The question asks how to convert local coordinates into
geocentric coordinates. As an example, What would be the
geocentric coordinates of a point displaced 250 meters west and
250 meters south of -108.619987 degrees longitude, 36.234600
degrees latitude (at sea level)? (For the reason why only six
decimal places are used here, please see
http://gis.stackexchange.com/a/8674.) This encompasses two
geometrical questions: What are the geocentric coordinates
corresponding to a given location given in terms of latitude and
longitude (at a nominal height of zero)? What rotation will
bring the local frame (in which x points east, y points north,
and z points up) into the geocentric frame (in which x points
towards the Prime Meridian at the Equator, z points at the North
Pole, and x-y-z form a positively oriented orthogonal coordinate
system)? Let's solve these geometrically. Because the math for
the first one has been done elsewhere on this site, the new part
is the second. To begin with, consider the geocentric fra