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The article discusses new kinematic surfaces that can be attributed to the class of surfaces of congruent cross sections. The surfaces of congruent cross sections were first identified in a separate class by Professor I.I. Kotov. Circular, elliptical and parabolic cylinders are taken as the guiding surfaces, and circles and parabolas are taken as generating plane curves, which can be located in the plane of the generating curve of the guiding cylinder or in a plane parallel to its longitudinal axis. The introduction of a new independent parameter helped to solve the set geometric problems. The analytical formulas are presented in generalized form, so the shape of the flat generatrix curve can be arbitrary. Two types of surfaces are considered: 1) when the local axes of the generating curves remain parallel during their movement; 2) when these axes rotate. The resulting surfaces can be of interest to architects, or can find application in machine-building thin-walled structures or in the study of the trajectories of bodies during their oscillatory-translational motion.

We study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be tropically planar, and compute the number of tropically planar graphs up to genus $7$. We provide non-trivial upper and lower bounds on the number of tropically planar graphs, and prove that asymptotically $0\%$ of connected trivalent planar graphs are tropically planar., Comment: 28 pages, 29 figures; updated to correct erroneous counts at the end of Section 3

We investigate the problem of classifying pencils of plane curves of degree d up to projective equivalence. We obtain explicit stability criteria in terms of the log canonical threshold by relating the stability of a pencil to the stability of the curves lying on it.

At every point, a smooth plane curve can be approximated, to second order, by a circle; this circle is called osculating. One may think of the osculating circle as passing through three infinitesimally close points of the curve. A vertex of the curve is a point at which the osculating circle hyper-osculates: it approximates the curve to third order. Equivalently, a vertex is a critical point of the curvature function. Consider a (necessarily non-closed) curve, free from vertices. The classical Tait-Kneser theorem [13, 5] (see also [3, 10]), states that the osculating circles of the curve are pairwise disjoint, see Figure 1. This theorem is closely related to the four vertex theorem of S. Mukhopadhyaya [8] that a plane oval has at least 4 vertices (see again [3, 10]). Figure 1 illustrates the Tait-Kneser theorem: it shows an annulus foliated by osculating circles of a curve.

This paper introduces a geometric generalization of signed distance fields for plane curves. We propose to store simplified geometric proxies to the curve at every sample. These proxies are constructed based on the differential geometric quantities of the represented curve and are used for queries such as closest point and distance calculations. We investigate the theoretical approximation order of these constructs and provide empirical comparisons between geometric and algebraic distance fields of higher order. We apply our results to font representation and rendering.

The implementation of tasks related to the development of the transportation network as a whole and logging roads as an integral part of it requires scientifically based theoretical studies of the patterns of formation of spatial curves when combining elements of the plan and the longitudinal profile, since the rational laying of the route for many years determines its most important transport and operational characteristics (speed, traffic safety, traffic capacity). Consideration of the visual perception of the road by the driver will improve the quality of design decisions, which will allow to avoid emergencies in the future after setting the route into service. On the other hand, a decrease in speed before seemingly sharp turns of the road affects the efficiency of logging road transport. Therefore, the view of the road ahead should strongly orient the driver, i.e. be visually clear and clearly changing, ensuring the constancy or smooth reduction of the traffic flow mode. At the same time, the need for a successful spatial solution of the road increases. In the designs of logging roads, straight lines, transition curves, described in recent years most often according to the clotoid, and circular curves are found as elements of the route plan. It is found that the road view in perspective correctly orients the driver of the car, i.e. it is visually clear, provided that the lines describing the edges of the roadway and the edges of the roadway in the perspective image are curved in the same direction as in the road plan. The purpose of the work is to determine a set of quantitative indicators (curvature, radius of the curve in the plan, maximum curvature, maximum rate of change of curvature) for optimization of the visual smoothness and clarity of the central projections of elementary spatial and plane curves. The performed studies allow us to fully characterize the visual smoothness and clarity of the central projections of elementary spatial and plane curves. The above algorithm makes it possible to compile a computer program to determine the mentioned indicators. The indicators determined by this algorithm allow us to evaluate both the visual smoothness and clarity of curves on logging roads. For citation: Borovlev A.O., Skrypnikov A.V., Kozlov V.G., Tyurikova T.V., Tveritnev O.N., Nikitin V.V. Mathematical Modeling of the Route of Logging Roads. Lesnoy Zhurnal [Russian Forestry Journal], 2021, no. 4, pp. 150−161. DOI: 10.37482/0536-1036-2021-4-150-161.

