[Paper Review] On tangents to quadric surfaces

Ciprian S. Borcea, Xavier Goaoc+2 more|ArXiv.org|Feb 24, 2004

Mathematics and Applications10 references3 citations

TL;DR

This paper investigates the geometric conditions under which up to four quadric surfaces in complex projective 3-space share a continuum of common tangents, introducing the concept of a 'basket'—a quadric tangent to others along a conic. It establishes that four real spheres have infinitely many common real tangents if and only if their centers are collinear and they share at least one common real tangent, which occurs when they intersect in a common circle or are tangent to a common surface of revolution (e.g., a cone, cylinder, or hyperboloid).

ABSTRACT

We study the variety of common tangents for up to four quadric surfaces in projective three-space, with particular regard to configurations of four quadrics admitting a continuum of common tangents. We formulate geometrical conditions in the projective space defined by all complex quadric surfaces which express the fact that several quadrics are tangent along a curve to one and the same quadric of rank at least three, and called, for intuitive reasons: a basket. Lines in any ruling of the latter will be common tangents. These considerations are then restricted to spheres in Euclidean three-space, and result in a complete answer to the question over the reals: ``When do four spheres allow infinitely many common tangents?''.

Motivation & Objective

• To determine the geometric conditions under which four quadric surfaces in complex projective 3-space admit a continuum of common tangents.
• To formalize the concept of a 'basket'—a quadric of rank at least three tangent to others along a conic—providing a unifying framework for degenerate configurations of common tangents.
• To analyze the real case of four spheres, identifying the precise necessary and sufficient conditions for infinitely many common real tangents.
• To characterize the algebraic and geometric structure of configurations admitting common baskets using Grassmannians, duality, and Veronese embeddings.
• To prove the uniqueness, up to projective equivalence, of a 'double-four' configuration of quadrics where each quadric in one group is a common basket for the other group, linking to the Reye configuration.

Proposed method

• Uses the Grassmannian $G(2,4)$ to represent the variety of lines in $\mathbb{P}^3$, embedding it via Plücker coordinates as a quadric in $\mathbb{P}^5$, enabling the study of common tangents as intersections.
• Applies duality and the Veronese embedding to relate the geometry of quadrics to their duals, particularly focusing on pencils containing rank-one (double-plane) or rank-two (cone) quadrics.
• Introduces the notion of a 'basket' as a quadric tangent to another along a conic, equivalent to the pencil of the two quadrics containing a double-plane.
• Employs algebraic geometry techniques to analyze the irreducible varieties $B^k_{n}$ parameterizing $k$ quadrics sharing a common basket $b$, computing their dimensions and characterizing generic configurations.
• Reduces the real case of four spheres to a system of quadratic and quartic equations in the direction vector of a common tangent line, using symmetry and translation to simplify the system.
• Completes the system by eliminating variables and analyzing the intersection of a conic and a quartic in the direction space, deriving necessary conditions for a one-dimensional family of solutions.

Experimental results

Research questions

• RQ1 Under what geometric conditions do four quadric surfaces in $\mathbb{P}^3(\mathbb{C})$ admit a continuum of common tangents?
• RQ2 When is a quadric $q_1$ considered a 'basket' for another quadric $q_2$, and what algebraic conditions characterize this tangency along a conic?
• RQ3 What is the precise condition on four real spheres in $\mathbb{R}^3$ for them to have infinitely many common real tangents?
• RQ4 How do the configurations of four quadrics sharing a common basket relate to classical configurations like the Reye or Desargues configuration?
• RQ5 What is the structure of the variety $B^4_{25}$ parameterizing four quadrics and a common basket, and how is it characterized geometrically?

Key findings

• Four non-singular complex quadrics in general position have exactly 32 common tangents, as the intersection of their tangent line varieties in the Grassmannian.
• A continuum of common tangents arises if and only if the four quadrics share a common basket—a quadric of rank at least three tangent to each along a conic.
• The variety $B^2_{16}$ of pairs of quadrics sharing a common basket is irreducible of dimension 16, and generic such pairs are characterized by their pencil containing a two-plane.
• The variety $B^4_{25}$ of four quadrics sharing a common basket is irreducible of dimension 25, and generic such configurations correspond to a complete quadrilateral of pencils of cones in the span of the four quadrics.
• For four real spheres, there are infinitely many common real tangents if and only if their centers are collinear and they share at least one common real tangent, which occurs when they intersect in a common circle or are tangent to a common surface of revolution (e.g., cone, cylinder, hyperboloid).
• In the real case, if the four centers are coplanar but not collinear, there are at most twelve common real tangents, and if three centers are collinear, rotational symmetry restricts the existence of common tangents to configurations with a shared conic or basket.