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Marcos Craizer

Publications and source records attributed to Marcos Craizer.

At least 19 recordsLinked to original sources
arXiv2025

Stable umbilical points of a W-congruence

The class of $W$-congruences is a central object of Projective Differential Geometry. Nevertheless, their singularities has not been extensively studied. In this paper we prove a characterization of $W$-congruences that allow us to study their umbilical points. We describe examples of isolated stable umbilical points of type $A_m$, $m\in\{1,2,3,4\}$, and discuss some partial results concerning the generalization of these results to any $m\in\mathbb{N}\cup\{\infty\}$.

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arXiv2024

Moutard hyperquadrics and generalized Darboux directions

The higher order contact of a quadric with a surface in $3$-space at a non-degenerate point is obtained by the Moutard quadric in the Darboux direction. In this paper, we discuss the extension of this result to hypersurfaces in arbitrary dimensions.

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arXiv2024

Singularities of Discrete Indefinite Affine Minimal Surfaces

A smooth affine minimal surface with indefinite metric can be obtained from a pair of smooth non-intersecting spatial curves by Lelieuvre's formulas. These surfaces may present singularities, which are generically cuspidal edges and swallowtails. By discretizing the initial curves, one can obtain by the discrete Lelieuvre's formulas a discrete affine minimal surface with indefinite metric. The aim of this paper is to define the singular edges and vertices of the corresponding discrete asymptotic net in such a way that the most relevant properties of the singular set of the smooth version remain valid.

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arXiv2022

Discrete Asymptotic Nets with Constant Affine Mean Curvature

In this paper we define the class of constant affine mean curvature (CAMC) discrete asymptotic nets, which contains the well-known classes of affine spheres and affine minimal asymptotic nets. This class is defined by considering fields of compatible interpolating quadrics, i.e., quadrics that have common tangent planes at the edges of the net. We show that, for CAMC asymptotic nets, ruled discrete asymptotic nets is equivalent to ruled compatible interpolating quadrics. Moreover, we prove discrete counterparts of some known properties of the Demoulin transform of a smooth CAMC surface.

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arXiv2022

Singularities of Discrete Improper Indefinite Affine Spheres

We consider in this paper discrete improper affine spheres based on asymptotic nets. In this context, we distinguish the discrete edges and vertices that must be considered singular. The singular edges can be considered as discrete cuspidal edges, while some of the singular vertices can be considered as discrete swallowtails. The classification of singularities of discrete nets is a quite difficult task, and our results can be considered as a fisrt step in this direction. We also prove some characterizations of ruled discrete improper affine spheres which are analogous to the smooth case.

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arXiv2022

A discrete analog of Segre's theorem on spherical curves

We prove a discrete analog of a certain four-vertex theorem for space curves. The smooth case goes back to the work of Beniamino Segre and states that a closed and smooth curve whose tangent indicatrix has no self-intersections admits at least four points at which its torsion vanishes. Our approach uses the notion of discrete tangent indicatrix of a (closed) polygon. Our theorem then states that a polygon with at least four vertices and whose discrete tangent indicatrix has no self-intersections admits at least four flattenings, i.e., triples of vertices such that the preceding and following vertices are on the same side of the plane spanned by this triple.

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arXiv2022

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.

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arXiv2021

Singularities of Generic Line Congruences

Line congruences are $2$-dimensional families of lines in $3$-space. The singularities that appear in generic line congruences are folds, cusps and swallowtails. In this paper we give a geometric description of these singularities. The main tool used is the existence of an equiaffine pair defining a generic line congruence.

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arXiv2020

Isoperimetric Inequalities in Normed Planes

The classical isoperimetric inequality can be extended to a general normed plane. In the Euclidean plane, the defect in the isoperimetric inequality can be calculated in terms of the signed areas of some singular sets. In this paper we consider normed planes with smooth by parts unit balls and the corresponding class of admissible curves. For such an admissible curve, the singular sets are defined as projections in the subspaces of symmetric and constant width admissible curves. In this context, we obtain some improved isoperimetric inequalities whose equality hold for symmetric or constant width curves.

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arXiv2019

Singular Improper Affine Spheres from a given Lagrangian Submanifold

Given a Lagrangian submanifold $L$ of the affine symplectic $2n$-space, one can canonically and uniquely define a center-chord and a special improper affine sphere of dimension $2n$, both of whose sets of singularities contain $L$. Although these improper affine spheres (IAS) always present other singularities away from $L$ (the off-shell singularities studied in our previous paper), they may also present singularities other than $L$ which are arbitrarily close to $L$, the so called singularities "on shell". These on-shell singularities possess a hidden $\mathbb Z_2$ symmetry that is absent from the off-shell singularities. In this paper, we study these canonical IAS obtained from $L$ and their on-shell singularities, in arbitrary even dimensions, and classify all stable Lagrangian/Legendrian singularities on shell that may occur for these IAS when $L$ is a curve or a Lagrangian surface.