A point p in projective plane is said to be quasi-Galois for a plane curve if the curve admits a non-trivial birational transformation which preserves the fibers of the projection $$\pi _p$$ from the point p. A number of quasi-Galois points for smooth plane curves of degree d are studied. In this paper, we will study the relationship between dihedral groups and smooth plane curves with many quasi-Galois points.

Any smooth tropical plane curve contains a distinguished trivalent graph called its skeleton. In 2020 Morrison and Tewari proved that the so-called big face graphs cannot be the skeleta of tropical curves for genus $12$ and greater. In this paper we answer an open question they posed to extend their result to the prism graphs, proving that they are the skeleton of a smooth tropical plane curve precisely when the genus is at most $11$. Our main tool is a classification of lattice polygons with two points than can simultaneously view all others, without having any one point that can observe all others., 11 pages, 13 figures

In previous studies, particles, such as ellipsoids and super-ellipsoids, are mainly described by implicit functions. However, parametric functions can define more surfaces that can be used to represent a wider variety of particle shapes. In this study, parametric functions are used to construct particles, and the algorithm to determine the multi-point contact between concave particles is also given. Furthermore, the geometric parameters at the contact point, for instance, normal vectors, curvatures, and overlapping, are formulated by the parameters in the parametric function. To verify the proposed method, we use the discrete element method to simulate the systems of torus-shaped particles nested within each other, and analyze the momentum and kinetic energy changes with time. The equilibrium state with approximately zero energy is obtained, which means that the algorithm for multi-point contact of concave particles is suitable and stable. In addition to the torus-shaped particle, another two concave particles defined by parametric function are modeled. The simulation results indicate that the method is universal. Any plane curve can be used to construct surfaces by using the method provided in this work, and computational efficiency of simulations of particles defined by parametric function is higher because calculations of the inverse Jacobian matrix in the Newton-Raphson method are unnecessary. The parametric function method extends the scope of previous studies on particle shape.

In the present paper we describe the family of all closed convex plane curves of class $$C^1$$ C 1 which have circles as their isoptics. In the first part of the paper we give a certain characterization of all ellipses based on the notion of isoptic and we give a geometric characterization of curves whose orthoptics are circles. The second part of the paper contains considerations on curves which have circles as their isoptics and we show the form of support functions of all considered curves.

First we give a sharp upper bound for the cardinal m of a minimal set of generators for the module of Jacobian syzygies of a complex projective reduced plane curve C. Next we discuss the sharpness of an upper bound, given by A. du Plessis and C.T.C. Wall, for the global Tjurina number of such a curve C, in terms of its degree d and of the minimal degree $$r\le d-1$$ of a Jacobian syzygy. We give a homological characterization of the curves whose global Tjurina number equals the du Plessis-Wall upper bound, which implies in particular that for such curves the upper bound for m is also attained. A second characterization of these curves in terms of the 0-th Fitting ideal of their Jacobian module is also given. Finally we prove the existence of curves with maximal global Tjurina numbers for certain pairs (d, r). We conjecture that such curves exist for any pair (d, r), and that, in addition, they may be chosen to be line arrangements when $$r\le d-2$$ . This conjecture is proved for degrees $$d \le 11$$ .

