Darboux cyclides and webs from circles

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.Comment: 34 pages, 20 figure

A surface containing a line and a circle through each point is a quadric

We prove that a surface in real 3-space containing a line and a circle through each point is a quadric. We also give some particular results on the classification of surfaces containing several circles through each point.Comment: Improved exposition, 4 figures adde

Dupin Cyclides as a Subspace of Darboux Cyclides

Dupin cyclides are interesting algebraic surfaces used in geometric design and architecture to join canal surfaces smoothly and construct model surfaces. Dupin cyclides are special cases of Darboux cyclides, which in turn are rather general surfaces in R^3 of degree 3 or 4. This article derives the algebraic conditions (on the coefficients of the implicit equation) for recognition of Dupin cyclides among the general implicit form of Darboux cyclides. We aim at practicable sets of algebraic equations describing complete intersections inside the parameter space.Comment: 20 pages, 1 figur

Enumerating the morphologies of non-degenerate Darboux cyclides

International audienceWe provide an enumeration of all possible morphologies of non-degenerate Darboux cyclides. Based on the fact that every Darboux cyclide in R 3 is the stereographic projection of the intersection surface of a sphere and a quadric in R 4 , we transform the enumeration problem of morphologies of Darboux cyclides to the enumeration of the algebraic sequences that characterize the intersection of a sphere and a quadric in R 4

Offsets, Conchoids and Pedal Surfaces

We discuss three geometric constructions and their relations, namely the offset, the conchoid and the pedal construction. The offset surface F d of a given surface F is the set of points at fixed normal distance d of F. The conchoid surface G d of a given surface G is obtained by increasing the radius function by d with respect to a given reference point O. There is a nice relation between offsets and conchoids: The pedal surfaces of a family of offset surfaces are a family of conchoid surfaces. Since this relation is birational, a family of rational offset surfaces corresponds to a family of rational conchoid surfaces and vice versa. We present theoretical principles of this mapping and apply it to ruled surfaces and quadrics. Since these surfaces have rational offsets and conchoids, their pedal and inverse pedal surfaces are new classes of rational conchoid surfaces and rational offset surfaces