Plane Curves Research Articles

GIT quotient of holomorphic foliations on ℂℙ2 of degree 2 and quartic plane curves

Abstract We study the quotient variety of the space of foliations on ℂ ⁢ ℙ 2 {\mathbb{CP}^{2}} of degree 2 up to change of coordinates. We find the intersection Betti numbers of this variety. As a corollary, we have that these intersection Betti numbers coincide with the intersection Betti numbers of the quotient variety of quartic plane curves. Finally, we give an explicit isomorphism between the space of foliations of degree 2 with different singular points, without invariant lines and the space of smooth quartic plane curves.

Envelopes of Bisection Lines of Polygons

A bisection line divides a convex planar curve into two parts with equal areas. It is natural to study the envelope of these lines, which in general present singularities. The polygonal case is particularly inte\-resting, since there are several different notions of a discrete envelope. In this paper, we study three different notions of discrete envelopes of bisection lines and the connections between them.

Bi-Lagrangian structures and the space of rays

This paper focuses on local curvature invariants associated with bi-Lagrangian structures. We establish several geometric conditions that determine when the canonical connection is flat, building on our previous findings regarding divergence-free webs (Domitrz and Zubilewicz 2023 Anal. Math. Phys. 13 4). Addressing questions raised by Tabachnikov (1993 Differ. Geom. Appl. 3 265–84), we provide complete solutions to two problems: the existence of flat bi-Lagrangian structures within the space of rays induced by a pair of hypersurfaces, and the existence of flat bi-Lagrangian structures induced by tangents to Lagrangian curves in the symplectic plane.

Reconstructing Curves from Sparse Samples on Riemannian Manifolds

AbstractReconstructing 2D curves from sample points has long been a critical challenge in computer graphics, finding essential applications in vector graphics. The design and editing of curves on surfaces has only recently begun to receive attention, primarily relying on human assistance, and where not, limited by very strict sampling conditions. In this work, we formally improve on the state‐of‐the‐art requirements and introduce an innovative algorithm capable of reconstructing closed curves directly on surfaces from a given sparse set of sample points. We extend and adapt a state‐of‐the‐art planar curve reconstruction method to the realm of surfaces while dealing with the challenges arising from working on non‐Euclidean domains. We demonstrate the robustness of our method by reconstructing multiple curves on various surface meshes. We explore novel potential applications of our approach, allowing for automated reconstruction of curves on Riemannian manifolds.

The Gradient Flow for Entropy on Closed Planar Curves

In this paper we consider the steepest descent L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^2$$\\end{document}-gradient flow of the entropy functional. The flow expands convex curves, with the radius of an initial circle growing like the square root of time. Our main result is that, for any initial curve (either immersed locally strictly convex of class C2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C^2$$\\end{document} or embedded of class W2,2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$W^{2,2}$$\\end{document} bounding a strictly convex body), the flow converges smoothly to a round expanding multiply-covered circle.

Symmetry group detection of point clouds in 3D via a decomposition method

Analyzing the symmetries present in point clouds, which represent sets of 3D coordinates, is important for understanding their underlying structure and facilitating various applications. In this paper, we propose a novel decomposition-based method for detecting the entire symmetry group of 3D point clouds. Our approach decomposes the point cloud into simpler shapes whose symmetry groups are easier to find. The exact symmetry group of the original point cloud is then derived from the symmetries of these individual components. The method presented in this paper is a direct extension of the approach recently formulated in Bizzarri et al. (2022a) for discrete curves in plane. The method can be easily modified also for perturbed data. This work contributes to the advancement of symmetry analysis in point clouds, providing a foundation for further research and enhancing applications in computer vision, robotics, and augmented reality.

Classification of unimodal parametric plane curve singularities in positive characteristic

In 2011 Hefez and Hernandes completed Zariski's analytic classification of plane branches belonging to a given equisingularity class by creating “very short” parameterizations over the complex numbers. Their results were used by Mehmood and Pfister to classify unimodal plane branches in characteristic 0 by giving lists of normal forms. The aim of this paper is to give a complete classification of unimodal plane branches over an algebraically closed field of positive characteristic. Since the methods of Hefez and Hernandes cannot be used in positive characteristic, we use a different approach and, for some sporadic singularities in small characteristic, computations with Singular. Our methods are characteristic-independent and provide a different proof of the classification in characteristic 0 showing at the same time that this classification holds also in large characteristic. The main theoretical ingredients are the semicontinuity of the semigroup and of the modality, which we prove and which may be of independent interest.

Fundamental group of complex hypocycloids

The Zariski–van Kampen theorem allows us to provide a presentation of the fundamental group for the complement of algebraic plane curves. However, certain computations require arduous work, as exemplified in the case of hypocycloids. In this paper we present the following result: Theorem 1.The fundamental group of any complex hypocycloid withNcusps is the Artin group of theN-gon. The main idea of the proof is take advantage of the symmetries inherent in the hypocycloid, allowing us to partition the domain to determine the generators of the fundamental group.