The Nagata Conjecture is one of the most intriguing open problems in the area of curves in the plane. It is easily stated. Namely, it predicts that the smallest degree d of a plane curve passing through r ≥ 10 general points in the projective plane P 2 with multiplicities at least l at every point, satisfies the inequality d > √ r · l. This conjecture has been proven by M. Nagata in 1959, if r is a perfect square greater than 9. Up to now, it remains open for every non-square r ≥ 10, after more than a half century of attention by many researchers. In this paper, we formulate new transcendental versions of this conjecture coming from pluripotential theory and which are equivalent to a version in C n of the Nagata Conjecture.

For a fixed integer $$d\ge 4$$ d ≥ 4 , the list of groups that appear as automorphism groups of smooth plane curves whose degree is d is unknown, except for $$d=4$$ d = 4 or 5. Harui showed a certain characteristic about structures of automorphism groups of smooth plane curves. Badr and Bars began to study for certain orders of automorphisms and try to obtain exact structures of automorphism groups of smooth plane curves. In this paper, based on the result of T. Harui, we extend Badr–Bars study for different and new cases, mainly for the cases of cyclic groups that appear as automorphism groups.

Several non equivalent definitions exist for the envelope of a 1-parameter family of plane curves. Another notion, often considered as related to envelopes, is the offset at a given distance of a plane curve. Using the so-called analytic definition, we study and compare the envelope of a 1-parameter family of circles centered on a parabola and an offset of this parabola. Then we perform a similar study for an astroid. This illustrates the non equivalence of the two notions. The work is performed with networking with a Computer Algebra System and a Dynamical Geometry System, bringing them into a certain form of dialog. The obtained curves are sextics, and their discovery as envelopes leads to an automated exploration of Talbot curves. Another output shows these curves within a unifying framework involving a Maltese Cross.

In this paper, we study the families of relatives of pedals and evolutes in the Minkowski spacetime plane R 2 1 . We obtain some relationships between these families which turn out to be different from Euclidean plane. Also, we classify and generalize these notions to the category of frontal curves in R 2 1 . Finally, some computational examples in support of our main results are given and plotted.

Given an oval C in the plane, the $$\alpha $$ α -isoptic $$C_\alpha $$ C α of C is the plane curve composed of the points from which C can be seen under the angle $$\pi -\alpha $$ π - α . We consider isoptics of ovals parametrized with the support function $$p(t)=a+\cos n t$$ p ( t ) = a + cos n t , $$n\in \mathbb {N}$$ n ∈ N , and present an example of an oval such that when $$\alpha $$ α increases, the $$\alpha $$ α -isoptics begin to be convex, then lose their convexity and finally are convex again along a curve intersecting the isoptics orthogonally. Next we give an example of a curve from the same family, for which the curvature of the isoptics changes its sign three times. These changes occur on the symmetry axes of the oval C and coincide with the orthogonal trajectories which start at the points with extremal curvature. Finally, we formulate the hypothesis concerning the general case where we expect $$n-1$$ n - 1 convexity limit angles for the isoptics of an oval parametrized by $$p(t)=a+\cos n t$$ p ( t ) = a + cos n t .

By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property (also known as the strong evolution property) if and only if the foliation comes from a Liouville net. A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net. Let us say that a strictly geodesically convex curve on a Riemannian surface has the Poritsky property if it can be parametrized in such a way that all of its string diffeomorphisms are shifts with respect to this parameter. In 1950, Poritsky has shown that the only closed plane curves with this property are ellipses. In the present article we show that a curve on a Riemannian surface has the Poritsky property if and only if it is a coordinate curve of a Liouville net. We also recall Blaschke’s derivation of the Liouville property from the Ivory property and his proof of Weihnacht’s theorem: the only Liouville nets in the plane are nets of confocal conics and their degenerations. This suggests the following generalization of Birkhoff’s conjecture: If an interior neighborhood of a closed strictly geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is one of the coordinate lines.

