Study centro-affine invariants on ellipses using canonical Lorentz metric.
problem Understanding centro-affine invariants on ellipses.
method Using the canonical Lorentz structure on the space of ellipses centered at zero.
result Described centro-affine invariants in terms of the canonical Lorentz structure.
In this paper we present a new method for motion tracking of tumors in liver ultrasound image sequences. Our algorithm has two main steps. In the first step, we apply mean shift algorithm with multiple features to estimate the center of the target in each frame. Target in the first frame is defined using an ellipse. Ed…
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.
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…
Method detects intersections between ellipses for Borromean linking.
problem Detecting Borromean linking between ellipses.
method Transforming one ellipse to a unit circle, examining intersections.
result Efficiently determines Borromean linking between ellipses.
Pedal curves derived from ellipses are invariant in area.
problem Finding invariant areas of pedal curves derived from ellipses.
method Analytical proof and explicit area expressions.
result Pedal curves derived from ellipses are invariant in area.
Billiard motion in ellipses analyzed with canonical coordinates.
problem Understanding billiard motion in ellipses.
method Canonical coordinates and kinematic analysis.
result Explicit parametrization of billiard motions using Jacobian elliptic functions.
Proves properties of periodic billiard orbits in ellipses.
problem Understanding periodic orbits in ellipses.
method Geometric and complex analytic methods.
result Sum of cosines of angles remains constant in one-parameter family of polygons.
A Steiner deltoid maintains constant area across all boundary points of an ellipse.
problem Finding curves associated with ellipses with constant area.
method Negative Pedal Curve of the Ellipse with respect to a boundary point M.
result The Steiner deltoid has constant area over all boundary points.
The paper extends Santaló's ellipse measures to hitting probabilities for circle lattices.
problem Calculating hitting probabilities for ellipses intersecting circles.
method Deriving measures for all positions of a moving ellipse inside a fixed circle and calculating hitting probabilities for circle lattices.
result Hitting probabilities for lattices of circles are deduced from ellipse measures.
Study calculates Mather β-function for ellipses and applies it to rigidity problems.
problem Calculating Mather β-function for ellipses and its application to rigidity.
method Used non-standard generating function of billiard problem to derive Mather β-function for ellipses. Applied to rigidity problems.
result Explicit formula for Mather β-function for ellipses and its application to rigidity.
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.
problem Approximating convex bodies by ellipses with respect to symmetric difference metric.
method Analyzing maximal intersection (MI) ellipsoids, proving uniqueness in 2D.
result Continuous family of MI ellipses interpolating John and Loewner ellipses in 2D.
Study on knots formed by gluing ellipses, defining gluing degree.
problem Properties of glued knots.
method Defined gluing degree to relate to knot properties.
result Classified all knots up to gluing degree 6.
Dan Reznik discovered conserved quantities for ellipses using billiard maps.
problem Conservation laws in periodic billiard trajectories.
method Non-standard generating function for the billiard ball map.
result Proved identities valid for all smooth convex billiard tables.
The cone projection fR(z)=z/(1+∣z∣/R) maps lines to conic arcs with specific properties.
problem Mapping lines to conic arcs with specific properties.
method Using a reciprocal lens identity and radial homeomorphism.
result The Self-Directrix Theorem and Confocal--Codirectrix Theorem.
Discussing rigidity properties of conics, inspired by billiards in ellipses.
problem Rigidity properties of conics and billiards in ellipses.
method Analog of polar duality and circle map properties.
result Two rigidity properties of conics.
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…
This paper connects billiards in ellipses to focal billiards in ellipsoids.
problem Proving the existence of isometric counterparts between billiards in ellipses and focal billiards in ellipsoids.
method Continuous transition via isometric focal billiards in a fixed ellipsoid.
result Established the connection between planar and spatial billiards.
Paper relates curvature ellipse and parabola of surface projections.
problem Relating local geometry of surfaces in different dimensions.
method Analyzes curvature ellipses and parabolas in R4 and R3. result Relates curvature geometry of surfaces to their projections.
The paper proves the conjecture about the number of cusps in a caustic formed by reflecting rays in a circle.
problem Proving the conjecture about the number of cusps in a caustic formed by reflecting rays in a circle.
method Analyzing the family of rays emanating from a non-focal point inside an elliptic billiard table, focusing on the caustic formed after multiple reflections.
result A proof of the conjecture that a caustic formed by reflecting rays in a circle has exactly four cusps.
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…
Paper constructs multivalued harmonic functions on R^3 using twistor methods.
problem Constructing multivalued harmonic functions on R^3.
method Twistor methods to construct multivalued harmonic functions.
result Found a family of multivalued harmonic functions with branching sets as ellipses and quadratic growth at infinity.
