On the convex hull of a space curve

TL;DR

This paper investigates the algebraic and geometric properties of the convex hull boundary of space curves, providing formulas, algorithms, and examples for understanding its structure and degree.

Contribution

It introduces formulas relating the convex hull boundary surface degree to the curve's invariants and offers algorithms for computing defining polynomials.

Findings

The boundary surface is reducible, composed of tritangent planes and stationary bisecants.

Degree of the boundary surface is expressed in terms of the curve's degree, genus, and singularities.

Algorithms are provided for computing the defining polynomials of the boundary surface.

Abstract

The boundary of the convex hull of a compact algebraic curve in real 3-space defines a real algebraic surface. For general curves, that boundary surface is reducible, consisting of tritangent planes and a scroll of stationary bisecants. We express the degree of this surface in terms of the degree, genus and singularities of the curve. We present algorithms for computing their defining polynomials, and we exhibit a wide range of examples.