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arXiv2019

Affine curvature lines of surfaces in 3-space

In this work we study the affine principal lines of surfaces in 3-space. We consider the binary differential equation of the affine curvature lines and obtain the topological models of these curves near the affine umbilic points (elliptic and hyperbolic). We also describe the generic behavior of affine curvature lines in the neighborhood of points with double eigenvalues (but not umbilics) of the affine shape operator and parabolic points

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arXiv2018

Centroaffine Duality for Spatial Polygons

In this paper, we discuss centroaffine geometry of polygons in $3$-space. For a polygon $X$ that is locally convex with respect to an origin together with a transversal vector field $U$, we define the centroaffine dual pair $(Y,V)$ similarly to [6]. We prove that vertices of $(X,U)$ correspond to flattening points for $(Y,V)$ and also that constant curvature polygons are dual to planar polygons. As an application, we give a new proof of a known $4$ flattening points theorem for spatial polygons.

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arXiv2018

Dual relations between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$

There is a natural duality between line congruences in $\mathbb{R}^3$ and surfaces in $\mathbb{R}^4$ that sends principal lines into asymptotic lines. The same correspondence takes the discriminant curve of a line congruence into the parabolic curve of the dual surface. Moreover, it takes the ridge curves to the flat ridge curves, while the subparabolic curves of a line congruence are taken to certain curves on the surface that we call flat subparabolic curves. In this paper, we discuss these relations and describe the generic behavior of the subparabolic curves at the discriminant curve of the line congruence, or equivalently, the parabolic curve of the dual surface. We also discuss Loewner's conjectures under the duality viewpoint.

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arXiv2017

Quadratic Points of Surfaces in Projective 3-Space

Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the surface. We show that generically the index of the 3-web at a quadratic point is 1/3 or -1/3, while the index of the line field is 1 or -1. Moreover, for an elliptic quadratic point whose cubic form is semi-homogeneous, we can use Loewner's conjecture to show that the index is at most 1. From the above local results we can conclude some global results: A generic compact elliptic surface has at least 6 quadratic points, a compact elliptic surfaces with semi-homogeneous cubic forms has at least 2 quadratic points and the number of quadratic points in a hyperbolic disc is odd. By studying the behavior of the cubic form in a neighborhood of the parabolic curve, we also obtain a relation between the indices of the quadratic points of a generic surface with non-empty elliptic and hyperbolic regions.

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arXiv2017

Envelope of Mid-Hyperplanes of a Hypersurface

Given 2 points of a smooth hypersurface, their mid-hyperplane is the hyperplane passing through their mid-point and the intersection of their tangent spaces. In this paper we study the envelope of these mid-hyperplanes (EMH) at pairs whose tangent spaces are transversal. We prove that this envelope consists of centers of conics having contact of order at least 3 with the hypersurface at both points. Moreover, we describe general conditions for the EMH to be a smooth hypersurface. These results are extensions of the corresponding well-known results for curves. In the case of curves, if the EMH is contained in a straight line, the curve is necessarily affinely symmetric with respect to the line. We show through a counter-example that this property does not hold for hypersurfaces.

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arXiv2017

Discrete Cycloids from Convex Symmetric Polygons

Cycloids, hipocycloids and epicycloids have an often forgotten common property: they are homothetic to their evolutes. But what if use convex symmetric polygons as unit balls, can we define evolutes and cycloids which are genuinely discrete? Indeed, we can! We define discrete cycloids as eigenvectors of a discrete double evolute transform which can be seen as a linear operator on a vector space we call curvature radius space. We are also able to classify such cycloids according to the eigenvalues of that transform, and show that the number of cusps of each cycloid is well determined by the ordering of those eigenvalues. As an elegant application, we easily establish a version of the four-vertex theorem for closed convex polygons. The whole theory is developed using only linear algebra, and concrete examples are given.

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arXiv2016

Affine geometry of equal-volume polygons in 3-space

Equal-volume polygons are obtained from adequate discretizations of curves in 3-space, contained or not in surfaces. In this paper we explore the similarities of these polygons with the affine arc-length parameterized smooth curves to develop a theory of discrete affine invariants. Besides obtaining discrete affine invariants, equal-volume polygons can also be used to estimate projective invariants of a planar curve. This theory has many potential applications, among them evaluation of the quality and computation of affine invariants of silhouette curves.

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