Self-intersection Points Research Articles

Cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points*

Cyclicity of slow–fast cycles with one self-intersection point and two nilpotent contact points*

Mapping structured Laguerre–Gaussian beam states onto the orbital Poincaré sphere in the form of controllable spatial trajectories

The structured Laguerre–Gaussian (LG) beam is a two-parameter superposition of 2n+ℓ+1 Hermite–Gaussian modes (where n and ℓ are a radial number and a topological charge of the initial LG beam) whose orbital angular momentum oscillations are controlled by phases and amplitude parameters. But we succeeded in reducing its representation to a simple sum of a standard LG mode and a hybrid Hermite–Laguerre–Gaussian (HLG) beam that is a key point in understanding a hidden geometry of the structured LG (sLG) beams and implementations of its unique prosperities. In assents, the hybrid HLG beam is mapped onto the orbital Poincaré sphere in the form of a plane trajectory along a main meridian of the sphere. However, the most intriguing thing is as follows. First, once we slightly perturb the HLG beam with a single LG mode, the flat trajectory turns into a complex multi-petalled tracery with multiple self-intersections due to cyclic variation of the phase parameter of the sLG beam. Moreover, the shape of the tracery as well as the birth and destruction of the self-intersection points can be controlled with the amplitude parameter. However, it is worth noting that when changing the beam parameters cyclically, the area outlined by the trajectory on the sphere is directly related to the geometric phase acquired by the sLG beam that can be treated as an additional degree of freedom for transmitting big data. In the article, we study the sLG beam properties and its mapping onto the orbital Poincarè sphere in the framework of a symplectic 4×4 matrix formalism while the orbital Stokes parameters are experimentally measured, and we have found good agreement between theory and experiment.

Cyclicity of slow-fast cycles with two canard mechanisms.

In this paper, we study the cyclicity of some degenerate slow-fast cycles with two canard mechanisms in planar slow-fast systems. One canard mechanism originates from a slow-fast Hopf point and the other from a point of self-intersection where the so-called entry-exit relation can be used. By studying the difference map, we show that the cyclicity of such slow-fast cycles is at most two (the associated slow divergence integral is nonzero or vanishes). As an example, we apply this result to the modified Holling-Tanner model.

On Self-Intersections of Cubic Bézier Curves

Cubic Bézier curves are widely used in computer graphics and geometric modeling, favored for their intuitive design and ease of implementation. However, self-intersections within these curves can pose significant challenges in both geometric modeling and analysis. This paper presents a comprehensive approach to detecting and computing self-intersections of cubic Bézier curves. We introduce an efficient algorithm that leverages both the geometric properties of Bézier curves and numerical methods to accurately identify intersection points. The self-intersection problem of cubic Bézier curves is firstly transformed into a quadratic problem by eliminating trivial solutions. Subsequently, this quadratic system is converted into a linear system that may be easily analyzed and solved. Finally, the parameter values corresponding to the self-intersection points are computed through the solution of the linear system. The proposed method is designed to be robust and computationally efficient, making it suitable for real-time applications.

Topological structure of optimal flows on the Girl's surface

We investigate the topological structure of flows on the Girl's surface which is one of two possible immersions of the projective plane in three-dimensional space with one triple point of self-intersection. First, we describe the cellular structure of the Boy's and Girl's surfaces and prove that there are unique images of the project plane in the form of a $2$-disk, in which the opposite points of the boundary are identified and this boundary belongs to the preimage of the $1$-skeleton of the surface. Second, we describe three structures of flows with one fixed point and no separatrices on the Girl's surface and prove that there are no other such flows. Third, we prove that Morse-Smale flows and they alone are structurally stable on the Boy's and Girl's surfaces. Fourth, we find all possible structures of optimal Morse-Smale flows on the Girl's surface. Fifth, we obtain a classification of Morse-Smale flows on the projective plane immersed on the Girl's surface. And finally, we describe the isotopic classes of these flows.

