Darboux cyclides and webs from circles

Abstract

Motivated by potential applications in architecture, we study Darboux cyclides. These algebraic surfaces of order a most 4 are a superset of Dupin cyclides and quadrics, and they carry up to six real families of circles. Revisiting the classical approach to these surfaces based on the spherical model of 3D Moebius geometry, we provide computational tools for the identification of circle families on a given cyclide and for the direct design of those. In particular, we show that certain triples of circle families may be arranged as so-called hexagonal webs, and we provide a complete classification of all possible hexagonal webs of circles on Darboux cyclides.

34 pages, 20 figures

Cited by in corpus (16)

• Surfaces containing two circles through each point and Pythagorean 6-tuples
• Surfaces containing two circles through each point
• Surfaces that are covered by two pencils of circles
• Kinematic generation of Darboux cyclides
• Surfaces containing two isotropic circles through each point
• Translational and great Darboux cyclides
• Algorithms for singularities and real structures of weak Del Pezzo surfaces
• Webs of rational curves on real surfaces and a classification of real weak del Pezzo surfaces
• Using Algebraic Geometry to Reconstruct a Darboux Cyclide from a Calibrated Camera Picture
• Möbius automorphisms of surfaces with many circles
• Minimal degree rational curves on real surfaces
• Dupin cyclides passing through a fixed circle
• Unlocking Euclidean Problems with Isotropic Initialization
• Shapes of surfaces that contain a great and a small circle through each point
• Dupin Cyclides as a Subspace of Darboux Cyclides
• Self-intersections of surfaces that contain two circles through each point