Question

Let’s say I have a small area mapped out using local planar coordinates in meters. e.g. A rectangular warehouse that’s 300m x 450m. I use some GPS device to find the WGS 84 lat/lon of one corner of the warehouse.

How can I project my plane coordinates onto the WGS 84 geoid to find the lat/lon values for the 3 other corners of the warehouse?

I understand this is a complicated problem since values vary on different parts of the earth. Do I need to deal with finding some local coordinate system first or can I somehow use the relationship between my known points to go direcly to WGS84?

I don’t have a lot of experience with this sort of problem so forgive me if parts of my question don’t make any sense.

Answer 1

Your question makes perfect sense, but the solution is far from trivial.

The solution depends on whether your local coordinate system is north-south aligned (relatively easy) or if completely arbitrary (prohibitively harder).

Your local system is accurately north-south aligned:

1. convert the known lat/lon coordinate to a cartesian coord (easting,northing).
2. determine the x and y differences between (1) and the local coordinate of that point.
3. apply this coord difference to to each of the 3 corners local coordinate.
4. convert (3) back to geographical lat/lon.

This works because both systems have the same unit scale (metric) and rotation (north-south), all that’s left is translation which is the coordinate difference.

Your local system is arbitrary, and not north-south aligned:

1. You need two more lat/lon known points. This is a fact, no way around it, otherwise rotation can’t be determined and no solution.
2. convert these known lat/lon coordinates to cartesian.
3. helmert transformation using the 3 known/unknown pairs to solve for the remaining unknown local coordinate(s). (Search for an online tool)
4. convert results back to geographical.

This works because the helmert transformation is determining both the translation and rotation between the two systems.

Converting between lat/lon and cartesian and back again is in itself not trivial, but necessary. Either do it accurately using a projection library like proj4, or approximately assuming a spherical earth, e.g. this answer.

Hope this hasn’t put you off!