The term “Elliptic Curve” in the context of mathematics may seem like a misnomer since the curve described by an elliptic curve equation doesn’t resemble an ellipse. The naming convention, however, has historical roots in the study of elliptic integrals. Here’s a detailed explanation:

1. Elliptic Integrals:

   • Elliptic integrals first appeared in studies related to calculating the arc length of an ellipse.
   • These integrals were utilized in the 17th and 18th centuries to express the arclength of an ellipse and were found to be non-elementary and non-trivial.

2. From Elliptic Integrals to Elliptic Functions:

   • In an effort to understand these integrals more deeply, mathematicians began to study the inverse functions of elliptic integrals, which became known as elliptic functions.
   • Elliptic functions, like Jacobi’s elliptic functions, were identified and studied by mathematicians such as Carl Gustav Jacobi.

3. Development of Elliptic Curves:

   • In studying the properties of elliptic functions, the connection to the study of certain cubic and quartic curves in the complex plane was established.
   • These curves are given by cubic equations like (y^2 = x^3 + ax + b), known today as elliptic curves.
   • The link to the original elliptic integrals comes through the study of the functions' properties and the integrals performed along the curves.

4. No Ellipse in the Curve:

   • The cubic equation used for elliptic curves does not generally describe an ellipse, and the confusion often arises from this discrepancy.
   • Despite the fact that the curve itself is not an ellipse, the terminology persists because of the historical link to elliptic integrals.

So the name “Elliptic Curve” originates not from the curve’s geometric shape but from its historical connection to elliptic integrals and elliptic functions. The name has been carried over from the study of these concepts.

References:

• McKean, H., & Moll, V. (2012). Elliptic Curves: Function Theory, Geometry, Arithmetic. Cambridge University Press.
• Silverman, J. H. (2009). The Arithmetic of Elliptic Curves (2nd ed.). Springer.
• Koblitz, N. (1993). Introduction to Elliptic Curves and Modular Forms. Springer.