I want to take a photo and use the stored GPS coordinates and the orientation of the phone to do some calculations. How can I access that information?
Alternatively if there is an API to do that I could make my own app.
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Of course! You can use Metadata viewer to see the GPS coordinates and other information of your photos. Android has Handful of APIs for anything you want. If you a developer you can make use of that. Here is all you need.
This is one of those problems that I believe requires a documented standard: the same reason every platform trivially accesses the GPS location of a photo is because of EXIF and it's standardized tags.
EXIF is not user-extensible, and side car files are too namespace-y (proprietary).
Unless I'm wrong about there not being a standard place for Euler angles or even the quaternion that establishes device attitude.