Formulae for null curves deriving from elliptic curves

Abstract

Any elliptic curve can be realised in the tangent bundle of the complex projective line as a double cover branched at four distinct points on the zero section. Such a curve generates, via classical osculation duality, a null curve in C 3 and thus an algebraic minimal surface in R 3 . We derive simple formulae for the coordinate functions of such a null curve.

A representation formula for curves in C3 with preset infinitesimal arc length

“…3) The recursion procedure (4.1) and the resulting 'free Weierstraß formula' (4.3) occurred in a number of other papers, for instance [3], [8], [10], [11].…”