Ah, but in a circle the circumference is always equidistant to the centre, which is never true of an ellipse.
I suppose there is only one circle in the world.
It's never true of an ellipse that isn't a circle. i.e., this is a--ahem--circular argument.
Why do circles need to be ellipses anyway, why can't they be absolutely different? If they were absolutely different, then circles would be purely ideal, and yet an organizing principle (or as the say in Greek, an ἀρχιτεκτονική, from when we receive the word architecture). The only way to understand this, ontologically, is if we take the world to be in a constant tension with the "earth," as Heidegger puts it (cf. The Origin of the Work of Art), the thing in which the "rifts," which is the actual discourse of idealism, come about.
You know, I thought about it for a moment, and I don't think the visual circle is even universal. The schema of the circle may be, but the circle itself never appears. See this article below[0].
https://en.wikipedia.org/wiki/Ellipse
Functionally, they might not be the same if you're programming them. For example, if you have a circle class with members detailing its center and its radius, it might be more efficient to draw an instance of it than an instance of an ellipse class that has two foci members that just happen to have the same values.
https://plato.stanford.edu/entries/goedel/goedel-phenomenolo...
Also, that definition of a circle as an ellipse via locus points (which you mentioned) still requires defining ellipses by an imaginary, ideal circle--even if transcendental (and I've mentioned elsewhere, the circle never appears). The circle that appears is always already an ellipse, empirically, so in pure mathematical terms as I've said the ellipse is actually absolutely different from the circle, which always exceeds it. In order for the circle and ellipse to be set in relation one must be forced to be analytically composed under the other.
Edit: Its clear that phenomenology was the most important philosophical influence on the modern theory of computation, but its curious to me that those that study pure math and physics haven't, for the most part, realized that yet, even as they employ computers to do so much of their work.
