Local Map Projections and Coordinate Systems

The Earth exists in three-dimensions but, other than globes, most representations of the earth are two dimensional. The process of converting a three-dimensional space to a two-dimensional map is called projection.

While there are hundreds of different projections used around the world for local areas, three collections of projections are commonly used in the United States:

Planar Coordinates

The process of projection involves distorition of area, distance, shape, and angles. With projections of large areas like the world, these distortions are significant, and dramatic compromises are required depending on the purpose of the map and the aesthetic objectives.

However, on smaller areas of the surface of the earth, our lived experience is of the earth as a flat plane. Accordingly, projections can be designed that minimize distortions and reflect that experience. These projections define a coordinate system that represents locations as X and Y coordinates, which are commonly referred to a eastings and northings. This provides the ability to accurately find specific locations and measure distances, such as when surveying property lines or engaging in military activity.

X/Y Coordinate System

The State Plane Coordinate System

The State Plane Coordinate System (SPCS) was developed in the 1930s by the US Coast and Geodetic Survey (predecessor of the National Ocean Service) to enable surveyors, mappers, and engineers to connect their land or engineering surveys to a common reference system. You will commonly see government data distributed with coordinates in SPCS.

There are 124 different zones in different parts of the US, usually aligned to county borders as a convenience.

State Plane Coordinate System Zones

Zone Projections

A State Plane Coordinate System zone is based on either a Lambert conic or transverse Mercator projection, depending on the shape of the zone.

The zones are specifically chosen to be small enough that distortion of distance can be greatly minimized to 1 part in 10,000 throughout the zone. This means you can measure a distance of two miles (10,760 feet) within around a foot of accuracy.

Zones that are wider than they are tall use a coordinate system based on a Lambert conic projections. Examples include the zones in Iowa, Pennsylvania, and California.

Lambert Conformal Conic Projection with a Single Secant Line

Zones that are taller than they are wide use a coordinate system based on transverse Mercator projections that are conformal. Examples include Illinois, New Mexico, and Mississippi.

Transverse Mercator With a Central Meridian

State Plane Coordinates

The SPCS projections turn the surface of the earth into a flat plane (hence, the name).

Dimension in a State Plane Coordinate System Zone

State Plane Coordinate System Variants

When selecting a projection to use with a particular state plane zone, you will be presented in GIS software with a number of different variants based on geographic coordinate systems (datums) that have been updated over the past century to reflect new geodetic measurement technology:

Further confusing matters, State Plane Coordinate Systems can express their coordinates in different length units as determined by the policies of the different state governments:

The choice of which of combination of datum and length unit to use should be based on the version being used on your project and / or in your organization. If there are no constraints on your choices, the safest option is to use meters with the most recent NAD 83 (2011) datum.

The following is data that may be helpful when working with the State Plane Coordinate System:

Example Calculations Using SPCS Northings and Eastings

As an example of calculations with SPCS coordinates, we use the Farmingdale, NY train station:

Using the NOAA Coordinate Convertor

This means that the Farmingdale train station is 63,337 meters north and 47,159 meters (347,159 easting - 300,000 false easting) east of Belmar Beach, NJ, which is the approximate location of the central meridian and latitude of origin.

Northing and Easting From the New York Long Island SPCS Zone Origin Point

Universal Transverse Mercator

Universal Transverse Mercator (UTM) is a coordinate system devised by the US Army Corps of Engineeers in the 1940s to make land navigation easier, and UTM is still used as the basis for the military's Military Grid Reference System. Unlike the State Plane Coordinate System which only covers the US, UTM covers the whole world.

UTM divides the world into 60 six-degree-wide strips, with separate zones for the northern and southern hemispheres, resulting in 120 total zones.

Universal Transverse Mercator Zones

UTM Zone Projection Parameters

Each zone is converted to a plane using a transverse Mercator projection. The use of this conformal projection reduces distortions of distance, while making distance calculations easier when conducting military operations within fairly small areas. The accuracy of UTM is 1 part in 2,500, or around two feet per mile.