A Classification of Closed Hypocycloids and Epicycloids

Summary The paper describes a classification of closed epicycloids and hypocycloids into three classes: “odd/odd,” “even/odd,” “odd/even.” A subset of “perfect” epicycloids and hypocycloids that do not have self-intersection points has been identified. A new composite geometric object is constructed: the Euler Ring of Rings, consisting of a perfect epicycloid and a perfect hypocycloid with the same indices.

A Sweep-Line Algorithm and Its Application to Spiral Pocketing.(Dept.M)

This paper presents an efficient line-offset algorithm for general polygonal shapes with islands. A developed sweep-line algorithm (SL) is introduced to find all sell-intersection points accurately and quickly. The previous work is limited to handle polygons that having no line-segments in parallel to $Weep-line directions. An invalid loop detection and removing (ILDR) algorithm is proposed. The invalid loops detection algorithm divides the polygon al self-intersection points into a set of small polygons, and re-polygonized them. The polygons are checked for direction; invalid polygons are always having inverse direction with the boundary polygon. The proposed algorithm has been implemented in Visual C++ and applied to offset point scquence curves, which contain several islands.

Invariance of immersed Floer cohomology under Maslov flows

We show that immersed Lagrangian Floer cohomology in compact rational symplectic manifolds is invariant under Maslov flows such as coupled mean curvature/Kaehler-Ricci flow in the sense of Smoczyk as a pair of self-intersection points is born or dies at a self-tangency, using results of Ekholm-Etnyre-Sullivan. This proves part of a conjecture of Joyce. We give a lower bound on the time for which the Floer cohomology is invariant under the (forward or backwards) flow, if it exists.

State selection in frustrated magnets

Magnets with frustration often show accidental degeneracies, characterized by a large classical ground-state space (CGSS). Quantum fluctuations may ``select'' one of these ground states---a phenomenon labeled ``order by (quantum) disorder'' in literature. In this article, we examine the mechanism(s) by which such state selection takes place. We argue that a magnet, at low energies, maps to a particle moving on the CGSS. State selection corresponds to localization of the particle at a certain point on this space. We distinguish two mechanisms that can bring about localization. In the first, quantum fluctuations generate a potential on the CGSS space. If the potential has a deep enough minimum, then the particle localizes in its vicinity. We denote this as ``order by potential'' (ObP). In the second scenario, the particle localizes at a self-intersection point due to bound-state formation---a consequence of geometry and quantum interference. Following recent studies by the present authors, we denote this scenario as ``order by singularity'' (ObS). In either case, localization leads to an energy gap between the ground state(s) and higher-energy states. This pseudo-Goldstone gap behaves differently in the two mechanisms, scaling differently with the spin length. We place our discussion within the context of the one-dimensional spin-$S$ Kitaev model. We map out its CGSS which grows systematically with increasing system size. It resembles a network where the number of nodes increases exponentially. In addition, the number of wires that cross at each node also grows exponentially. This self-intersecting structure leads to ObS, with the low-energy physics determined by a small subset of the CGSS, consisting of ``Cartesian'' states. A contrasting picture emerges when an additional XY antiferromagnetic coupling is introduced. The CGSS simplifies dramatically, taking the form of a circle. Spin wave fluctuations generate a potential on this space, giving rise to state selection by ObP under certain conditions. Apart from contrasting ObS and ObP, we discuss the possibility of ObS in macroscopic magnets.

On the symmetry of finite sums of exponentials

In this note we are interested in the rich geometry of the graph of a curve $\gamma_{a,b}: [0,1] \rightarrow \mathbb{C}$ defined as \begin{equation*} \gamma_{a,b}(t) = \exp(2\pi i a t) + \exp(2\pi i b t), \end{equation*} in which $a,b$ are two different positive integers. It turns out that the sum of only two exponentials gives already rise to intriguing graphs. We determine the symmetry group and the points of self intersection of any such graph using only elementary arguments and describe various interesting phenomena that arise in the study of graphs of sums of more than two exponentials.