We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called $C_{ab}$ curves, as well as algorithms for inverting the encoding map, which we call “unencoding”. Some $C_{ab}$ curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity $ \tilde { \mathcal {O}}(\text {n}^{3/2})$ resp. $ \tilde { \mathcal {O}}({\it\text { qn}})$ for AG codes over any $C_{ab}$ curve satisfying very mild assumptions, where n is the code length and q the base field size, and $ \tilde { \mathcal {O}}$ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity $ \tilde { \mathcal {O}}(\text {n})$ for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding and unencoding algorithms have complexities $ \tilde { \mathcal {O}}(\text {n}^{5/4})$ and $ \tilde { \mathcal {O}}(\text {n}^{3/2})$ respectively.

Several non-local curvature flows for plane curves with a general rotation number are discussed in this work. The types of flows include the area-preserving flow and the length-preserving flow. We have a relatively good understanding of these flows for plane curves with the rotation number one. In particular, when the initial curve is strictly convex, the flow exists globally in time, and converges to a circle as time tends to infinity. Even if the initial curve is not strictly convex, a global solution, if it exists, converges to a circle. Here, we deal with curves with a general rotation number, and show, not only a similar result for global solutions, but also a blow-up criterion, upper estimates of the blow-up time, and blow-up rate from below. For this purpose, we use a geometric quantity which has never been considered before., Comment: 24 pages

In this article we give an expression of the motivic Milnor fiber at infinity and the motivic nearby cycles at infinity of a polynomial f in two variables with coefficients in an algebraic closed field of characteristic zero. This expression is given in terms of some motives associated to the faces of the Newton polygons appearing in the Newton algorithm at infinity of f without any condition of convenience or non degeneracy. In the complex setting, we compute the Euler characteristic of the generic fiber of f in terms of the area of the surfaces associated to faces of the Newton polygons. Furthermore, if f has isolated singularities, we compute similarly the classical invariants at infinity $$\lambda _{c}(f)$$ which measures the non equisingularity at infinity of the fibers of f in $${\mathbb {P}}^2$$ , and we prove the equality between the topological and the motivic bifurcation sets and give an algorithm to compute them.

We consider the parabolic polyharmonic diffusion and the L 2 {L^{2}} -gradient flow for the square integral of the m-th arclength derivative of curvature for regular closed curves evolving with generalised Neumann boundary conditions. In the polyharmonic case, we prove that if the curvature of the initial curve is small in L 2 {L^{2}} , then the evolving curve converges exponentially in the C ∞ {C^{\infty}} topology to a straight horizontal line segment. The same behaviour is shown for the L 2 {L^{2}} -gradient flow provided the energy of the initial curve is sufficiently small. In each case the smallness conditions depend only on m.

In this paper we give a positive answer to a question of Dimca and Greuel about the quotient between the Milnor and the Tjurina numbers for any irreducible germ of plane curve singularity. This result is based on a closed formula for the minimal Tjurina number of an equisingularity class in terms of the sequence of multiplicities of the strict transform along a resolution. The key points for the proof are previous results by Genzmer [6], and by Wall and Mattei [11, 13]., The first and third authors were supported by Spanish Ministerio de Ciencia, Innovación y Universidades MTM2015-69135-P, Generalitat de Catalunya 2017SGR-932 projects, and they are with the Barcelona Graduate School of Mathematics (BGSMath), through the project MDM-2014-0445. The second and fourth authors were supported by Spanish Ministerio de Ciencia, Innovación y Universidades MTM2016-76868-C2-1-P

Let C be an irreducible plane curve of PG ( 2 , K ) where K is an algebraically closed field of characteristic p ≥ 0 . A point Q ∈ C is an inner Galois point for C if the projection π Q from Q is Galois. Assume that C has two different inner Galois points Q 1 and Q 2 , both simple. Let G 1 and G 2 be the respective Galois groups. Under the assumption that G i fixes Q i , for i = 1 , 2 , we provide a complete classification of G = 〈 G 1 , G 2 〉 and we exhibit a curve for each such G. Our proof relies on deeper results from group theory.

