Study of discrete Koenigs nets and their properties.
problem Characterization and properties of discrete Koenigs nets.
method Generalization of inscribed conics to inscribed quadrics and study of Koenigs d-grids.
result Established a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves.
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.
problem Discretizing surfaces with spherical curvature lines.
method Lie-geometric discretisation in terms of principal contact element nets.
result Circular nets with two families of spherical parameter lines are related to Darboux cyclides.
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.
problem Bounding strictly-convex regions with rhombs inscribed in spheres.
method Combines recent work with earlier results on sphere-rhomb inscribing.
result Extends class of inscribing spheres to more directions.
Study examines Hilbert area of inscribed polygons in projective geometry.
problem Understanding Hilbert area of inscribed polygons in projective geometry.
method Examined correspondence between Fock-Goncharov and Cartesian coordinates, analyzed degeneration and Hilbert area of inscribed quadrilaterals, developed microlocal condition.
result Sequence of strictly convex domains with bounded Hilbert area and divergent Goldman parameters.
Every curve can fit countless rhombuses.
problem Finding many rhombi within any curve.
method No curve regularity assumed.
result Uncountably many rhombi fit every curve.
Dancing polygons and rolling balls linked via a special geometric distribution.
problem Understanding the geometric and mechanical relationship between dancing polygons and rolling balls.
method Mapping dancing polygons to trajectories of a rolling ball on a 3D surface, both described by a specific geometric distribution.
result Non-degenerate dancing pairs of polygons exist for all n≥6 and correspond to rolling ball trajectories. 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.
problem Finding inscribed triangles in arbitrary planar domains with specific angle constraints.
method Proving the existence of uniformly fat triangles and not-too-fat triangles in bounded open sets.
result Existence of a maximal number Θ (between 0 and 60) for inscribed triangles with angles ≥ Θ degrees.
Derives conformal parameters of curves using inscribed circular polygons.
problem Characterizing conformal invariants of smooth curves in 3D.
method Limiting process with inscribed circular polygons, based on elementary geometry.
result Derives conformal length, curvature, and torsion via a novel method.
Paper generalizes inscribed radius estimate to hyperbolic Einstein manifolds.
problem Bounding inscribed radius in asymptotically hyperbolic Einstein manifolds.
method Generalized inscribed radius estimate to AH Einstein manifolds, combining recent work.
result Rigidity result achieved for upper bound of relative volume.
We prove a sharp estimate for the inscribed radius under certain fully nonlinear curvature flows. This estimate is asymptotically sharp on cylinders.
Study on non-orientable surfaces and inscribed rectangles in 4-manifolds.
problem Understanding non-orientable surfaces and their inscribed rectangles.
method Analyzing smooth and locally-flat non-orientable surfaces in 4-ball with specific knots, comparing results.
result Established differences between smooth and locally-flat non-orientable 4-genus of torus knots.
Square inscribed in a curve made of two graph functions.
problem Finding inscribed squares in curves formed by graph functions.
method Analysis of spectral invariants of Jordan Floer homology under curve perturbations.
result Existence of inscribed squares in curves with specific Lipschitz constants.
Smooth knots with odd Conway polynomial terms have inscribed trefoils.
problem Finding inscribed trefoils for smooth knots with specific polynomial terms.
method Using a perturbation of the double-cover of the orientation class and analyzing planar configurations.
result Smooth knots with odd quadratic terms of the Conway polynomial have inscribed trefoils.
Similar simplices can be inscribed in most smoothly embedded spheres.
problem Inscribing families of similar simplices in spheres.
method Diffeomorphic mapping and techniques from previous work on inscribing triangles.
result A dense family of spheres allows inscribing similar simplices of every pose.
Study on volumes of random inscribed polytopes in projective geometries.
problem Estimating volumes of random inscribed polytopes in projective geometries.
method Central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
result Established central limit theorems and normal approximation for volumes and dual volumes of random inscribed polytopes.
Transformed quadrics from 2D to higher dimensions.
problem Generalizing quadric transformations to higher dimensions.
method Bianchi's Hazzidakis transformation method.
result Generalization to higher dimensional quadrics.
