I'm working on sphere packings. When I write, I'm confused with basic definitions. I'm hesitating between the terms "sphere", "ball" or "oriented sphere".

For example, on the wikipedia page of [circle packing theorem][1]

> A circle packing is a connected collection of circles whose interiors are disjoint.

But rigorously, a circle (1-sphere) is the boundary of a disk (2-ball). A sphere has no interior, a ball has. The interior is important for the definition of packings.

The problem becomes serious if I want to include hyperplanes as generalised spheres, or if I want to use the usual exterior as the interior (negatively curved balls)

I also read in literature definitions like "a sphere packing is a collection of balls", 
I prefer this one, but why not call it a "ball packing" directly as [here][2]?
I tend to use "ball packing", but since almost every others is using "sphere packing", I'm wondering if I missed something important by calling these objects "balls". (For example, google with "Apollonian ball packing" returns no result.)

Another solution is "oriented sphere", as in [Lie sphere geometry][3]. Then my question is, what's the difference (in practice) between a sphere with an orientation, and a ball with an interior?


  [1]: http://en.wikipedia.org/wiki/Circle_packing_theorem
  [2]: http://intlpress.com/_newsite/site/pub/files/_fulltext/journals/mrl/1996/0003/0001/MRL-1996-0003-0001-A-005.pdf
  [3]: http://en.wikipedia.org/wiki/Lie_sphere_geometry