Surfaces that are covered by two pencils of circles

TL;DR

This paper classifies real surfaces covered by multiple circle pencils, analyzing their geometric properties, and applies these findings to confirm conjectures and classify webs of circles, including new examples beyond three dimensions.

Contribution

It provides a comprehensive classification of surfaces covered by two circle pencils, including their degrees, embeddings, and incidences, with applications to conjectures and circle webs.

Findings

Classified surfaces covered by two pencils of circles and their geometric properties.

Confirmed Blum's conjecture in higher-dimensional space.

Discovered new hexagonal circle webs not embeddable in 3D.

Abstract

We list up to M\"obius equivalence all possible degrees and embedding dimensions of real surfaces that are covered by at least two pencils of circles, together with the number of such pencils. In addition, we classify incidences between the contained circles, complex lines and isolated singularities. Such geometric characteristics are encoded in the N\'eron-Severi lattices of such surfaces and is of potential interest to geometric modelers and architects. As an application we confirm Blum's conjecture in higher dimensional space and we address the Blaschke-Bol problem by classifying surfaces that are covered by hexagonal webs of circles. In particular, we find new examples of such webs that cannot be embedded in 3-dimensional space.