[Paper Review] Convex Hull of Two Circles in R^3

Evan D. Nash, Ata Firat Pi̇r+2 more|arXiv (Cornell University)|Dec 30, 2016

Polynomial and algebraic computation9 references3 citations

TL;DR

This paper investigates the convex hulls of two circles in ℝ³, showing that when the circles are disjoint in complex projective space, their edge surface is an irrational ruled surface of degree eight. The authors classify the face lattices of these convex hulls, determine when they are spectrahedra (only when the circles lie on a quadratic cone), and use projective duality to analyze their structure, correcting a prior claim that the edge surface is always a cylinder.

ABSTRACT

We describe convex hulls of the simplest compact space curves, reducible quartics consisting of two circles. When the circles do not meet in complex projective space, their algebraic boundary contains an irrational ruled surface of degree eight whose ruling forms a genus one curve. We classify which curves arise, classify the face lattices of the convex hulls, and determine which are spectrahedra. We also discuss an approach to these convex hulls using projective duality.

Motivation & Objective

• To classify the convex hulls of two circles in ℝ³ based on their geometric arrangements.
• To determine the structure of the edge surface and edge curve of stationary bisecants for pairs of circles.
• To identify when such convex hulls are spectrahedra, particularly under which geometric conditions.
• To correct the prior claim in Ranestad and Sturmfels (2013) that the edge surface is always a cylinder.
• To apply projective duality to analyze the dual of the convex hull and extract information about exposed and nonexposed faces.

Proposed method

• Uses projective duality to analyze the dual body of the convex hull, where faces of the dual correspond to exposed faces of the original convex hull.
• Applies secant varieties and the theory of algebraic boundaries to describe the edge surface as a ruled surface in ℝ³.
• Analyzes the edge curve as a bidegree (2,2) curve in the product of the two conics C₁ × C₂, showing it is reduced and irreducible under generic conditions.
• Employs algebraic geometry techniques to compute the degree of the edge surface, proving it is eight when the circles are disjoint in ℂℙ³ and not tangent to each other's planes.
• Uses the fact that the convex hull is the projection of a spectrahedron, leveraging the existence of a linear matrix representation.
• Applies the duality framework from [6] and [7] to compute the dual of the convex hull as an intersection of dual cones, with geometric interpretation of boundary components.

Experimental results

Research questions

• RQ1 What is the structure of the edge surface in the convex hull of two disjoint circles in ℝ³?
• RQ2 When is the convex hull of two circles in ℝ³ a spectrahedron?
• RQ3 Which bidegree (2,2) curves in C₁ × C₂ arise as edge curves of two conics in ℂℙ³?
• RQ4 How does projective duality reveal information about exposed and nonexposed faces in the convex hull?
• RQ5 Why does the edge surface fail to be a cylinder in general, contrary to a prior claim?

Key findings

• The edge surface of two disjoint circles in ℂℙ³ is an irrational ruled surface of degree eight, not a cylinder as previously claimed.
• The edge curve of stationary bisecants is a reduced curve of bidegree (2,2) in C₁ × C₂, and all such curves arise except a rational curve with a cusp and a maximally reducible curve.
• The convex hull is a spectrahedron if and only if the two circles lie on a common quadratic cone.
• The dual body of the convex hull is bounded and convex, with its boundary encoding exposed faces and supporting hyperplanes.
• Nonexposed faces, such as nonexposed stationary bisecants, are not clearly visible in the dual body, highlighting a limitation of duality in capturing nonexposed structure.
• The dual of the convex hull is the intersection of two dual cones (or ellipsoids, in the case of ellipsoids), and this intersection encodes tritangent planes, bitangent edges, and tangent planes to the original circles.