# “full coverage” in hyperspherical space

I'm working on a computer algorithm that considers the relationships between data points in a theoretical n-dimensional space.

I am "looking" from the origin in all directions in a programmatic way that will help me cull out some of the unneeded data points. Each time I do this I will have an angle range that has been "covered". I will know that I am done when all possible angles have been covered.

What I need to know is how to determine if every direction has been covered. Intuitively I think that for each subdivision of one angle I have to have covered all subdivisions of the other possible angles but I need to know for certain before I start writing implementation for this program.

Edit: I noticed an ambiguity in my question after the first answer. I'm looking at this feature space using the description of the hyperspherical coordinate system as described on Wikipedia. It's similar to the spherical coordinate system but for each extra dimension you add another angle with a range [0, $\pi$). It seemed like the best option for my purposes.

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Your notion of angle is not defined in $n$-dimensional space. Consider data that are in $3$-space. How are you going to point your camera from the origin (literally look) in every possible direction? Your camera's aperture will determine an angle" which is a percentage of the sphere. You can move your eyes around and mentally stitch together an image of the surrounding world, or you can use a series of photos that can be stitched together to show what is known as a 360 degree view. What you want to do is to cover the entire observational sphere with disks. Probably the easiest conceptual way to do so is to make observations in the directions of an $n$-cube. If so you need $2^n$ observations. More economically, you can use an $n$-simplex, but still you need the aperture to be wide enough to view beyond a certain radius.