Enumeration of non-nodal real plane rational curves

Welschinger invariants enumerate real nodal rational curves in the plane or in another real rational surface. We analyze the existence of similar enumerative invariants that count real rational plane curves having prescribed non-nodal singularities and passing through a generic conjugation-invariant configuration of appropriately many points in the plane. We show that an invariant like this is unique: it enumerates real rational three-cuspidal quartics that pass through generically chosen four pairs of complex conjugate points. As a consequence, we show that through any generic configuration of four pairs of complex conjugate points, one can always trace a pair of real rational three-cuspidal quartics.

Extending the Hough transform to recognize and approximate space curves in 3D models

Feature curves are space curves identified by color or curvature variations in a shape, which are crucial for human perception (Biederman, 1995). Detecting these characteristic lines in 3D digital models becomes important for recognition and representation processes. For recognizing plane curves in images, the Hough transform (HT) provided a very good solution to the problem. It selects the best-fitting curve in a dictionary of families of curves through a voting procedure that makes it robust to noise and missing parts. Since 3D digital models are often obtained by scanning real objects and may have many defects, the HT has been extended to recognize and approximate space curves in 3D models that correspond to relevant featuresThis work overviews three HT-based different approaches for identifying and approximating spatial profiles of points extracted from point clouds or meshes. A first attempt at this extension involved projecting the spatial points onto the regression plane, thus reducing the problem to planar recognition and using families of plane curves. A second approach has been proposed to recognize spatial profiles that cannot be projected onto the regression plane, using two types of space curve families. Unfortunately, the main drawback of methods based on traditional HT is that it requires prior knowledge of which family of curves to look for.To overcome this limitation, a third method has been developed that provides a piecewise space curve approximation using specific parametric polynomial curves. Additionally, free-form curves that a parametric or implicit form cannot express can be represented using this technique.In the paper, we also analyze the pros and cons of the various approaches and how they managed and reduced the HT's computational cost, given the large number of parameters introduced when families of space curves are considered.

On G2 approximation of planar algebraic curves under certified error control by quintic Pythagorean-hodograph splines

The Pythagorean-Hodograph curve (PH curve) is a valuable curve type extensively utilized in computer-aided geometric design and manufacturing. This paper presents an approach to approximate a planar algebraic curve within a bounding box by employing piecewise quintic PH spline curves, while maintaining G2 smoothness of the approximating curve and preserving second-order geometric details at singularities. The bounding box encompasses all x-coordinates of key topological points, ensuring accurate representation. The paper explores the analysis of the G2 interpolation problem for quintic PH curves with invariant convexity, transforming the quest for interpolation solutions into identifying positive roots within a set of algebraic equations. Through infinitesimal order analysis, it is established that a solution necessarily exists following adequate subdivision, laying the groundwork for practical application. Finally, the paper introduces a novel algorithm that integrates prior research to construct the approximating curve while maintaining control over the desired error levels.

End-rotation-driven kinetic elastica structures: Concept and morphology design

Euler’s elastica, a family of planar curves describing the equilibrium shapes of a thin elastic beam subjected to end forces and moments, demonstrates both geometric versatility and mechanical functionality, leading to its diverse applications across multiple disciplines. Despite its roots in ancient structural engineering practices, the kinetic potential of Euler’s elastica remains largely unexplored in this domain. To address this gap, this paper proposes the concept of end-rotation-driven kinetic elastica (ERKE) structures. This novel kinetic form consists of a sequence of discrete elastica arches transformed from initially straight members by elastic bending. The kinetic behavior arises from manipulating the end rotations of each elastica arch, prompting a continuous flow of motions resulting from adaptations in the arch shape. This adaptive quality opens up limitless possibilities for creating visually captivating dynamic patterns in the overall structure by coordinating the motions of individual arches. As the inaugural exploration of ERKE structures, this paper centers around their conceptualization, with an emphasis on morphology design, which refers to the creation of desirable dynamic patterns. To this end, the paper provides an exposition of the general design procedure and criteria, along with demonstrative examples. The paper also envisions potential application scenarios and identifies research needs, paving the way for future investigations.

A graphical user interface tool to teach plane curves in pandemic situations through a flipped classroom

During the pandemic and lockdown situations of the last few years, we transitioned to online, physical, and hybrid education approaches at various times. We had difficulty teaching multiple calculus and analytical geometry units online. In light of the current situation, a graphical user interface (GUI) teaching tool has been developed to teach university students many plane curve modules through a flipped classroom. With a few inputs and a single click as output, these executable spreadsheets can assist physics and mathematics students in practising several principles of analytical geometry. Spreadsheets can readily draw planes in three-dimensional geometry due to their widespread use. Some online vector GUIs present the results; however, this GUI conveniently illustrates the plot. The well-designed user interface also helps create new questions with answer keys and gives access to the teachers of physics and mathematics, which can make online teaching very easy. The effectiveness of this GUI was checked through Kolmogorov-Smirnov and Shapiro-Wilk tests, and a good agreement has been found in the results.Contribution: This article contributes to Mathematics teaching through GUIs.

On the Algebraic Geometry of Multiview

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras.

