> So, our ellipse E is the set of all points y such that y = Ax where x is in the unit ball [and A is a transformation matrix].
So an ellipse is only something that exists following a transformation of a unit ball? So, these "unit balls" are the elemental atoms of this ellipsis physics? Technically speaking at least.
Every ellipse can be encoded by the matrix A (and the geometric concept is generalized to arbitrary dimensions).
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.
So I think the answer to your question is ‘yes*’. In the sense that the ellipse is a function of the unit ball && the matrix A. In this sense you could say that all ellipsoids are linearly-deformed spheres.
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
Then, could we say that the general or abstract form of the ellipse is a set of vectors? And with that set, there is an identity element, namely the unit ball. And that the identity is what produces the uninteresting basis vectors. Maybe what makes the vector space (that is, the set of vectors) interesting is equipped linear operators on all the vector space's elements?
Maybe this whole structure, a set equipped with some operations, has distinguishable features. Or maybe it can perform as underlying machinery for practical endeavors.
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.
The ellipse can be also encoded by just the lengths along each axis and then by a rotation in R^n (which is just a unitary matrix multiplication). So in essence, for the problem in the original post there are three pieces of information needed: a translation, a rotation, and the deformations of the unit ball along the axes.
So an ellipse is only something that exists following a transformation of a unit ball? So, these "unit balls" are the elemental atoms of this ellipsis physics? Technically speaking at least.