I want to take a series of latitude, longitude pair like and return Cartesian coordinates for it. HOWEVER, I don't want the origin to be at the center of the earth. Instead, I want it to be an arbitrarily defined point somewhere in the United States (all my coordinates are on the continental US). For example, if we defined Chicago, IL as 0,0 and we had an imaginary city that was 20 miles south and 40 miles east of Chicago, it would be at (40,-20).
I don't care about z/altitude.
I suggest to put up a custom coordinate system, centered on your place of interest.
If your data is all over the US, try an equirectangular projection. You need to know the lat/lon of your place, and the distance from the equator. The PROJ.4 projection string for Chicago would be:
+proj=eqc +lat_ts=41.836944 +lon_0=-87.684722 +x_0=0 +y_0=-4662266 +datum=NAD83 +units=m +no_defs
and the coordinate grid looks like this:
For smaller areas of study, a transverse mercator projection fits as well. You don't need the distance to the equator for that:
+proj=tmerc +lat_0=41.836944 +lon_0=-87.684722 +x_0=0 +y_0=0 +datum=NAD83 +units=m +no_defs
The y-axis is almost the same, but the x-axis cuts the West Coast at a different point:
Other popular projections, like Albers Equal Area, Lambert Conformal Conic or Equidistant Conic, have basically the same problem.
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It sounds like what you are searching for is a coordinate system in general, and in particular a projected coordinate system. ESRI has a good introduction to coordinate systems here, but I will attempt to summarize which one you should use and why. I would highly suggest reading through the above link if you are attempting a project that requires some sort of coordinate system, however.
