Plane Curves Research Articles

Limits of conformal images and conformal images of limits for planar random curves

We address the scaling limits of random curves arising from, e.g., planar lattice models, especially in rough domains. The well-known precompactness conditions of Kemppainen and Smirnov (2017) show that certain crossing probability estimates guarantee the subsequential weak convergence of the random curves in the topology of unparametrized curves, as well as in a topology inherited from the unit disc via conformal maps. We complement this result by proving that proceeding to weak limit commutes with changing topology, i.e., limits of conformal images are conformal images of limits, with minimal boundary regularity assumptions on the domains where the random curves lie. Such rough boundaries are especially interesting if, in the context of multiple random curves, a limit candidate is defined in terms of iterated SLE-type processes with \kappa \in (4, 8) , and one hence needs to study (boundary-touching) curves in domains slit by other random curves.

Online measurement of birefringence of sensing fibers in the AFOCT system

AbstractThis paper presents a novel scheme to online measure the birefringence of the sensing fiber in all‐fiber optic current transformer (AFOCT) system, which monitors the current on direct current (DC) transmission lines using the Faraday magneto‐optic effect in sensing fibers. In this scheme, by purposefully varying the difference of square‐wave modulated phases via the phase modulator in the AFOCT system, different system transmittances can be obtained. On the basis of the relationship between the system transmittance and linear and circular birefringences, an online measurement scheme of both the linear and circular birefringences of the sensing fibers can be established. In this scheme, the AFOCT system is described by the transmission model based on the Jones matrices, with which the current on DC transmission lines can be measured from the phase shift caused by the Faraday effect. Then, under different square‐wave modulated phases, the system transmittance models are constructed with respect to the linear and circular birefringences of the sensing fiber. For a given sensing fiber with certain linear and circular birefringences, the specific system transmittances are determined and are the curves in space in the respective system transmittance models. Subsequently, these curves in space can be projected onto the coordinate plane composed of the linear and circular birefringence coordinate axes, where the intersection point of the projected plane curves gives the to‐be‐measured values of the linear and circular birefringences of the sensing fiber. The proposed scheme is validated by the preset linear and circular birefringences of the sensing fiber, with different linear birefringences of the polarization‐maintaining fiber. The measurement results show the values consistent with the preset ones.

Anisotropic area-preserving nonlocal flow for closed convex plane curves

Abstract We consider an anisotropic area-preserving nonlocal flow for closed convex plane curves, which is a generalization of the model introduced by Pan and Yang (J. Differential Equations 266 (2019), 3764–3786) when τ = 1. Under this flow, the evolving curve maintains its convexity and converges to a homothety of a smooth symmetric strictly convex plane curve in the C ∞ sense. The analysis of the asymptotic behavior of this flow implies the possibility of deforming one curve into another within the framework of Minkowski geometry.

Learning curve of ultrasound-guided surgeon-administered transversus abdominis plane (UGSA-TAP) block on a porcine model

BackgroundSurgeons commonly perform ultrasound-guided Transversus Abdominis Plane blocks to manage acute pain following abdominal surgeries. There is no consensus on whether surgeons should undergo basic hands-on training to perform TAP blocks or if video-based learning is sufficient. We theorized that simulation-based learning is superior to video-based learning. In the present study, we present the analysis of technical skills of UGSA-TAP block performance on a live porcine model by general surgery trainees after undergoing video or simulation-based learning. MethodsWe performed a prospective, double-blinded, randomized study. Ten surgery residents and two surgery critical-care fellows (n = 12) without prior experience in performing the TAP block were recruited. The participants were randomized either into a video-based or simulation-based training group. After that, all participants performed a TAP block on a live anesthetized pig, which was recorded and scored by three blinded anesthesiologists. All participants completed a post-performance survey to assess their confidence in gaining competency in the UGSA-TAP block. Statistical analyses were performed to assess the differences between the two groups. P < 0.05 was considered statistically significant. ResultsAll simulation-based learning participants successfully performed a survey scan, identified the three muscular layers of the abdominal wall, and identified the transversus abdominis plane compared to 50 %, 50 %, and 33 % video-based learning group participants for the respective parameters (p < 0.05). While some performance metrics showed no statistically significant differences between the groups, substantial effect sizes (Cohen's ℎ up to 1.07) highlighted notable differences in participants' performance. Both groups exhibited confidence in core competencies, with varied rates of satisfactory skill execution. Performance assessed using a global rating scale revealed a higher passing rate for the simulation group (83 % vs. 33 %). Participant feedback via the Likert scale reflected confidence post-training. Inter-rater reliability (0.83–1) confirmed the robustness of study evaluations. ConclusionThe UGSA-TAP block curriculum should be introduced into the surgical residency programs with an emphasis on simulation-based learning to enhance the procedural skills of the trainees before transitioning to surgical patients.

