TITLE: The term “elliptic”
QUESTION [23 upvotes]: There are many things which are called “elliptic” in various branches of mathematics:

Elliptic curves
Elliptic functions
Elliptic geometry
Elliptic hyperboloid
Elliptic integral
Elliptic modulus
Elliptic paraboloid
Elliptic partial differential equation

I take it that most of them relate to ellipses in one way or another, but the relation is often unclear to me. Does anyone know the etymology of these various terms; i.e. which was derived from which, and why? For example, this answer suggests that elliptic curves derive from elliptic integrals, since the first known elliptic curves were found while studying elliptic integrals. How about the rest?
I don't expect answers to exhaustively address all terms, but hope that eventually enough answers will contribute to form a cohesive picture of the whole term.

REPLY [2 votes]: When mathematical equations/ models distinctly trifurcate on basis of discriminant /characteristic sign of its principal arguments (< 0, 0, > 0) in a group ( i.e., when there is a transformation possible between positive and negative sign types ) we have three classes/types (elliptic, parabolic, hyperbolic ) which need not generally be associated with an ellipse curve.
Regarding the partitioning nomenclature of constant Gauss curvature surfaces ( K = +1 and -1) labelling/appellation left me never so comfortable. Hope it is appropriate to mention it here.
I think we should call them as
1)Elliptic sphere
2)Riemann sphere
3)Hyperbolic sphere
and
4)Elliptic pseudosphere
5)Central or Beltrami pseudosphere
6)Hyperbolic pseudosphere
In the following article ( I have  no access at present)
Felix Klein," Vorlesungen über nicht-euklidische Geometrie" 3rd ed. (Berlin, 1928). German. Crelle's Journal.
iirc the names are like: 1)spindles 2)sphere 3) like Cheese "tires" 4) Conoid 5)Pseudosphere 6)Rings
Const_Gauss_Curvtr_names