Print
Decreasing distortion is a constant and elusive goal in map projection. It can be done in several ways. Most involve reducing the distance between the map projection surface and the ellipsoidal surface. One way this is done is to move the mapping surface from tangency with the ellipsoid and make it actually cut through it. This strategy produces what is known as a secant projection. A secant projection is one way to shrink the distance between the map projection surface and the ellipsoid. Thereby, the area where distortion is in an acceptable range on the map can be effectively increased. The illustrations here are intended to show how the tangent case distorts a larger area than does the secant case. The tangent on the top shows a flat projection plane touching the Earth at one point. This is the approach used in polar protections (North Pole, South Pole). The area in orange has high distortion. The low distortion, the area in the light beige, is much smaller. However, if the projection plane cuts the Earth rather than just touching it at one point, there's a larger area of relatively low distortion. The less the distance between the plane and the Earth, the less the distortion. (The areas on the Earth closest to the plane have less distortion than the areas farther away from the plane.) The secant case has lower distortion over a larger area than does the tangent case. This is why State Plane coordinate systems in the United States use secant projections. In the case of Lambert projection, there are two parallels of latitude where the mapping plane cuts the Earth. In the case of Transverse Mercator, there are two approximately north-south lines that are not meridian of longitude. In both cases, these are lines of exact scale.
Both cones and cylinders have an advantage over a flat map projection plane. They are curved in one direction and can be designed to follow the curvature of the area to be mapped in that direction. Also, if a large portion of the ellipsoid is to be mapped, several cones or several cylinders may be used together in the same system to further limit distortion. In that case, each cone or cylinder defines a zone in a larger coverage. This is the approach used in State Plane Coordinate systems. Generally, one cone or one cylinder is not sufficient to cover an entire state (though there are exceptions). Usually a collection of cones or cylinders, one for each zone, is used. That's the approach that was used in the original design of most State Plane coordinate systems in this country.