arXiv research

Study of discrete Koenigs nets and their properties.

We study convex polyhedra in three-space that are inscribed in a quadric surface. Up to projective transformations, there are three such surfaces: the sphere, the hyperboloid, and the cylinder. Our main result is that a planar graph $Γ$ is realized as the $1$-skeleton of a polyhedron inscribed in the hyperboloid or cyl…

Circular nets with spherical parameter lines have geometric properties related to Darboux cyclides and terminating Laplace sequences.

A planar graph is inscribable if it is combinatorial equivalent to the skeleton of a polyhedra which is inscribed in a sphere. For an inscribable graph, in its combinatorial equivalent class, if we could always find polyhedra inscribed in any given convex surface which is sufficiently close to the sphere, then we call …

Extends sphere-rhomb inscribing to more directions.

Study examines Hilbert area of inscribed polygons in projective geometry.

Every curve can fit countless rhombuses.

Dancing polygons and rolling balls linked via a special geometric distribution.

A regular $n$-gon inscribing a knot is a sequence of $n$ points on a knot, such that the distances between adjacent points are all the same. It is shown that any smooth knot is inscribed by a regular $n$-gon for any $n$.

New bounds on inscribed triangles in arbitrary planar domains.

Derives conformal parameters of curves using inscribed circular polygons.

Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.

We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.

Square inscribed in a curve made of two graph functions.

Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.

Smooth knots with odd Conway polynomial terms have inscribed trefoils.

Study on volumes of random inscribed polytopes in projective geometries.

Similar simplices can be inscribed in most smoothly embedded spheres.

Transformed quadrics from 2D to higher dimensions.

We study convex polyhedra in $\mathbb{R}\mathbb{P}^3$ with all their vertices on a sphere. We do not require, in particular, that the polyhedra lie in the interior of the sphere, hence the term "weakly inscribed". Such polyhedra can be interpreted as ideal polyhedra, if we regard $\mathbb{R}\mathbb{P}^3$ as a combinati…

Sharp upper bounds on inscribed radius for metric spaces with convex boundary.

We prove that any cyclic quadrilateral can be inscribed in any closed convex $C^1$-curve. The smoothness condition is not required if the quadrilateral is a rectangle.

Floer homology applied to inscribing rectangles into curves.

We show that for every positive integer n there is a simple closed curve in the plane (which can be taken infinitely differentiable and convex) which has exactly n inscribed squares.

We study CR quadrics satisfying a symmetry property $(\tilde S)$ which is slightly weaker than the symmetry property $(S)$, recently introduced by W. Kaup, which requires the existence of an automorphism reversing the gradation of the Lie algebra of infinitesimal automorphisms of the quadric. We characterize quadrics s…

We consider a family of embedded, mean convex hypersurfaces which evolve by the mean curvature flow. It follows from general results of White that the inscribed radius at each point on the surface is at least $\frac{c}{H}$, where $c$ is a constant that depends only on the initial data. Andrews recently gave a new proof…

Square can fit inside curves close to smooth ones.

Classifies real rational knots and curves in a specific quadric space.

In the present article, we provide examples of fake quadrics, that is, minimal complex surfaces of general type with the same numerical invariants as the smooth quadric in $\PP ^3$ which are quotients of the bidisc by an irreducible lattice of automorphisms. Moreover, we list classes of arithmetic lattices over a real …

Study finds Stäckel equivalence for superintegrable systems via invariant quadrics.

A fake quadric is a smooth projective surface that has the same rational cohomology as a smooth quadric surface but is not biholomorphic to one. We provide an explicit classification of all irreducible fake quadrics according to the commensurability class of their fundamental group. To accomplish this task, we develop …

Extends Moutard quadric concept to higher dimensions.

We provide a generalization of Bianchi's Bäcklund transformation from 2-dimensional quadrics to higher dimensional quadrics. The starting point of our investigation is the higher dimensional (infinitesimal) version of Bianchi's main four theorems on the theory of deformations of quadrics and Bianchi's treatment of the …

Square-like quadrilaterals inscribed in space curves proven for finite total curvature.

The abstract proves polygon inscriptions in curves with specific edge ratios.

Generalizes embedding complex Grassmannians into quadrics.

Study holomorphic isometric embeddings of a Grassmannian into quadrics.

Discrete Laplacians defined for spherical and hyperbolic surfaces.

In trying to provide explicit deformations of quadrics the starting point of our investigation is to use Bianchi's link between real deformations of totally real regions of real paraboloids and various totally real forms of the sine-Gordon equation coupled with Bianchi's simple observation that the vacuum soliton of th…

The abstract shows how embeddings inscribe trapezoids or map three points to a line, proving nonexistence of certain maps.

The paper classifies Hopf hypersurfaces in complex quadrics with commuting Jacobi operators.

We find bounds on the difference between the writhing number of a smooth curve, and the writhing number of a polygon inscribed within. The proof is based on an extension of Fuller's difference of writhe formula to the case of polygonal curves. The results establish error bounds useful in the computation of writhe.

Researchers find explicit Bäcklund transforms for specific quadrics.

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics which are isometric along asymptotic lines and …

It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley (BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess solutions whose level surfaces are central quadrics in the space of independent variables (the so-called central quadric ansatz). It was demonstrated that generic solutio…

The paper classifies Hopf hypersurfaces with constant curvatures on complex quadrics.

We establish a link between Archimedes' method of integration for calculating areas, volumes and centers of mass of segments of parabolas and quadrics of revolution by factorization via the moments of a balance and an integration technique for a particular integrable system, namely Bianchi's Bäcklund transformation for…

We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.