Related papers: Surfaces containing two circles through each point
We find all analytic surfaces in space R^3 such that through each point of the surface one can draw two circular arcs fully contained in the surface. The proof uses a new decomposition technique for quaternionic matrices.
We study analytic surfaces in 3-dimensional Euclidean space containing two circular arcs through each point. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We reduce finding all such surfaces to…
We prove (under some technical assumptions) that each surface in $\mathbb R^3$ containing two arcs of parabolas with axes parallel to $Oz$ through each point has a parametrization…
In this paper we define and construct a new class of algebraic surfaces in three-dimensional Euclidean space generated by a curve and a congruence of circles. We study their properties and visualize them with the program Mathematica.
We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with…
A surface that is the pointwise sum of circles in Euclidean space is either coplanar or contains no more than 2 circles through a general point. A surface that is the pointwise product of circles in the unit-quaternions contains either 2,…
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.
Comessatti proved that the set of real points of a rational real algebraic surface is either a nonorientable surface, or the two-sphere, or the torus. Conversely, it is easy to see that all of these surfaces admit a rational real algebraic…
We present a variety of geometrical and combinatorial tools that are used in the study of geometric structures on surfaces: volume, contact, symplectic, complex and almost complex structures. We start with a series of local rigidity results…
A theta surface in affine 3-space is the zero set of a Riemann theta function in genus 3. This includes surfaces arising from special plane quartics that are singular or reducible. Lie and Poincar\'e showed that theta surfaces are precisely…
We classify real two-dimensional orbits of conformal subgroups such that the orbits contain two circular arcs through a point. Such surfaces must be toric and admit a M\"obius automorphism group of dimension at least two. Our theorem…
Results of number of geometric operations (often used in technical practise, as e.g. the operation of blending) are in many cases surfaces described implicitly. Then it is a challenging task to recognize the type of the obtained surface,…
There are two problems Analytical Geometry with facing anyone who studies this discipline: define the nature of the locus represented by the general equation 2do degree in two or three variables: That curve represents the plane? What…
We prove the filling area conjecture in the hyperelliptic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu's result in genus 0. We translate the problem into a question about closed…
The task of recognizing an algebraic surface from a single apparent contour can be reduced to the recovering of a homogeneous equation in four variables from its discriminant. In this paper, we use the fact that Darboux cyclides have a…
This note is about an old conjecture of Voisin, which concerns zero--cycles on the self-product of surfaces of geometric genus one. We prove this conjecture for surfaces with $p_g=1$ and $q=2$.
We classify the topological types of surfaces in the 3-dimensional unit sphere that contain both a great and a small circle through each point. In particular, these surfaces are homeomorphic to one of five normal forms and are either the…
This thesis deals with automorphisms of real algebraic surfaces, which are polynomial transformations with a polynomial inverse. The main concern is whether their restriction to the real locus reflects all the richness of the complex…
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…
We classify the singular loci of real surfaces in three-space that contain two circles through each point. We characterize how a circle in such a surface meets this loci as it moves in its pencil and as such provide insight into the…