The term “elliptic”

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.

A partial answer, going from "easy" to "hard":

"Elliptic hyperboloid" and "Elliptic paraboloid"

• these are of course quadric surfaces with the cross sections implied by their respective names; one could rightly ask why they weren't "hyperbolic ellipsoids" and "parabolic ellipsoids", tho.

"Elliptic integral"

• The "second kind" studied by Legendre is precisely what turns up in deriving the arclength function of an ellipse. (Euler, Fagnano, and other mathematicians have certainly studied related integrals earlier, since they turn up in deriving the arclength functions of curves like the hyperbola and lemniscate.) Why Legendre generalized this to "integrals involving the square root of a cubic or quartic polynomial" is something I still need to look up.

"Elliptic functions"

• these turned up as inverses of the "first kind" integral. Confusingly, Legendre first termed his elliptic integrals as "elliptic functions", and it was not until the work of Abel and Jacobi (who had the insight to invert the integrals being studied by Legendre) that "elliptic function" was used for these objects, and "elliptic integral" became the accepted term for the ones studied by Legendre. The proof that they were the only doubly periodic functions (one of Jacobi's great contributions to the theory; see this) led to "doubly periodic" and "elliptic" being synonymous adjectives, at least in this context.

"Elliptic curves"

• most of these things can be parametrized by (hyper)elliptic functions, e.g. the cubic and quartic forms can be birationally transformed into the "Weierstrass form", which then admits a parametrization in terms of the Weierstrass elliptic functions. As reuns also wanted me to note, another connecting thread is that elliptic curves over $\mathbb C/L$ with $L$ an appropriate period lattice are intimately related to modular and elliptic functions (as was once conjectured by Taniyama and Shimura, and subsequently proven by Wiles and others).

"Elliptic modulus"

• It's one possible argument for an elliptic integral or an elliptic function. I had talked about this at some length here, so no need to repeat it for this answer.

Elliptic geometry

• Here's the odd one out. "ellipsis" (ἔλλειψις) means a "deficit" or "falling short" in the original Greek. In the conic context, this is related to the eccentricity "falling short" from unity (and a mathematically similar condition leads to the adjective "elliptic" for some second order linear PDEs); in the sense of non-Euclidean geometry, "elliptic geometry" "falls short" since it does not exhibit the parallel postulate (this is due to Klein, if memory serves), as opposed to Euclidean geometry where you only have one parallel, or hyperbolic ("excess") geometry where you can have infinitely many parallels.

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