In this note we investigate the behavior of the polar tangential angle of a general plane curve, and in particular prove its monotonicity for certain curves of monotone curvature. As an application we give (non)existence results for an obstacle problem involving free elasticae., 11 pages, 5 figures

In this paper, we present a mathematical model and numerical method designed for the segmentation of satellite images, namely to obtain in an automated way borders of Natura 2000 habitats from Sentinel-2 optical data. The segmentation model is based on the evolving closed plane curve approach in the Lagrangian formulation including the efficient treatment of topological changes. The model contains the term expanding the curve in its outer normal direction up to the region of habitat boundary edges, the term attracting the curve accurately to the edges and the smoothing term given by the influence of local curvature. For the numerical solution, we use the flowing finite volume method discretizing the arising advection-diffusion intrinsic partial differential equation including the asymptotically uniform tangential redistribution of curve grid points. We present segmentation results for satellite data from a selected area of Western Slovakia (Zahorie) where the so-called riparian forests represent the important European Natura 2000 habitat. The automatic segmentation results are compared with the semi-automatic segmentation performed by the botany expert and with the GPS tracks obtained in the field. The comparisons show the ability of our numerical model to segment the habitat areas with the accuracy comparable to the pixel resolution of the Sentinel-2 optical data.

In this paper the author devotes to studying a logarithmic type nonlocal plane curve flow. Along this flow, the convexity of evolving curve is preserved, the perimeter decreases, while the enclosed area expands. The flow is proved to exist globally and converge to a finite circle in the C∞ metric as time goes to infinity.

In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of $n\times n$ Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is $2n-3$., Comment: v1 14 pages; v2 19 pages

By a length-preserving flow, we provide a new proof of a conjecture on the reverse isoperimetric inequality composed by Pan et al. (Math Inequal Appl 13:329–338, 2010), which states that if $$\gamma $$ is a convex curve with length L and enclosed area A, then the best constant $$\varepsilon $$ in the inequality $$\begin{aligned} L^2\le 4\pi A+\varepsilon |{\tilde{A}}| \end{aligned}$$ is $$\pi $$ , where $${\tilde{A}}$$ denotes the oriented area of the locus of its curvature centers.

In this paper, new plane curves (multidirectorial curves as counterpart to multifocal curves; and focal-directorial curves as curves of a transitory type) are considered through geometric genesis, and their form variation for special choices of foci and directrices disposition. Furthermore, the form variation of Weberian focal-directorial curves (WFDC) for different values of accompanying parameter S and Weberian coefficients is considered.

We construct a topological invariant of algebraic plane curves, which is in some sense an adaptation of the linking number of knot theory. This invariant is shown to be a generalization of the I-invariant of line arrangements developed by the first author with Artal and Florens. We give two practical tools for computing this invariant, using a modification of the usual braid monodromy or using the connected numbers introduced by Shirane. As an application, we show that this invariant distinguishes several Zariski pairs, i.e. pairs of curves having same combinatorics, yet different topologies. The former is the well known Zariski pair found by Artal, composed of a smooth cubic with 3 tangent lines at its inflexion points. The latter is formed by a smooth quartic and 3 bitangents., Comment: Entirely new version. Exposition revised and simplified; new application added. (10 pages and 2 figures)

This paper deals with a 1 / κ α -type length-preserving nonlocal flow of convex closed plane curves for all α > 0 . Under this flow, the convexity of the evolving curve is preserved. For a global flow, it is shown that the evolving curve converges smoothly to a circle as t → ∞ . Some numerical blow-up examples and a sufficient condition leading to the global existence of the flow are also constructed.

Tropical curves in $${\mathbb {R}}^2$$ R 2 correspond to metric planar graphs but not all planar graphs arise in this way. We describe several new classes of graphs which cannot occur. For instance, this yields a full combinatorial characterization of the tropically planar graphs of genus at most five.

