisen among prospective 

users of these systems concerning their applicability in connection with Geological Survey 

maps on the polyconic projection. Some of the questions deal with the basic properties of 

the several projections; some question the legitimacy of combining, in a single map, data 

based on more than one projection; some concern the ease of matching adjoining sheets to 

form a map of a larger area. An attempt wi11 be made in this discussion to clarify the 

relationship between the projections and to answer some of the questions that have been 

propounded and some that are expected to recur from time to time. 

As it is impossible to represent a portion of a spherical or spheroida~ surface on a 

plane sheet without some distortion, many types of projection have been devised for mapping 

the earth's surface and at the same time keeping the resulting distortion within permissible 

limits. Each projection has its own advantages and disadvantages. No one system is superior 

to all others in every respect, and the choice of projection for a particular map must 

depend onthesize, shape, and location of the area to be mapped, the scale of the proposed 

map, and the use to which it is to be put. For a small area, many projections are available 

in which the distortion can be kept within such small limits that it is impossible to 

distinguish between them. As the area to be mapped increases in size, the distortions 

inherent in the various projections become increasingly significant, and the map maker is 

confronted with the necessityof choosing a projection that will best suit his own particular needs. When the area is too large to be represented conveniently on a single sheet at 

the map scale, it must be divided into smaller areas. It is desirable, then, that the projection be such as to permit the joining of adjacent sheets so that corresponding edge 

details will match properly. 
POLYCONIC PROJECTION 

Infinite number of cones tangent to spheroid 

(Each strip of infinitesimal width) 

LAMBERT PROJECTION 

Single cone cutting spheroid 

Area of compressi scale too small 

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Area of stretch in scale too large 

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Area of stretching, scale too large 

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TRANSVERSE MERCATOR PROJECTION 

Cylinder cutting spheroid 

FIGURE I. DIAGRAMS SHOWING DEVELOPMENT OF RESPECTIVE PROJECTIONS AND TYPE AND LOCATION OF PRINCIPAL DISTORTIONS. (ALL VERY HIGHLY EXAGGERATED) 
3 

Only three projections will be considered in this discussion: the polyconic projection, used for all quadrangle sheets of the Geological Survey; the Lambert conformal 

conic projection, used for many of the State plane-coordinate systems; and the transverse 

Mercator projection, used for the remainder of the State plane-coordinate systems. 

POLYCONIC PROJECTION 

The polyconic projection nas been generally acceptedas the best for a small, regularshaped area such as the standard quadrangle. In this projection, the earth's surface is 

projected onto an infinite number of cones, each tangent to the ellipsoid at its own 

parallel of latitude. The developed parallels are therefore true circles, each of a 

different radius, and each centered at a different point on the extension of the central 

meridian of the projection. The central meridian is a straight line; all other meridians 

are slightly concave toward it. Scale is true along the central meridian and along each 

parallel of latitude; at all other points, and in other directions, the scale is slightly 

exaggerated. Although the polyconic projection does not comply strictly with most of the 

desirable specifications for a map projection--it is not conformal, scale is not uniform, 

shapes and areas are not retained exactly--it comes closer to compliance with more of 

these specifications than any other available projection. The distortions are kept to a 

minimum in all parts of the map sheet, and, for small areas, the earth's surface may be 

represented with a greater fidelity on this projection than on any of the others. On a 

standard quadrangle sheet, the distortions will be so small that it is impossible to plot 

or measure them at map scale, so that, for practical purposes, a polyconic projection of a 

single quadrangle may be considered a true portrayal of that portion of the earth's surface. 

The principal weaknesses of the polyconic projection are two: it cannot be used on a large 

area without noticeable distortion; and, although adjacent sheets can be matched continuously in one direction or strip, two or more strips cannot be matched for any great 

distance without developing gaps along the abutting edges. The situation is somewhat 

analogous to that of an orange peel. If the peel is cut into a large number of tiny 

quadrangular pieces, each may be placed on a flat surface without much stretching; but it 

is impossible to reassemble a large number of pieces on a flat surface, except in a single 

strip. Similarly, a single large piece cannot be laid flat without stretchingor tearing it. 

LAMBERT PROJECTION 

The Lambert projection portrays a portion of the earth's surface on the developed 

surface of secant cone. Along the two parallels of latitude at which the cone cuts the 

spheroid, the scale is exact. In the area between these parallels, the map scale will be 

too small, and in the area outside these parallels, it will be too great. The mapped area 

may be extended without limit in the east-west direction, but is restricted within narrow 

limits in the north-south direction. Scale distortions increase rapidly as the distance 

increases beyond the two standard parallels. Therefore the use of this projection is 

limited to relatively narrow east-west bands and to States of that shape or States that 
N-S Strips E-W Strips 

POL YGONIC PROJECTION 

I \ I I 

LAMBERT PROJECTION TRANSVERSE MERCATOR PROJECTION 

FIGURE 2. DIAGRAMS SHOWING MATCHING OF QUADRANGLE SHEETS AND ZONES ON THE RESPECTIVE PROJECTIONS 
5 

can be cut into zones of that shape. The principal advantages of this projection are that 

the scale distortions are systematic and can be predetermined and that a map sheet of any 

portion of a zone can be matched perfectly with an adjoining sheet in the same zone. 