Proves Birkhoff-Poritsky conjecture for centrally-symmetric billiards.
problem Proves conjecture for centrally-symmetric billiards.
method Uses non-standard generating function, invariant curve structure, and integral-geometry approach.
result Billiard curve is an ellipse under given conditions.
The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.
problem Finding critical configurations and their properties for polygons on circles or ellipses.
method Computing Morse indices and gradient vector fields for isolated critical points, relating to eigenvalue questions.
result Computed Morse indices and relationships to eigenvalue questions for polygons on circles or ellipses.
Employing the affine normal flow, we prove a stability version of the p-affine isoperimetric inequality for p≥1 in R2 in the class of origin-symmetric convex bodies. That is, if K is an origin-symmetric convex body in R2 such that it has area π and its p-affine perimeter is close en…
The paper studies the number of normals to ellipsoids and their intersections with caustics.
problem The number of normals to an ellipsoid passing through a given point.
method Intersection points of the ellipsoid and its caustics are used to study the problem in 3D space.
result The number of normals is dependent on the position of the given point with respect to the caustics of the ellipsoid.
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…
Minimal surfaces of general type in Euclidean 4-space are characterized with the conditions that the ellipse of curvature at any point is centered at this point and has two different principal axes. Any minimal surface of general type locally admits geometrically determined parameters - canonical parameters. In such pa…
The study finds the best elliptical trajectory for planets using a variation of the hodograph theorem.
problem Finding the best elliptical trajectory for planets.
method Using a variation of the circular hodograph theorem, the study finds the best fitting ellipse for planetary trajectories by minimizing the sum of square distances from the points to the plane.
result The study finds that the best fitting ellipse for planetary trajectories minimizes the sum of square distances from the points to the plane.
We prove that the only compact convex ancient solutions of the planar affine normal flow are contracting ellipses.
New method studies moving points on curves using rotating frames.
problem Understanding the motion of points on curves.
method Constructing rotating frames for curves and analyzing the motion of points within these frames.
result A new binary mathematical formation mechanism for curves based on linear and rotational motion.
Curvature flow and inverse curvature flow solutions on 2D light cone identified.
problem Identifying self-similar solutions to curvature flow and inverse curvature flow on 2D light cone.
method Proved correspondence between CF and ICF solutions, analyzed ellipses and hyperboles, and characterized self-similar solutions.
result Ellipses and hyperboles are the only curves evolving under homotheties on the 2D light cone.
New parametrization handles sextactic points on closed curves.
problem Parametrizing closed projective plane curves with sextactic points.
method Introducing an additional scalar parameter α to define a 2π-periodic global parametrization.
result The balanced parametrization is unique up to a shift of the parameter and is a global projective invariant.
R. Schwartz's inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill's equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke-Santalo inequality …
The indicatrix or curvature ellipse and the characteristic curve of a surface in R4 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.
Unified view of surfaces in R^n using Gauss map, caustics, and quadratic forms.
problem Understanding smooth surfaces in R^n via various geometric perspectives.
method Combining evolute, curvature ellipse, Gauss map, and pseudo-Euclidean geometry of quadratic forms.
result Intersection of caustic with normal space of a surface yields polar dual of curvature ellipse.
Study finds a minimum volume for vector fields on a punctured sphere.
problem Finding the minimum volume of unit vector fields on a punctured sphere.
method Analyzes the volume of vector fields tangent to an antipodally punctured unit 2-sphere.
result Provides a lower bound for the volume of unit vector fields.
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 …
New minimal surfaces in 4D space discovered using complex rotations.
problem Discovering new minimal surfaces in 4D space.
method Complex parabolic rotations of holomorphic null curves in 4C space.
result Existence of minimal surfaces foliated by conic sections in 4D space.
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…
New periodic solution found in 4-body problem, not part of expected geometrical family.
problem Finding new periodic solutions in the 4-body problem not fitting the expected geometrical family.
method Analytic continuation of a numerical solution to discover a new family of solutions.
result Existence of a non-planar periodic solution for any pair of masses and integer n.
The paper examines soliton surfaces using a parallel transport frame field in 4D space.
problem Geometric properties of soliton surfaces associated with the Betchov-Da Rios equation.
method Parallel transport frame field approach in four-dimensional Euclidean space.
result Characterization of soliton surfaces as flat, minimal, semi-umbilic, or Wintgen ideal.
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.
problem Investigating minimal submanifolds with nullity properties in Euclidean spheres.
method Analyzing submanifolds with index of relative nullity at least \(m-2\) and providing a complete local parametric description using 1-isotropic surfaces.
result Any complete submanifold is either totally geodesic or has dimension three.
A cone projection maps complex plane arcs to conic sections with fixed focus and directrix.
problem Mapping complex plane arcs to conic sections with fixed focus and directrix.
method Elementary spatial construction and reciprocal lens identity.
result The cone projection maps every line not through the origin onto an arc of a conic with focus at the origin and directrix the line itself.