arXiv research

Pedal curves derived from ellipses are invariant in area.

Paper relates curvature ellipse and parabola of surface projections.

We classify the Lagrangian orientable surfaces in complex space forms with the property that the ellipse of curvature is always a circle. As a consequence, we obtain new characterizations of the Clifford torus of the complex projective plane and of the Whitney spheres in the complex projective, complex Euclidean and co…

We consider smooth plane curves which are convex with respect to the origin. We describe centro-affine invariants (that is, GL_+(2,R)-invariants), such as centro-affine curvature and arc length, in terms of the canonical Lorentz structure on the three dimensional space of all the ellipses centered at zero, by means of …

Curvature flow and inverse curvature flow solutions on 2D light cone identified.

At any point of a surface in the four-dimensional Euclidean space we consider the geometric configuration consisting of two figures: the tangent indicatrix, which is a conic in the tangent plane, and the normal curvature ellipse. We show that the basic geometric classes of surfaces in the four-dimensional Euclidean spa…

Method detects intersections between ellipses for Borromean linking.

Billiard motion in ellipses analyzed with canonical coordinates.

Proves properties of periodic billiard orbits in ellipses.

A Steiner deltoid maintains constant area across all boundary points of an ellipse.

The paper extends Santaló's ellipse measures to hitting probabilities for circle lattices.

Study calculates Mather β-function for ellipses and applies it to rigidity problems.

We estimate the second order linking invariants of Lipschitz maps from an n-dimensional ellipse. The estimate uses a new directionally-dependent version of the isoperimetric inequality for cycles inside the ellipse. Using this work, we prove new lower bounds for the k-dilation of maps from one ellipse to another.

We estimate the linear isoperimetric constants of an n-dimensional ellipse. Using these estimates and a technique of Gromov, we estimate the Hopf and linking invariants of Lipschitz maps from ellipses to round spheres. Using these estimates, we give a lower bound for the k-dilation of degree non-zero maps between ellip…

New MI ellipses interpolate between John and Loewner ellipses in 2D.

The indicatrix or curvature ellipse and the characteristic curve of a surface in $\mathbf R^4$ are presented, as well as the projective duality connecting them. The characterisation of points in the surfaces as elliptic, parabolic and hyperbolic points, and the inflection points, are also discussed.

Study on knots formed by gluing ellipses, defining gluing degree.

Unified view of surfaces in R^n using Gauss map, caustics, and quadratic forms.

Dan Reznik discovered conserved quantities for ellipses using billiard maps.

Discussing rigidity properties of conics, inspired by billiards in ellipses.

We deduce a recent theorem by R. Schwartz on the structure of the so-called Poncelet grid from complete integrability of the billiard in an ellipse

We characterize subgroups of the mapping class group that stabilize a Teichmueller disk in terms of ellipses and strips that are immersed in the associated translation surface. In particular, we show that the space of immersed ellipses/strips that meet at least three cone points is naturally a (non-manifold) 2-dimensio…

The paper examines soliton surfaces using a parallel transport frame field in 4D space.

This paper connects billiards in ellipses to focal billiards in ellipsoids.

We discuss Darboux-Staude type of thread configurations for the ellipsoid similar to Chasles-Graves type of thread configurations for the ellipse. These threads are formed by rectilinear segments, geodesic and line of curvature segments on the considered ellipsoid and with tangents tangent to the given ellipsoid and a …

The paper proves the conjecture about the number of cusps in a caustic formed by reflecting rays in a circle.

We define two transforms between minimal surfaces with non-circular ellipse of curvature in the 5-sphere, and show how this enables us to construct, from one such surface, a sequence of such surfaces. We also use the transforms to show how to associate to such a surface a corresponding ruled minimal Lagrangian submanif…

Wintgen proved in [P. Wintgen, Sur l'inégalité de Chen-Willmore, C. R. Acad. Sci. Paris, 288 (1979), 993--995] that the Gauss curvature $K$ and the normal curvature $K^D$ of a surface in the Euclidean 4-space $E^4$ satisfy $K+|K^D|\leq H^2,$ where $H^2$ is the squared mean curvature. A surface $M$ in $\E4$ is called …

Wintgen ideal surfaces in E^4 form an important family of surfaces, namely surfaces with circular ellipse of curvature. Obviously, Wintgen ideal surfaces satisfy the pointwise equality K+K_N=H^2. In the present study we consider the Wintgen ideal surfaces in n-dimensional Euclidean space E^4. We have shown that Wintgen…

The paper studies minimal submanifolds in spheres with specific nullity properties.

Superconformal surfaces in Euclidean space are the ones for which the ellipse of curvature at any point is a nondegenerate circle. They can be characterized as the surfaces for which a well-known pointwise inequality relating the intrinsic Gauss curvature with the extrinsic normal and mean curvatures, due to Wintgen (\…

The paper proves an inequality and describes a curve flow in centro-affine geometry.

In this paper are studied the simplest patterns of axial curvature lines (along which the normal curvature vector is at a vertex of the ellipse of curvature) near a critical point of a surface mapped into R4. These critical points, where the rank of the mapping drops from 2 to 1, occur isolated in generic one parameter…

Paper constructs multivalued harmonic functions on R^3 using twistor methods.

Adapts stereographic projection for ellipsoid and elliptic paraboloid.

We consider two types of $p$-centro affine flows on smooth, centrally symmetric, closed convex planar curves, $p$-contracting, respectively, $p$-expanding. Here $p$ is an arbitrary real number greater than 1. We show that, under any $p$-contracting flow, the evolving curves shrink to a point in finite time and the only…

Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.

The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.

Employing the affine normal flow, we prove a stability version of the $p$-affine isoperimetric inequality for $p\geq1$ in $\mathbb{R}^2$ in the class of origin-symmetric convex bodies. That is, if $K$ is an origin-symmetric convex body in $\mathbb{R}^2$ such that it has area $π$ and its $p$-affine perimeter is close en…

The cone projection $f_R(z) = z/(1 + |z|/R)$ maps lines to conic arcs with specific properties.

The paper studies the number of normals to ellipsoids and their intersections with caustics.

The study finds the best elliptical trajectory for planets using a variation of the hodograph theorem.

We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses.

We prove that the only compact, origin-symmetric, strictly convex ancient solutions of the planar $p$ centro-affine normal flows are contracting origin-centered ellipses.

Solitons are special polygon midpoints under affine transformations.

New method studies moving points on curves using rotating frames.

It is well-known that in any codimension a simply connected Euclidean minimal surface has an associated one-parameter family of minimal isometric deformations. In this paper, we show that this is just a special case of the associated family to any simply connected elliptic surface for which all curvature ellipses of a …

Suppose curves are moving by curvature in a plane, but one embeds the plane in $R^3$ and looks at the plane from an angle. Then circles shrinking to a round point would appear to be ellipses shrinking to an ``elliptical point,'' and the surface energy would appear to be anisotropic as would the mobility. The result of …