We consider the traveling wave solutions of the degenerate nonlinear parabolic equation $$u_{t}=u^{p}(u_{xx}+u)$$ which arises in the model of heat combustion, solar flares in astrophysics, plane curve evolution problems and the resistive diffusion of a force-free magnetic field in a plasma confined between two walls. We also deal with the equation $$v_{\tau }=v^{p}(v_{xx}+v-v^{-p+1})$$ related with it. We first give a result on the whole dynamics on the phase space $${\mathbb {R}}^{2}$$ with including infinity about two-dimensional ordinary differential equation that introduced the traveling wave coordinates: $$\xi = x-ct$$ by applying the Poincare compactification and dynamical system approach. Second, we focus on the connecting orbits on it and give a result on the existence of the weak traveling wave solutions with quenching for $$c>0$$ and $$p\in 2{\mathbb {N}}$$ . Moreover, we give the detailed information about the asymptotic behavior of $$u(\xi )$$ , $$u'(\xi )$$ , $$v(\xi )$$ and $$v'(\xi )$$ for $$p\in 2{\mathbb {N}}$$ . In the case that $$p \in 2 {\mathbb {N}}+1$$ , it is too complicated to determine the dynamics near the singularities on the Poincare disk, however, we classify the connecting orbits and corresponding traveling wave solutions and obtain their asymptotic behavior.

We study the arithmetic (geometric) progressions in the $x$-coordinates of quadratic points on smooth projective planar curves defined over a number field $k$. Unless the curve is hyperelliptic, we prove that these progressions must be finite. We, moreover, show that the arithmetic gonality of the curve determines the infinitude of these progressions in the set of $\overline{k}$-points with field of definition of degree at most $n$, $n\ge 3$.

This paper presents a new algorithm for computing the topology of an algebraic space curve. Based on an efficient weak generic position-checking method and a method for solving bivariate polynomial systems, the authors give a first deterministic and efficient algorithm to compute the topology of an algebraic space curve. Compared to extant methods, the new algorithm is efficient for two reasons. The bit size of the coefficients appearing in the sheared polynomials are greatly improved. The other is that one projection is enough for most general cases in the new algorithm. After the topology of an algebraic space curve is given, the authors also provide an isotopic-meshing (approximation) of the space curve. Moreover, an approximation of the algebraic space curve can be generated automatically if the approximations of two projected plane curves are first computed. This is also an advantage of our method. Many non-trivial experiments show the efficiency of the algorithm.

In this paper, we study noncommutative surface singularities arising from orders. The singularities we study are mild in the sense that they have finite representation type or, equivalently, are log terminal in the sense of the Mori minimal model program for orders (Chan and Ingalls in Invent Math 161(2):427–452, 2005). These were classified independently by Artin (in terms of ramification data) and Reiten–Van den Bergh (in terms of their AR-quivers). The first main goal of this paper is to connect these two classifications, by going through the finite subgroups $$G \subset {{{\,\mathrm{GL}\,}}_2}$$ , explicitly computing $$H^2(G,k^*)$$ , and then matching these up with Artin’s list of ramification data and Reiten–Van den Bergh’s AR-quivers. This provides a semi-independent proof of their classifications and extends the study of canonical orders in Chan et al. (Proc Lond Math Soc (3) 98(1):83–115, 2009) to the case of log terminal orders. A secondary goal of this paper is to study noncommutative analogues of plane curves which arise as follows. Let $$B = k_{\zeta } \llbracket x,y \rrbracket $$ be the skew power series ring where $$\zeta $$ is a root of unity, or more generally a terminal order over a complete local ring. We consider rings of the form $$A = B/(f)$$ where $$f \in Z(B)$$ which we interpret to be the ring of functions on a noncommutative plane curve. We classify those noncommutative plane curves which are of finite representation type and compute their AR-quivers.