However, a map of one zone cannot be matched with a map of an adjoining zone. 

TRANSVERSE MERCATOR PROJECTION 

For the transverse Mercator projection, picture a portion of the earth's surface 

projected onto the developed surface of a tangent cylinder with its axis in the plane of 

the equator, so that the line of tangency is the central meridian of the area. Then, 

reduce the scale of the projection slightly so that it will be too small in the region of 

the central meridian. The scale will then be exact along two lines parallel to the central 

meridian and spaced symmetrically on both sides of it. In the entire region between these 

two parallel lines, the scale will be too small; in the regions beyond them, it will be 

too great. The mapped area may be extended without limit in the north-south direction, but 

is restricted to narrow limits in the east-west direction. Scale distortions increase 

rapidly as the distance increases beyond these parallel lines. This projection is limited 

to relatively narrow north-south bands, and to States and zones of that shape. Its advantages and disadvantages are essentially the same as those of the Lambert projection. 

COMPARISON OF PROJECTIONS 

The Lambert and transverse Mercator projections were selected as the bases of the 

State plane-coordinate systems because of their mathematical simplicity, because the scale 

factor at any point can be readily determined, and because the limiting scale factor can 

be controlled by proper choice of zone width. On either of these systems, there is a 

unique relationship between grid coordinates and geographic coordinates, and the interchange between coordinates is relatively easy. Both projections are conformal. As already 

explained, a polyconic projection for a single quadrangle map of 1 degr~e extent, or less, 

contains so little distortion that it may be considered a true representation of the 

earth's surface. Therefore, a polyconic projection of a quadrangle may be fitted into the 

appropriate portion of any State coordinate grid by a mere adjustment of the scale. 

For the mapping of a large area, such as an entire State or State zone, two methods 

are available when the scale is too large to permit representation of the entire area on a 

single sheet. Each section, quadrangle, or county may be mappedon the polyconic projection, 

with the State grid shown, without perceptible error. Then, when several sections are 

assembled to compile the larger map, gaps may be expected to develop along the abutting 

sheet edges. The other available method is to adopt the projection of the State grid and 

to map each smaller portion of the area as a part of the larger projection. With this 

method, all the sheets will fit together perfectly, but the internal distortion will be 

greater because of the variation in map scale. A map compiled by this method would not 

match with a similar map of an adjoining State or zone. 
6 

For a picture of the magnitude of the distortions involved, consider some of the 

State coordinate systems actually in use. These systems (with one exception) have been 

designed so that the greatest scale distortion factor at any point in the widest zones 

will not exceed 1 part in 10,000. For a map distance of 20 inches, this represents a map 

error of only 0.002 inch. The maximum cumulative error in a map distance of 100 inches 

(8ft. 4 in.) is only 0.01 inch. It may be stressed that these extreme errors can exist 

only in the widest of the zones adopted, and only at critical portion~ of these zones. 

In general, the distortions at other points in the zones will be considerably smaller. 

It follows, therefore, that map detail may be transferred readily between a polyconic 

quadrangle map and either of the State coordinate systems, in any part of a State zone, 

without perceptible error. Furthermore, the State coordinate grid may be shown on a 

polyconic quadrangle map without any qualms about the inconsistency of mixing projections. 

While differences will exist theoretically, they will be of infinitesimal magnitude and 

may be considered nonexistent. 

DISTORTIDNS 

At this point, mention should be made of the distortions that may be developed in a 

map through the physical instability of the medium on which it is drawn or printed. It is 

well known that paper will shrink or expand with changes in the surrounding atmospheric 

conditions, as well as from other causes. The change in dimensions of any sheet of paper 

will not necessarily be the same in all directions, and changes of shape, as well as of 

size, may result. These changes can be so great as to obliterate completely the theoretical distortions and scale changes in the map projection. It is not unusual for a 

dimensional change as great as 1/8 inch to occur in a single quadrangle map sheet on 

ordinary paper. Under such conditions, it is manifestly impossible to assemble perfectly 

a number of adjoining sheets, each with its own unpredictable paper distortion, regardless 

of the projection on which the maps were prepared. Many media are available in which 

distortions can be reduced or controlled, and such improved materials should always be 

used, when feasible, in the preparation of maps. Unless materials of exceptional stability are employed, it may be assumed that the theoretical differences between the 

projections referred to will always be less than the probable distortions of the paper, 

or other materials on which the map may be produced. 

PROCEDURE 

In conclusion, the following procedure is offered as a simple, yet accurate, method 

of transferring topographic detail from a polyconic quadrangle map to a State coordinate 

grid: 

1. Construct a rectangular grid to the desired scale, to represent the State grid. 

2. Compute the plane coordinates, on the State system, of the projection corners of 

the quadrangle, and plot these corners on the grid. 
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3· Connect the contiguous plotted corners, if desired, to form a graticule of meridians and parallels. 

4. By the usual drafting methods, adjust the scale of the quadrangle map to that of 

the graticule and transfer the detail as desired. 

Conversely, to show the State grid on a quadrangle map, compute the geographic coordinates of the appropriate grid intersections, plot the computed points on the quadrangle map, and connect the appropriate plotted points by straight lines. 

INT.-DUP. SEC., WASH., D.C. 911'J6