State Plane Coordinate Systems are built on *map projections*. Map projection means representing a portion of the actual Earth on a plane. Done for hundreds of years to create paper maps, it continues, but map projection today is most often really a mathematical procedure done in a computer. Nevertheless, even in an electronic world, it cannot be done without distortion.

Orange Peel

The problem is often illustrated by trying to flatten part of an orange peel. The orange peel stands in for the surface of the earth. A small part can be pushed flat without much noticeable deformation.

Distortion

However, when the portion of the orange peel gets larger, problems appear. Suppose the whole orange peel is involved; as the center is pushed down, the edges tear or stretch, or both. And if the peel gets even bigger, the tearing gets more severe. So, if a map is drawn on the orange before it is peeled, the map gets distorted in unpredictable ways when it is flattened. And it is difficult to relate a point on one torn piece with a point on another in any meaningful way.

These are the problems that a map projection needs to solve to be useful. The first problem is the surface of an ellipsoid, like the orange peel, is *non-developable*. In other words, flattening it inevitably leads to distortion. So, a useful map projection ought to start with a surface that is *developable*, a surface that may be flattened without all that unpredictable deformation. It happens that a paper cone or cylinder both illustrate this idea nicely. They are illustrations only, models for thinking about the issues involved. If a right circular cone is cut perpendicularly from the base to its apex, the cone can then be made completely flat without trouble. The same may be said of a cylinder cut perpendicularly from base to base. A developable surface can be split up an element and flattened as you see in the two illustrated developable surfaces, a cylinder and a cone shown in the figure above. The cylinder is used in Mercator and Transverse Mercator map projections, and the cone is used in Lambert Conic map projections. These include the two that are most common in State Plate coordinate systems. There is one oblique portion of State Planes along the panhandle of Alaska. The right circular cone, if cut up one of its elements that is perpendicular from the base to the apex, can be completely flattened without trouble. And the same can be said of a cylinder cut up perpendicular from base to base.

Or one could use the simplest case, a surface that is already developed. A flat piece of paper is an example. If the center of a flat plane is brought tangent to the earth, a portion of the planet can be mapped on it, that is, it can be projected directly onto the flat plane. In fact, this is the typical method for establishing an independent *local coordinate system*. These simple Cartesian systems are convenient and satisfy the needs of small projects. The method of projection, onto a simple flat plane, is based on the idea that a small section of the earth, as with a small section of the orange mentioned previously, conforms so nearly to a plane that distortion on such a system is negligible. Subsequently, local tangent planes have been long used. Such systems demand little if any manipulation of the field observations, and the approach has merit as long as the extent of the work is small. It's convenient. It's easy. It can satisfy small projects fairly well, but the larger each of the planes grows, the more untenable it becomes. Another difficulty arises when the local coordinate systems are brought together—as you see in this image—they simply don't match. There are overlaps, there are gaps. The scale of each of the local systems are not the same, nor is their orientation. In other words, a local tangent coordinate system can be useful in a very limited extent, as long as one stays within the zone that it intends to cover. When you bring in another coordinate system from another direction, trouble ensues. So as the area being mapped grows, the reduction of observations becomes more complicated and must take account of the actual shape of the earth. This usually involves the ellipsoid, the geoid, and geographical coordinates, latitude and longitude. At that point, surveyors and engineers rely on map projections to mitigate the situation and limit the now troublesome distortion. This is one of the reasons State Plane coordinate systems were devised. A well-designed map projection can offer the convenience of working in plane Cartesian coordinates and while not eliminating the inevitable distortion keep it at manageable levels. They cover an entire state or a fairly good portion of the state with a zone or zones.