Name or Adjective for Ellipse with very different (or very similar) scales

geometry conic-sections ellipsoids

I am aware of no special name. But is a name was to be given, I would say very plainly "with high or low" eccentricity due to formula

$$e=\sqrt{1-\frac{b^2}{a^2}} \ \ \text{on a scale} \ \ 0 \le e < 1$$

(the minimum $0$ for a circle, $1$ cannot be attained because we would have then a parabola).

Besides, for the 3D case of an ellipsoid with semiaxes

$$a \ge b \ge c,$$

I would like to attract your attention on two specific names for the case of equality:

  • if $a = b > c$: "Prolate spheroid" (in fact ellipsoid) (rugby ball),

  • if $a > b = c$: "Oblate spheroid" (ideal Cinderella's pumpkin...).

The prolate spheroidals, in particular, are connected with various applications:

  • a 3D coordinates' system

  • orthogonal functions. See for example this article explaining how that these functions are at the same time orthogonal in $L^2(-1,1)$ and in $L^2(-\infty,+\infty)$...

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