Minimization of hypersurfaces

Let F ∈ Z [ x 0 , … , x n ] F \in \mathbb {Z}[x_0, \ldots , x_n] be homogeneous of degree d d and assume that F F is not a ‘nullform’, i.e., there is an invariant I I of forms of degree d d in n + 1 n+1 variables such that I ( F ) ≠ 0 I(F) \neq 0 . Equivalently, F F is semistable in the sense of Geometric Invariant Theory. Minimizing F F at a prime p p means to produce T ∈ M a t ( n + 1 , Z ) ∩ G L ( n + 1 , Q ) T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q}) and e ∈ Z ≥ 0 e \in \mathbb {Z}_{\ge 0} such that F 1 = p − e F ( [ x 0 , … , x n ] ⋅ T ) F_1 = p^{-e} F([x_0, \ldots , x_n] \cdot T) has integral coefficients and v p ( I ( F 1 ) ) v_p(I(F_1)) is minimal among all such F 1 F_1 . Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors w ∈ Z ≥ 0 n + 1 w \in \mathbb {Z}_{\ge 0}^{n+1} to F F . We show that for any dimension n n and degree d d , there is a complete set of weight vectors consisting of [ 0 , w 1 , w 2 , … , w n ] [0,w_1,w_2,\dots ,w_n] with 0 ≤ w 1 ≤ w 2 ≤ ⋯ ≤ w n ≤ 2 n d n − 1 0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1} . When n = 2 n = 2 , we improve the bound to d d . This answers a question raised by Kollár. These results are valid in a more general context, replacing Z \mathbb {Z} and p p by a PID R R and a prime element of R R . Based on this result and a further study of the minimization process in the planar case n = 2 n = 2 , we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree d d . We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma.

Detecting similarities of rational plane curves using complex differential invariants

We present a new and efficient method to detect whether or not two given rational plane curves are similar. If both curves are the same, the method finds the symmetries of the curve. The method relies on the introduction of a complex differential invariant that has a nice behavior with respect to Möbius transformations, which are the mappings lying behind the similarities in the parameter space. From a computational point of view, our algorithm only requires bivariate gcds and factoring. The algorithm is implemented in Maple™ [Maplesoft, a division of Waterloo Maple Inc. Waterloo, Ontario (2021)]. An extensive study of its practical performance is also provided.

Evaluating the index of panoramic X-ray image quality using K-means clustering method

BackgroundA panoramic X-ray image is generally considered optimal when the occlusal plane is slightly arched, presenting with a gentle curve. However, the ideal angle of the occlusal plane has not been determined. This study provides a simple evaluation index for panoramic X-ray image quality, built using various image and cluster analyzes, which can be used as a training tool for radiological technologists and as a reference for image quality improvement.ResultsA reference panoramic X-ray image was acquired using a phantom with the Frankfurt plane positioned horizontally, centered in the middle, and frontal plane centered on the canine teeth. Other images with positioning errors were acquired with anteroposterior shifts, vertical rotations of the Frankfurt plane, and horizontal left/right rotations. The reference and positioning-error images were evaluated with the cross-correlation coefficients for the occlusal plane profile, left/right angle difference, peak signal-to-noise ratio (PSNR), and deformation vector fields (DVF). The results of the image analyzes were scored for positioning-error images using K-means clustering analysis. Next, we analyzed the correlations between the total score, cross-correlation analysis of the occlusal plane curves, left/right angle difference, PSNR, and DVF. In the scoring, the positioning-error images with the highest quality were the ones with posterior shifts of 1 mm. In the analysis of the correlations between each pair of results, the strongest correlations (r = 0.7–0.9) were between all combinations of PSNR, DVF, and total score.ConclusionsThe scoring of positioning-error images using K-means clustering analysis is a valid evaluation indicator of correct patient positioning for technologists in training.