The standard approach to finding the envelope of a family of curves or a family of surfaces is shown to be a parameter optimization problem. This statement is first verified by discussing the envelope of a one-parameter family of plane curves. The standard approach is given, and the corresponding optimization problem is established. Extensions of the envelope problem to multiple parameters and families of surfaces are considered, as well as envelope problems that cannot be solved by the standard approach, for example, problems involving multiple parameters subject to constraints on the parameters.

International audience; PTOPO is a MAPLE package computing the topology and describing the geometry of a parametric plane curve. The algorithm behind PTOPO constructs an abstract graph that is isotopic to the curve. PTOPO exploits the benefits of the parametric representation and performs all computations in the parameter space using exact computing. PTOPO computes the topology and visualizes the curve in less than a second for most examples in the literature.

A Riemann surface [Formula: see text] having field of moduli ℝ, but not a field of definition, is called pseudo-real. This means that [Formula: see text] has anticonformal automorphisms, but none of them is an involution. A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d ≥ 4 in [Formula: see text]. We characterize pseudo-real-plane Riemann surfaces [Formula: see text], whose conformal automorphism group Aut+([Formula: see text]) is PGL3(ℂ)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of [Formula: see text]. In particular, we show that such pseudo-real-plane Riemann surfaces exist only if Aut+([Formula: see text]) is cyclic of even order n dividing the degree d. Explicit families of pseudo-real-plane Riemann surfaces are given for any degree d = 2pm with m > 1 odd, p prime and n = d/p.

T\^ete-\`a-t\^ete graphs were introduced by N. A'Campo in 2010 with the goal of modeling the monodromy of isolated plane curves. Mixed t\^ete-\`a-t\^ete graphs provide a generalization which define mixed t\^ete-\`a-t\^ete twists, which are pseudo-periodic automorphisms on surfaces. We characterize the mixed t\^ete-\`a-t\^ete twists as those pseudo-periodic automorphisms that have a power which is a product of right-handed Dehn twists around disjoint simple closed curves, including all boundary components. It follows that the class of t\^ete-\`a-t\^ete twists coincides with that of monodromies associated with reduced function germs on isolated complex surface singularities., Comment: 21 pages, 14 figures. Minor corrections. Version as accepted in journal. arXiv admin note: text overlap with arXiv:1706.05580

The automorphism groups of smooth plane curves of degree at least 4 is considered as a finite subgroup of the projective linear group. Using this fact, in this paper, we will classify subgroups whose actions are free of automorphism groups of smooth plane curves of degree at least 4.

Evolutes and pedals of plane curves have been well investigated since the beginning of the history of differential geometry. However, there might be no direct relationships between the pedal and the evolute of a curve. We introduce families of relatives of pedals and evolutes and investigate some relationships between these families curves. Moreover, we generalize these notions to the category of frontal curves. Then the relation can be completely described in this category.

application/pdf, We investigate Weierstrass semigroups of ramification points on double covers of plane curves of degree 7. We treat the cases where the Weierstrass semigroups are generated by at most 5 elements and the ramification point is on a total flex.

After a general discussion of group actions, orbifolds, and "weak orbifolds" this note will provide elementary introductions to two basic moduli spaces over the real or complex numbers: First the moduli space of effective divisors with finite stabilizer on the projective space ${\mathbb P}^1$ modulo the group ${\rm PGL}_2$ of projective transformations of ${\mathbb P}^1$; and then the moduli space of effective 1-cycles with finite stabilizer on ${\mathbb P}^2$ modulo the group ${\rm PGL}_3$ of projective transformations of ${\mathbb P}^2$.

We introduce the DeterminantalRepresentations package for Macaulay2, which computes definite symmetric determinantal representations of real polynomials. We focus on quadrics and plane curves of low degree (i.e. cubics and quartics). Our algorithms are geared towards speed and robustness, employing linear algebra and numerical algebraic geometry, without genericity assumptions on the polynomials., Comment: Code available at https://github.com/papridey/DeterminantalRepresentations