Related papers: On Curves and Surfaces of Constant Width

In this paper we introduce the Constant Width Measure Set, which measures the constant width property of an oval, i.e. the planar simple closed strictly convex curve. We study its geometrical properties. We find the exact relation between…

Extending Blaschke and Lebesgue's classical result in the Euclidean plane, it has been recently proved in spherical and the hyperbolic cases, as well, that Reuleaux triangles have the minimal area among convex domains of constant width $D$.…

The mean width is a measure on three-dimensional convex bodies that enjoys equal status with volume and surface area [Rota]. As the phrase suggests, it is the mean of a probability density f. We verify formulas for mean widths of the…

In this paper, we investigate constant breadth curves on a surface according to Darboux frame and give some characterizations of these curves.

In this paper, we study the curvature properties of random complex plane curves. We bound from below the probability that a uniform proportion of the area of a random complex degree $d$ plane curve has a curvature smaller than $-d/8$. Our…

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere…

It is known that an $n$-dimensional convex body which is typical in the sense of Baire category, shows a simple, but highly non-intuitive curvature behaviour: at almost all of its boundary points, in the sense of measure, all curvatures are…

For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of…

We study surfaces with one constant principal curvature in Riemannian and Lorentzian three-dimensional space forms. Away from umbilic points they are characterized as one-parameter foliations by curves of constant curvature, each of these…

We discuss the property of the number density of a fluid of particles living in a curved surface without boundaries to be constant in the thermodynamic limit. In particular we find a sufficient condition for the density to be constant along…

Packings of equal disks in the plane are known to have density at most $\pi/\sqrt{12}$, although this density is never achieved in the square torus, which is what we call the plane modulo the square lattice. We find packings of disks in a…

This is the fifth in a series of papers giving a proof of the Kepler conjecture, which asserts that the density of a packing of congruent spheres in three dimensions is never greater than $\pi/\sqrt{18}\approx 0.74048...$. This is the…

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global…

In this paper we obtain density estimates for compact surfaces immersed in R^n with total boundary curvature less than 4pi and with sufficiently small L^p norm of the mean curvature, p>2. Our results generalize the main results in [2]. We…

We present an algorithm to simulate random sequential adsorption (random "parking") of discs on constant-curvature surfaces: the plane, sphere, hyperboloid, and projective plane, all embedded in three-dimensional space. We simulate complete…

It is a widely observed phenomenon in computer graphics that the size of the silhouette of a polyhedron is much smaller than the size of the whole polyhedron. This paper provides, for the first time, theoretical evidence supporting this for…

We give necessary and sufficient conditions on the curvature and the torsion of a regular curve of the space forms $\h^3$ and $\s^3$ to be contained in a totally umbilical surface. In case that the curve has constant torsion, we obtain the…

We consider the sub-Riemannian metric $g_{h}$ on $\mathbb{S}^3$ provided by the restriction of the Riemannian metric of curvature 1 to the plane distribution orthogonal to the Hopf vector field. We compute the geodesics associated to the…

It is still an open question whether a compact embedded hypersurface in the Euclidean space R^{n+1} with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of…

A recent result of Chepoi, Estellon and Vaxes [DCG '07] states that any planar graph of diameter at most 2R can be covered by a constant number of balls of size R; put another way, there are a constant-sized subset of vertices within which…