Closed-form solution of repeat ground track orbit design and constellation deployment strategy

The present work proposes a closed-form solution for circular repeat ground track (RGT) orbit designs and constellations and its application strategy. The proposed solution is an equation of motion proving that the characteristic of a circular RGT orbit is similar to that of the parametric curve of Ptolemy's model and provides a low computational cost for the RGT orbit design and related numerical analysis. The application strategy is to coincide with the self-intersection point of an RGT orbit over a specific target of interest on Earth and make it the starting point of the first satellite orbit in the satellite group. The remaining satellites are designed to have the same RGT orbit as the first satellite; therefore, the entire group has twice the repeat cycle over a specific target. To this end, this study represents an algorithm to match the specific target and the intersection point through an iterative numerical analysis method centering on the RGT equation. A set of orbital elements for a satellite group is also derived herein. The satellite group with a repeat cycle that doubles for a specific target shows a significantly shorter revisit time compared with the conventional Walker constellation.

Necessary condition for the existence of a simple closed geodesic on a regular tetrahedron in the spherical space

In the spherical space the curvature of the tetrahedron’s faces equals 1, and the curvature of the whole tetrahedron is concentrated into its vertices and faces. The intrinsic geometry of this tetrahedron depends on the value α of faces angle, where π/3 < α ⩽ 2π/3. The simple (without points of self-intersection) closed geodesic has the type (p,q) on a tetrahedron, if this geodesic has p points on each of two opposite edges of the tetrahedron, q points on each of another two opposite edges, and (p+q) points on each edges of the third pair of opposite one. For any coprime integers (p,q), we present the number αp, q (π/3 < αp, q < 2π/3) such that, on a regular tetrahedron in the spherical space with the faces angle of value α > αp, q, there is no simple closed geodesic of type (p,q)

A general method for decomposing self-intersecting polygon to normal based on self-intersection points

Checking whether polygons are self-intersecting or not is an important step for GIS projects before they are published to the web. Automatically converting self-intersection polygons into normal ones is practically useful, especially there are numerous polygons need to be processed. Based on the relationships of self-intersection points, this paper presents an algorithm to convert a complex self-intersection polygon to a normal one which has no self-intersection part. Furthermore, using the relationships of the repeat points (original self-intersection points) of the decomposed polygon, the result of the only simple polygon can be split into independent sub-polygons bounded by those points. The algorithm is easy to understand and with high efficiency because we consider only the self-intersection point relationships of the polygon, and we do not pay attention to the edges and their directions. A point structure in which the relationships of the self-intersection points are defined is used in the algorithm.

Order by singularity in Kitaev clusters

The Kitaev model is a beautiful example of frustrated interactions giving rise to deep and unexpected phenomena. In particular, its classical version has remarkable properties stemming from exponentially large ground state degeneracy. Here, we present a study of magnetic clusters with spin-$S$ moments coupled by Kitaev interactions. We focus on two cluster geometries -- the Kitaev square and the Kitaev tetrahedron -- that allow us to explicitly enumerate all classical ground states. In both cases, the classical ground state space (CGSS) is large and self-intersecting, with non-manifold character. The Kitaev square has a CGSS of four intersecting circles that can be embedded in four dimensions. The tetrahedron CGSS consists of eight spheres embedded in six dimensions. In the semi-classical large-$S$ limit, we argue for effective low energy descriptions in terms of a single particle moving on these non-manifold spaces. Remarkably, at low energies, the particle is tied down in bound states formed around singularities at self-intersection points. In the language of spins, the low energy physics is determined by a distinct set of states that lies well below other eigenstates. These correspond to `Cartesian' states, a special class of classical ground states that are constructed from dimer covers of the underlying lattice. They completely determine the low energy physics despite being a small subset of the classical ground state space. This provides an example of order by singularity, where state selection becomes stronger upon approaching the classical limit.