Locations in UTM must contain the followign parts:

Dimension in a Northern UTM Zone
Dimension in a Southern UTM Zone

UTM Example Calculations

As an example of UTM coordinates, we use the Farmingdale, NY train station:

As with the State Plane Coordinate System, the UTM system has undergone modifications since its advent in the early 20th century. Early versions of UTM used by the US Army in mid 20th century were based on the Clarke Ellipsoid of 1866 or the International Ellipsoid (1910). With the advent of GPS (which uses WGS 84 based on the GRS 80 ellipsoid), contemporary UTM coordinates in the US are consistently based on WGS 84.

The Military Grid Reference System and the US National Grid

The Military Grid Reference System (MGRS) is a grid coordinate system that is based on UTM zones, but which specifies locations as grid coordinates rather than distance offsets from a zone origins.

An essentially identical system called the US National Grid (USNG) was introduced in 2005 by the Department of Homeland Security as a domestic rebranding of MGRS. MGRS and USNG provide the following advantages:

A MGRS location is then specified in the following parts:

MGRS Example Calculations

Again, using the example of the Farmingdale, NY train station, Google Maps gives an estimated WGS84 latitude and longitude of 40.735677, -73.441718. Using a convertor like this one from the National Geodetic Survey, gives a USNG location of:

18T XL 31578 10583

An interactive web map of MGRZ zones is provided by MappingSupport.com.

Calculating Euclidian Distance in Planar Coordinates

Since SPCS and UTM coordinates are given in ground distance, you can use the Pythagorean theorem to calculate distances. This is called Euclidian distance after the ancient Greek mathematician Euclid (ca 300 BCE):

A2 + B2 = C2

Distance = √(X2 - X2)2 + (Y1 - Y2)2

Euclidian Distance In A Planar Coordinate System Using the Pythagorean Theorem

For example: The Bethpage Railroad Station has WGS 84 coordinates of 40.742931, -73.483848.

Lat1 = 40.735677
Lon1 = -73.441718

Lat2 = 40.742931
Lon2 = -73.483848

North1 = 63337 meters
East1 = 347159 meters

North2 = 64120 meters
East2 = 343594 meters

X = East2 - East1 = 347159 - 343594 = 3565 meters
Y = North2 - North1 = 64120 - 63337 = 783 meters

Distance = SQRT(X^2 + Y^2) = SQRT(12709225 + 613089) = 3648 meters

3648 meters / 1609.34 meters/mile = 2.27 miles

We can do the same type of calculation using UTM eastings and northings, with results that are only about three one-hundredths of a percent different:

Lat1 = 40.735677
Lon1 = -73.441718

Lat2 = 40.742931
Lon2 = -73.483848

North1 = 4510583 meters (4510 km north of the equator)
East1 = 631578 meters (632 km)

North2 = 4511326 meters
East2 = 628007 meters

X = East2 - East1 = 628007 - 631578 = -3571 meters
Y = North2 - North1 = 4511326 - 4510583 = 743 meters

Distance = SQRT(X^2 + Y^2) = SQRT(12752041 + 552049) = 3647 meters

Miles = 3647 meters / 1609.34 meters/mile = 2.27 miles

% Difference = (3648 / 3647) - 1 = 0.0002741 = 0.027%

Likewise, if the points are in the same MGRS square, you can also use the grid X/Y locations:

Lat1 = 40.735677
Lon1 = -73.441718

Lat2 = 40.742931
Lon2 = -73.483848

MGRS1 = 18T XL 31578 10583

MGRS2 = 18 T XL 28010 11114

X = 31578 - 28010 = 3568 meters
Y = 11114 - 10583 = 531 meters

Distance = SQRT(X^2 + Y^2) = SQRT(12730624 + 281961) = 3607 meters

Miles = 3648 meters / 1609.34 meters/mile = 2.24 miles

% Difference = (3648 / 3607) - 1 = 0.0113 = 1.1%

Using the Google Maps distance calculation tool, we get an almost identical distance of 2.25 miles

Google Maps Measure Distance Tool

Manhattan Distance and Network Distance

Unless you are a bird or flying in a helicopter directly from one point to another, Euclidian distance will likely not reflect the actual distance you would need to travel to walk around buildings or navigate through a road network.