Reconstruction of a Human Spine Curve in the Sagittal Plane From the Measured Contour of the Human Back

Reconstruction of a Human Spine Curve in the Sagittal Plane From the Measured Contour of the Human Back

REDUCIBILITY OF MULTIPERIODIC LINEAR SYSTEMS WITH A DIAGONAL DIFFERENTIATION OPERATOR AND ITS APPLICATION TO CONDITIONALLY PERIODIC SYSTEMS

Abstract. The main theorem is proven by substantiating the reducibility of an equivalent multiperiodic linear system, with a differentiation operator in the direction of diagonal of independent variable space. It is shown that helical lines with an elevation angle of  / 4, when applied to a P-circular cylindrical surface, are  -periodic characteristics of the differentiation operator.The reducibility of a multiperiodic linear system is investigated near the helix line which starts from the initial point of the phase circle, following the scheme of substantiating the reducibility of linear periodic systems. The concept of a monodromy matrix is introduced, which is constant along the first integrals of the characteristic equations of the differentiation operator and has the properties of multiperiodicity and smoothness. The existence of localised positive eigenvalues with all the properties of the monodromy matrix is proved. It is assumed that the monodromy matrix accepts the maximum number of different eigenvalues at the initial point of the phase circle compared to other points. In the neighbourhood of these eigenvalues, the localisation of eigenvalues of the monodromy matrix, defined on a cylindrical surface, is realized in its entirety. The Gershgorin method is employed for this purpose. The non-singularity of the monodromy matrix allows the existence of a curve of the complex plane that does not cover zero and surrounds the spectrum of the monodromy matrix. This proves the existence of the logarithm of the monodromy matrix, and therefore the reducibility of a multiperiodic linear system. Hence, a proof of the main theorem for a conditionally periodic system generating a multiperiodic system is obtained as a consequence.

DETERMINING THE ACCELERATIONS OF POINTS OF A PLANAR EIGHT-LINK THIRD-CLASS MECHANISM USING GRAPH-ANALYTICAL METHOD

A characteristic feature of technological machines used in the fashion industry is the complexity of motion of the working links, both in terms of geometry and motion laws. Such technological motions are provided by complex mechanisms, the basis of which consists of structural groups of third-class and higher-class links. In multi-link structural groups of higher classes, complex motions of certain points along trajectories resembling planar linkage curves of various geometric shapes are observed. If the centers of external kinematic pairs of another structural group, which is attached to the previous group of links, are placed at such points, theoretically, any trajectory of the working point of the machine with the necessary motion laws required for performing a technological operation can be obtained. In practical application, the use of higher-class structural groups within the kinematic scheme of a planar mechanism leads to complexities in its further investigation. This is explained by the necessity of conducting studies using specially developed methods for their implementation. In cases where these methods cannot be applied, there arises the need for individual development of a sequence for conducting such studies in each specific case of such complex mechanisms, taking into account their structural characteristics. A sequence of actions has been developed and a kinematic analysis of an eight-link third-class mechanism has been conducted using a graph-analytical method. Acceleration plan has been graphically constructed, and the magnitudes and directions of angular accelerations of the mechanism’s links have been calculated. The selection of a conditionally different possible initial mechanism allowed transforming the eight-link third-class mechanism with two sequentially attached structural groups of links into a mechanism with sequentially-parallel attachment of second-order second-class link groups and performing the analysis of the third-class mechanism in a manner characteristic of the analysis of second-class mechanisms. The proposed method of analyzing higher-class mechanisms may be useful for conducting similar studies.Keywords: research of mechanism, analysis of mechanism, kinematic investigation, kinematic analysis, angular acceleration, linear acceleration vector, acceleration plan

On the embedded Nash problem

Abstract The embedded Nash problem for a hypersurface in a smooth algebraic variety is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal models of the pair contribute with such families. We solve the problem for unibranch plane curve germs, in terms of the resolution graph. These are embedded analogs of known results for the classical Nash problem on singular varieties.