Rogue curves in the Davey-Stewartson I equation.

We report new rogue wave patterns whose wave crests form closed or open curves in the spatial plane, which we call rogue curves, in the Davey-Stewartson I equation. These rogue curves come in various striking shapes, such as rings, double rings, and many others. They emerge from a uniform background (possibly with a few lumps on it), reach high amplitude in such striking shapes, and then disappear into the same background again. We reveal that these rogue curves would arise when an internal parameter in bilinear expressions of the rogue waves is real and large. Analytically, we show that these rogue curves are predicted by root curves of certain types of double-real-variable polynomials. We compare analytical predictions of rogue curves to true solutions and demonstrate good agreement between them.

Smooth limits of plane curves of prime degree and Markov numbers

Smooth limits of plane curves of prime degree and Markov numbers

Incidence of Cervical Kyphosis and Factors Associated with Improvement in Postoperative Cervical Spinal Alignment in Idiopathic Scoliosis with Major Thoracolumbar/Lumbar and Thoracic Curves.

Background: This study aimed to compare the incidence and severity of cervical kyphosis before and after surgery between patients with adolescent idiopathic scoliosis (AIS) with major thoracolumbar/lumbar curves (Lenke type 5C group) and those with major thoracic curves (Lenke type 1A group). Further, factors associated with cervical spinal alignment changes after surgery in the two groups were examined. Methods: This study included consecutive patients with AIS who underwent posterior spinal fusion for Lenke type 1A and 5C curves and who were followed up for at least 1 year. To measure changes in sagittal alignment, all patients underwent radiography before, immediately after, and at 1 year after surgery. The correlation coefficients change the value of the C2-C7 angle before and after surgery (ΔC2-ΔC7) and other spinopelvic parameters were examined. Results: In total, 19 of 30 patients in the Lenke type 1A group and 21 of 36 in the Lenke type 5C group presented with cervical kyphosis preoperatively. Hence, the incidence of cervical kyphosis did not significantly differ between the two groups. Further, the two groups had significantly higher thoracic kyphosis (TK) and greater C2-C7 angles postoperatively. The TK of the Lenke type 5C group further increased at 1 year postoperatively. The Lenke 1A type group presented with a significant re-decrease in the C2-C7 angle at 1 year postoperatively. However, the C2-C7 angle of the Lenke type 5C group did not change. The ΔTK was closely associated with the ΔC2-ΔC7 in the Lenke type 1A group, but not in the Lenke type 5C group. Conclusions: In thoracic AIS, postoperative cervical alignment should achieve an adequate TK and promote correction of the coronal plane curve. Moreover, selective corrective surgery can improve postoperative cervical alignment in lumbar AIS.

Simultaneous Galois points for a reducible plane curve consisting of nonsingular components

Simultaneous Galois points for a reducible plane curve consisting of nonsingular components

On free and plus-one generated curves arising from free curves by addition–deletion of a line

In a recent paper, after introducing the notion of plus-one generated hyperplane arrangements, Takuro Abe has shown that if we add (respectively, delete) a line to (respectively, from) a free line arrangement, then the resulting line arrangement is either free or plus-one generated. In this paper, we prove that the same properties hold when we replace the line arrangement by a free curve and add (respectively, delete) a line. The proof uses a new version of a key result due originally to H. Schenck, H. Terao and M. Yoshinaga, in which no quasi-homogeneity assumption is needed. Two conjectures about the Tjurina number of a union of two plane curve singularities are also stated. As a geometric application, we show that, under a mild numerical condition, the projective closure of a contractible, irreducible affine plane curve is either free or plus-one generated, using a deep result due to U. Walther.

SED-RCNN-BE: A SE-Dual channel RCNN network optimized binocular estimation model for automatic size estimation of free swimming fish in aquaculture

Size of fish body is one of the most important morphological characteristics for cultured fish and plays a key role to enable effective management in aquaculture. Automatic, accurate and real-time monitoring of fish body size is therefore essential for improving aquaculture production. Computer vision facilitated the non-contact estimation of fish size and reduce labor consumption. While, current computer vision estimation techniques are still restricted in multi-scale size and multi-pose-variant of free-swimming fish, which limit the relevance application to aquaculture. Hence, we propose a novel method to facilitate automatic and accurate size estimation of free-swimming fish for growth monitoring of fish in aquaculture. This method comprised of an improved keypoint detection module based on Keypoint RCNN (SE-D-KP-RCNN) network and size estimation module based on CREStereo matching and spatial plane curve fitting. In the detection module, a squeeze and excitation Dual channel pooling layer (SE-D) is designed for the backbone network to improve the keypoint detection accuracy for free-swimming fish. In the estimation module, the global parallax map of binocular image is produced by CREStereo matching to obtain the 3D information of detection keypoint and build curve fitting of multi-pose variant fish body for height and length estimation. We implement and test our proposed method in different sizes and poses fish and the comprehensive experimental results show that the average accuracy of size estimation can reached 93.18% in the real aquaculture scene. This is important to help breeders to obtain underwater fish size information accurately and improve the efficiency for aquaculture management.