One way to estimate this distance is to use Manhattan distance, which is the sum of the east/west and north/south distances between two points. If you were walking blocks in Manhattan's street grid, you usually have to walk east/west, then north/south.

You can calculate Manhattan distances using SPCS northings and eastings by adding the absolute value of the difference of the northings to the absolute value of the distance between the easting:

North1 = 63337 meters
East1 = 347159 meters

North2 = 64120 meters
East2 = 343594 meters

EastDistance = |East2 - East1| = |343594 - 347159| = 3565 meters
NorthDistance = |North2 - North1| = |64120 - 63337| = 783 meters

Distance = EastDistance + NorthDistance = 3565 + 783 = 4348 meters

Miles = 4348  meters / 1609.34 meters/mile = 2.7 miles

Difference = (2.7 / 2.27) - 1 = 19%

Google Maps uses a model of the street network to more-accurately calculate driving distance. This network distance in this example is 3.1 miles:

Driving Distance in Google Maps

The US Public Land Survey System

The ceding of vast land areas west of the Mississippi River to the US by the British following the Revolutionary War resulted in a demand for a partitioning scheme that could organize Euro-American settlement and private ownership. The Land Ordinance of 1785 established the US Public Land Survey System that provided a surveying methodology for clearly identifying parcels of land.

The PLSS is based around a set of initial points, of which 35 total were eventually chosen across the western and southern United States:

The Public Land Survey System (USGS)

A specific parcel is then specified with the following convention IN REVERSE ORDER:

While archaic and fraught with idiosyncrasies that reflect both the technical and political limitations of its era, the PLSS is still the basis for property records in some parts of the western US. Similar surveying systems also persist as part of property records even in locations not covered by the PLSS. Therefore, the PLSS still remains significant two and a half centuries after its creation.

Benchmarks

As demonstrated by the PLSS, surveying has traditionally used designated locations as reference points for marking locations in the surrounding area. Those locations are usually marked with survey marks, also referred to as benchmarks, monuments, control points, or geodetic marks.

Vertical Control Point on the University of Denver Campus

While the widespread availability of high-precision GPS receivers has made benchmarks less immediately useful for everyday surveying activity in most areas, benchmarks are still sometimes used as reference points with differential GPS as precisely-known locations that can be used to determine the atmospheric distortions in the GPS signal in a specific area, and transmit those offsets to improve the accuracy of professional-grade surveying GPS equipment used in the area.

State and local governments also maintain their own sets of benchmarks for determining property lines and precise locations during infrastructure construction and renewal.

The National Geodetic Survey Data Explorer is a web map that can be used to locate benchmarks in a given area, as well as data about that benchmark. Note that you must expand the Map Layers menu and select Find Mark to search in a given area. Also note that the database includes marks that have been destroyed.

National Geodetic Survey Data Explorer

The database lists multiple types of benchmarks:

Cadastral Maps

In the United States and most developed countries, public and private property ownership is mediated by the government. Extensive land records on individual parcels of land are usually maintained by county governments. These public records include (but are not limited to):

A set of parcel land records and an associated map that specifically defines the geographic boundaries of the parcels in those records is called a cadastre. While cadastres have traditionally been maintained on paper maps and books, many jurisdictions now use property information systems based on GIS technology to maintain cadastral data and make it available to the parties that need that information.

Most counties encompassing major real estate markets now maintain online interfaces for accessing cadastral data that include web maps. These web sites make it easy for developers, realtors, investors, lawyers, potential buyers, and other members of the public to access information needed for commercial real estate activities.

You can usually find the property information system web site for a city by doing a google search on the county name and parcel information. For example the figure below shows the property information system web site for Denver:

Denver Maps - Real Property