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How can I get information on height and X,Y coordinates using Google Earth data for a given terrain?

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You can get the elevation of individual points in Google Earth by hovering your mouse pointer over the location and looking at the status bar at the bottom, which should indicate the altitude at that point.

You can get an "elevation profile" graph in Google Earth Pro by drawing a line right-clicking on it and selecting "show elevation profile".

To get elevations programatically, you can use the Google Maps Elevation API to request elevation for a point or a list of points. There is a limit to how many points you can request for free, and the terms of service indicate that you should only do this for use of the data on a Google Maps API page or similar.

If you want a height/elevation grid for a whole area, then you should look for a Terrain dataset (often referred to as a Digital Elevation Model or DEM), since extracting bulk data like that from Google Earth and Maps is 1.) technically difficult and 2.) against the terms of service. For free DEM data, you can start with the SRTM dataset (15m resolution?).

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