## District of Columbia Boundary Stones: Theoretical vs. Actual Locations

The small blue markers on the map show the theoretical positions of DC Boundary Stones as computed by a simpleminded **locally-flat-Earth** model. The large red markers depict the actual GPS coordinates measured for many of the stones. See DC Boundary Stones for a map without the theoretical location markers.
Zoom in on a stone and change to other display modes *(e.g., "Hybrid")* to see the displacement of that stone from its theoretical site. With the exception of stone NW4 *(which has a theoretical location in or near the Potomac River channel)* and stone SW2 *(which has been replaced by a marker in a different location)* all of the theoretical locations are within a few tens of meters of the measured actual stone coordinates.

These theoretical stone locations were computed as follows:

- take the GPS coordinates
*(in DC_Boundary_Stones_waypoints.xml)* for the four corner stones *(SOUTH, WEST, NORTH, EAST)*
- define lat_center and lng_center as the average latitude and longitude of the corner stones
- define offsets:
- lat_offset = +10 for NORTH, +9 for NE1 and NW9, ... 0 for EAST and WEST, ... -10 for SOUTH
- lng_offset = +10 for EAST, ... -10 for WEST

- define C = 2 * (pi) * (Radius of Earth) / (360 degrees) = approximately 69.1 miles/degree
- The theoretical latitude and longitude of a stone is then:
- lat_theory = lat_center + lat_offset / (C * (sqrt(2)))
- lng_theory = lng_center + lng_offset / (C * (sqrt(2)) * cos(lat_center))

- The theoretical coordinates are stored in DC_Boundary_Stones_theoretical.xml

This is an oversimplified model which replaces an assumed-spherical Earth with a plane tangent to that sphere at the center point of the stones. The constant C converts miles to degrees. The latitude and longitude offsets are divided by sqrt(2) factors because the stones are located one mile apart along 45° lines relative to north-south and east-west. The cosine of the center latitude is a factor that roughly corrects for the convergence of longitude lines. The various approximations involved in this theoretical model are probably comparable in size to the GPS errors in coordinate measurements and may be significantly larger than the actual errors in DC Boundary Stone placement.

*The information on this page was collected, organized, analyzed, and presented by Mark Zimmermann, with considerable help from countless other people. For excellent historical and locational information on the stones see Mark Kennedy's Boundary Stones of the District of Columbia site, Tom Howder's Washington DC Boundary Stones page, and the Daughters of the American Revolution map and discussion; cf. DC Boundary Stones, DC Metro Area Trails and Google Map Experiments in the **^zhurnal*, or send email to z (at) his (dot) com ... last update 3 Sep 2006