We study convex polyhedra in RP3 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 RP3 as a combinati…
Sharp upper bounds on inscribed radius for metric spaces with convex boundary.
problem Bounding inscribed radius in metric measure spaces with convex boundary.
method Proves sharp upper bounds on inscribed radius for subsets with convex boundary.
result Sharp upper bounds on inscribed radius for subsets with convex boundary.
We prove that any cyclic quadrilateral can be inscribed in any closed convex C1-curve. The smoothness condition is not required if the quadrilateral is a rectangle.
Floer homology applied to inscribing rectangles into curves.
problem Determining if a Jordan curve can inscribe a square.
method Constructing Floer homology from inscribed rectangles and using spectral invariants.
result A Jordan curve inscribes a square if its enclosed area exceeds half a circle's area.
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 (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 Hc, 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.
problem Finding inscribed squares in nearly smooth curves.
method Using curvature and a map to relate curves, proving existence of inscribed squares.
result Curves close to smooth ones contain inscribed squares.
Classifies real rational knots and curves in a specific quadric space.
problem Classifying real rational knots and curves in a quadric space of signature (3,2). method Classification through a study of real rational curves of low degree in the quadric.
result Provides representatives of all real rational knots of degree ≤5 in the quadric. 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.
problem Understanding Stäckel equivalence in superintegrable systems.
method Using invariant quadrics to determine Stäckel classes of superintegrable systems.
result Stäckel classes of superintegrable systems can be derived from associated 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.
problem Higher order contact of quadrics with surfaces in 3D.
method Extension to hypersurfaces in arbitrary dimensions.
result Extension of 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.
problem Square-peg problem in space curves.
method Local curvature analysis and limiting argument on approximating curves.
result Every embedded curve with finite total curvature has an inscribed square-like quadrilateral.
The abstract proves polygon inscriptions in curves with specific edge ratios.
problem Proving the existence of polygons inscribed in Jordan curves with prescribed edge ratios.
method Using the properties of differentiable curves and proportional side lengths.
result Existence of polygons inscribed in Jordan curves with prescribed edge ratios.
Generalizes embedding complex Grassmannians into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannians into quadrics.
method Generalization of do Carmo-Wallach theory for moduli spaces.
result Moduli spaces of embeddings discussed.
Study holomorphic isometric embeddings of a Grassmannian into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannian into quadrics.
method Generalization of do Carmo--Wallach theory to study moduli space.
result Moduli space of embeddings up to equivalence discussed.
Discrete Laplacians defined for spherical and hyperbolic surfaces.
problem Defining discrete Laplacians for non-Euclidean geometries.
method Definitions close to Euclidean, structure-preserving properties proven.
result Connection between discrete and smooth Laplacians in non-Euclidean settings.
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 study shows that inscribed rectangles in smooth curves cover at least one third of all possible aspect ratios.
problem Determining the coverage of inscribed rectangles in smooth Jordan curves.
method Analyzing sets of disjoint homologically nontrivial projective planes and applying Kemperman's theorem.
result The Lebesgue measure of the set of aspect ratios is at least 1/3.
The abstract shows how embeddings inscribe trapezoids or map three points to a line, proving nonexistence of certain maps.
problem Proving the nonexistence of affinely 3-regular maps in infinitely many dimensions.
method Elementary proof using embeddings and nonsingular bilinear maps.
result Recovery of nonexistence results for affinely 3-regular maps without complex algebraic techniques.
The paper classifies Hopf hypersurfaces in complex quadrics with commuting Jacobi operators.
problem Characterizing Hopf real hypersurfaces with commuting Jacobi operators.
method Investigating the commuting property between normal and structure Jacobi operators.
result A remarkable classification of Hopf real hypersurfaces in the complex quadric 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.
problem Isometric deformations of diagonal higher dimensional quadrics without center.
method Explicitly found Bäcklund transforms using the Bianchi Permutability Theorem and 3-moving Möbius configuration.
result Explicit solutions can be iterated with arbitrary constants.
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.
problem Classifying Hopf hypersurfaces with specific curvature properties.
method Analyzing hypersurfaces on complex quadrics with at most five distinct constant principal curvatures.
result All classified hypersurfaces are open parts of homogeneous examples.
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…