Curvature diffusion of planar curves with generalised Neumann boundary conditions inside cones

Curvature diffusion of planar curves with generalised Neumann boundary conditions inside cones

Singularities of swept surfaces in Euclidean 3-space

&lt;p&gt;This study examines the local singularities of tube surfaces, especially those of swept surfaces $ (SS) $ in Euclidean 3-space $ \mathcal{E}^{3} $. $ SS $ is created by moving a planar curve through space such that the trajectory of any point on the surface remains perpendicular to the plane. The Type-2 Bishop frame is considered, and the singularities of these $ SS $ are analyzed. Examples are offered and illustrated.&lt;/p&gt;

POINTWISE 1-TYPE GAUSS MAP OF DEVELOPABLE SMARANDACHE RULED SURFACES IN $\mathbb{E}^{3}$

In this paper, we study developable TN, TB, and NB-Smarandache ruled surface with pointwise 1-type Gauss map. In particular, we obtain every such developable TN-Smarandache ruled surface has constant mean curvature, developable TB Smarandache ruled surface is minimal if and only if the curve is a planar curve or helix, and developable NB-Smarandache ruled surface is always plane. Also, we give some examples.

The adjoint map of Euclidean plane curves and curvature problems

The adjoint map of a pair of naturally parametrized curves in the Euclidean plane is studied from the point of view of the curvature. A main interest is when the given curve and its adjoint curve share the same natural parameter and the same curvature. For the general linear second order differential equation we introduce a function expressing the deformation of curvatures induced by the adjoint map.

Algebraic approach to the completeness problem for (k,n)-arcs in planes over finite fields

In a projective plane over a finite field, complete (k,n)-arcs with few characters are rare but interesting objects with several applications to finite geometry and coding theory. Since almost all known examples are large, the construction of small ones, with k close to the order of the plane, is considered a hard problem. A natural candidate to be a small (k,n)-arc with few characters is the set Ω(C) of the points of a plane curve C of degree n (containing no linear components) such that some line meets C transversally in the plane, i.e. in n pairwise distinct points. Let C be either the Hermitian curve of degree q+1 in PG(2,q2r) with r≥1, or the rational BKS curve of degree q+1 in PG(2,qr) with q odd and r≥1. Then Ω(C) has four and seven characters, respectively. Furthermore, Ω(C) is small as both curves are either maximal or minimal. The completeness problem is investigated by an algebraic approach based on Galois theory and on the Hasse-Weil lower bound. Our main result for the Hermitian case is that Ω(C) is complete for r≥4. For the rational BKS curve, Ω(C) is complete if and only if r is even. If r is odd then the uncovered points by the (q+1)-secants to Ω(C) are exactly the points in PG(2,q) not lying in Ω(C). Adding those points to Ω(C) produces a complete (k,q+1)-arc in PG(2,qr), with k=qr+q. The above results do not hold true for r=2 and there remain open the case r=3 for the Hermitian curve, and the cases r=3,4 for the rational BKS curve. As a by product we also obtain two results of interest in the study of the Galois inverse problem for PGL(2,q).

New algebraic and geometric characterizations of planar quintic Pythagorean-hodograph curves

The aim of this work is to provide new characterizations of planar quintic Pythagorean-hodograph curves. The first two are algebraic and consist of two and three equations, respectively, in terms of the edges of the Bézier control polygon as complex numbers. These equations are symmetric with respect to the edge indices and cover curves with generic as well as degenerate control polygons. The last two characterizations are geometric and rely both on just two auxiliary points outside the control polygon. One requires two (possibly degenerate) quadrilaterals to be similar, and the other highlights two families of three similar triangles. All characterizations are a step forward with respect to the state of the art, and they can be linked to the well-established counterparts for planar cubic Pythagorean-hodograph curves. The key ingredient for proving the aforementioned results is a novel general expression for the hodograph of the curve.

From Space to Maths and to Arts: Virtual Art in Space with Planetary Orbits

Relying on the students' cultural background to teach mathematics may be a strong incitement for the learning process. This background can include artistic creations, items from the news, among others. Because of the strong presence of space related news (in particular the almost simultaneous launching of three spacecrafts towards Mars), we present activities in a technology-rich environment, using both a Dynamic Geometry System and a Computer Algebra System. We start from a simplified model of planetary orbits and building new knowledge on plane curves. Interesting byproducts are the discovery of the strong influence of changing numerical approximations, non – trivial aspects of the study of periodicity, the plotting of interesting curves and their connections with existing pieces of visual art, including architecture and mandalas. The educative aspect of these activities is also discussed and is presented from the point of view of STEAM Education.

Applications of Differential Geometry of Curves in Roads Design

Abstract An important role in the roads design is played by plane curves, if the road lies on a plane surface and by space curves, if the road can not be embeded in a plane. This article presents some work done in the two and three dimensional roads design, by using the differential geometry methods. A study of a general case for plane roads is given.

Frobenius methods for analytic second order linear partial differential equations

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).