Not sure I follow the physics analogy though. A unit ball is a specific case of an ellipse where A is the identity matrix. Perhaps the entries of A would be the atoms in this case as they uniquely shape it?
A unit ball along with the possibility of doing linear transformations on the unit ball are dependencies for constructing an ellipse.
:)
The physics metaphor is meant to communicate that you can have different materials (like aluminum or air), depending on the combinations or parameters used for a particular ellipsis case. And the physics or dynamics of ellipses would be the study of how the combinations on the dependencies (that is, the atomic elements) create different ellipsis shapes and properties.
But more probably, you would consider the sphere part to be essentially your basis vectors and not terribly informative. All the information is in the matrix. The only important data is the extent along each axis, not the fact that if you set all extents to 1 it happens to be a unit sphere
Maybe this whole structure, a set equipped with some operations, has distinguishable features. Or maybe it can perform as underlying machinery for practical endeavors.
https://en.wikipedia.org/wiki/Ellipse?useskin=vector#Definit...
Or for the n-dimensional generalization: https://en.wikipedia.org/wiki/N-ellipse?useskin=vector
So in that sense an ellipse is a function `ellipse(p_1, ..., p_n, d)`. But given the foci you can get the vectors and vice versa. So one representation isn't necessarily 'more fundamental' than another. They just have different (equivalent) perspectives that lend more easily/naturally to certain problem contexts.
(for example, the set of points definition is a lot less directly useful than the matrix A if you're trying to do linear algebra)
The unit ball isn't really an inherent "identity element" for ellipses. It's more of a special case where the foci are equally spaced (eccentricity is zero).
If you want to go that route, the eccentricity is a more fundamental descriptor of all conic sections, of which ellipses and circles are just two. In fact, all conic sections can be described as `conic_section(p_1, ..., p_n, eccentricity)`. So maybe you'd argue that's more "fundamental" because it's more general.
https://en.wikipedia.org/wiki/Eccentricity_(mathematics)?use...