Forming of contour-parallel lines family with the detection of their non-working sections

The paper considers family of working contour-parallel lines formation used in the design of the toolpath that processes pocket surfaces. Contour-parallel working lines are the ones from which the lines-noises, i.e. non-working sections, are removed. Non-working sections include self-intersection loops and intersections of oncoming fronts in the case of multiply-connected domains. The spatial geometric model of forming contour-parallel lines is based on the cyclographic mapping. As a tool for detecting non-working sections for the case of oncoming fronts, the method of a testing ray is proposed. In case of self-intersections of contour-parallel lines, non-working sections are cut off by the parameter of these lines at the points of self-intersection. At the output of the proposed shaping algorithm of the working contour-parallel lines family, the parametric equations of these lines are formed. The algorithm successfully works for multiply-connected domains with polygonal and curved contours of the boundaries of the domain and islands in it. A comparative evaluation of the proposed method of forming a contour-parallel lines family with trimming non-working sections and the known methods that use the distance function is performed.

Formation of displacement curves with the identification of their non-working areas

The object of research is the shaping of a family of displacement curves used in designing the path of a tool that processes pocket surfaces. The subject of the study is the working displacement curves in the case of multiply connected areas. Working displacement curves are lines from which non-working sections have been removed. Non-working areas include self-intersecting loops of displacement curves and sections formed when intersecting displacement curves of opposing fronts. The paper presents methods for analyzing and cutting off non-working sections for cases of self-intersection and intersection of displacement curves of opposing fronts. The spatial geometric model of the formation of displacement curves is based on the cyclographic method of displaying space. As a tool for detecting non-working areas for the case of opposing fronts, a method of a testing beam is proposed. In the case of self-intersections of the displacement curves, non-working sections are cut off by the parameter of these lines at the points of self-intersection. The novelty of the study lies in the fact that the obtained mathematical model of the formation of displacement curves for multiply connected regions with contours of complex handicap curves makes it possible to obtain parametric equations of working lines at the output of the computational algorithm in a more reliable and simple way. This greatly simplifies the solution to the problem of automated design of the trajectory of the cutting tool. A comparative assessment of the proposed method of shaping a family of displacement curves with cutting off non-working sections and known methods using the distance function is performed.

A Functional Integral Approaches to the Makeenko–Migdal Equations

Makeenko and Migdal (Phys Lett B 88(1):135–137, 1979) gave heuristic identities involving the expectation of the product of two Wilson loop functionals associated to splitting a single loop at a self-intersection point. Kazakov and Kostov (Nucl Phys B 176(1):199–215, 1980) reformulated the Makeenko–Migdal equations in the plane case into a form which made rigorous sense. Nevertheless, the first rigorous proof of these equations (and their generalizations) was not given until the fundamental paper of Levy (2017). Subsequently Driver, Kemp, and Hall Commun. Math. Phys. 351(2), 741–774 (2017) gave a simplified proof of Levy’s result and then with Driver, Gabriel, Kemp, and Hall Commun. Math. Phys. 352(3), 967–978 (2017) we showed that these simplified proofs extend to the Yang–Mills measure over arbitrary compact surfaces. All of the proofs to date are elementary but tricky exercises in finite dimensional integration by parts. The goal of this article is to give a rigorous functional integral proof of the Makeenko–Migdal equations guided by the original heuristic machinery invented by Makeenko and Migdal. Although this stochastic proof is technically more difficult, it is conceptually clearer and explains “why” the Makeenko–Migdal equations are true. It is hoped that this paper will also serve as an introduction to some of the problems involved in making sense of quantizing Yang–Mill’s fields.

No immersed 2-knot with at most one self-intersection point has triple point number two or three

An embedded/immersed surface-knot is a closed and connected surface embedded/immersed in R4, respectively. The triple point number t(F) of an embedded/immersed surface-knot F is the minimum number of triple points required for a diagram of F. Satoh proved that (i) there does not exist an embedded surface-knot F such that t(F)=1, and (ii) there does not exist an embedded 2-knot F such that t(F)=2 or 3. In this paper, we prove similar results for immersed surface-knots with some conditions.

Exact computation for existence of a knot counterexample

Previously, numerical evidence was presented of a self-intersecting Bezier curve having the unknot for its control polygon. This numerical demonstration resolved open questions in scientic visualization, but did not provide a formal proof of self-intersection. An example with a formal existence proof is given, even while the exact self-intersection point remains undetermined.

Parity conditions for realizability of Gauss diagrams

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.