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/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...