arXiv subjects

Ricardo Uribe-Vargas

Publications and source records attributed to Ricardo Uribe-Vargas.

7 recordsLinked to original sources

arXiv2025

On Surfaces in R^n via Gauss Map, Caustics, Duality and Pseudo Euclidean Geometry of Quadratic Forms

We get new results (and rederive some know ones) on smooth surfaces in $\mathbb{R}^n$ by unifying several view points into a coherent general view. Namely, we show and use new relations of the evolute (caustic) with the curvature ellipse, the Gauss map and the pseudo-Euclidean geometry of the $3$-space of quadratic forms on $\mathbb{R}^2$. A key result (Th.3.3.1): for a surface $M$ in $\mathbb{R}^n$ the intersection of its caustic with the normal space $N_pM$ is the polar dual hypersurface (in $N_pM$) of the curvature ellipse at $p$. Moreover, all local objects $X$ (cf. the invariants and their relations) have a "paired" version $X^*$ (with ${X^*}^*=X$) -- this provides new results on the original objects.

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arXiv2024

Evolving Surfaces and Evolving Implicit Differential Equations Via Contact Geometry and Singularities

We present the list of unavoidable local phenomena (transitions) occurring on the configuration of the parabolic and flecnodal curves of evolving smooth surfaces in R^3 (or RP^3). We also present the list of transitions occurring on the curve of inflections of the solutions of evolving implicit differential equations (IDE). Our results are based on the properties of the contours of surfaces (in a contact 3-space) for projections all whose fibres are Legendrian. Keywords: Surface, flecnodal curve, contact geometry, implicit differential equations.

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arXiv2021

Contact with circles and Euclidean invariants of smooth surfaces in R^3

We investigate the vertex curve, that is the set of points in the hyperbolic region of a smooth surface in real 3-space at which there is a circle in the tangent plane having at least 5-point contact with the surface. The vertex curve is related to the differential geometry of planar sections of the surface parallel to and close to the tangent planes, and to the symmetry sets of isophote curves, that is level sets of intensity in a 2-dimensional image. We investigate also the relationship of the vertex curve with the parabolic and flecnodal curves, and the evolution of the vertex curve in a generic 1-parameter family of smooth surfaces.

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arXiv2020

Characteristic Points, Fundamental Cubic Form and Euler Characteristic of Projective Surfaces

We define local indices for projective umbilics and godrons (also called cusps of Gauss) on generic smooth surfaces in projective 3-space. By means of these indices, we provide formulas that relate the algebraic numbers of those characteristic points on a surface (and on domains of the surface) with the Euler characteristic of that surface (resp. of those domains). These relations determine the possible coexistences of projective umbilics and godrons on the surface. Our study is based on a "fundamental cubic form" for which we provide a closed simple expression.

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arXiv2019

On Projective Umbilics: a Geometric Invariant and an Index

We define a geometric invariant and an index (+1 or -1) for projective umbilics of smooth surfaces. We prove that the sum of the indices of the projective umbilics inside a connected component H of the hyperbolic domain remains constant in any 1-parameter family of surfaces if the topological type of H does not change. We prove the same statement for any connected component E of the elliptic domain. We give formulas for the invariant and for the index which do not depend on any normal form.

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arXiv2006

A New Projective Invariant Associated to the Special Parabolic Points of Surfaces and to Swallowtails

We show some generic (robust) properties of smooth surfaces immersed in the real 3-space (Euclidean, affine or projective), in the neighbourhood of a {\em godron} (term due to R.Thom): an isolated parabolic point at which the (unique) asymptotic direction is tangent to the parabolic curve. With the help of these properties and a projective invariant that we associate to each godron we classify the godrons and present all possible local configurations of the flecnodal curve at a generic swallowtail in $R^3$. We present some global results, for instance: {\em A closed parabolic curve bounding a hyperbolic disc has a positive even number of godrons, and the flecnodal curve lying in that disc has an odd number of transverse self-intersections.}.

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arXiv2005

On Vertices and Focal Curvatures of Space Curves

The {\em focal curve} of an immersed smooth curve $γ:s\mapsto γ(s)$, in Euclidean space $\R^{m+1}$, consists of the centres of its osculating hyperspheres. The focal curve may be parametrised in terms of the Frenet frame of $γ$ (${\bf t},{\bf n}_1, ...,{\bf n}_m$), as $C_γ(s)=(γ+c_1{\bf n}_1+c_2{\bf n}_2+...+c_m{\bf n}_m)(s)$, where the coefficients $c_1,...,c_{m-1}$ are smooth functions that we call the {\em focal curvatures} of $γ$. We discovered a remarkable formula relating the Euclidean curvatures $κ_i$, $i=1,...,m$, of $γ$ with its focal curvatures. We show that the focal curvatures satisfy a system of Frenet equations (not vectorial, but scalar!). We use the properties of the focal curvatures in order to give, for $k=1,...,m$, necessary and sufficient conditions for the radius of the osculating $k$-dimensional sphere to be critical. We also give necessary and sufficient conditions for a point of $γ$ to be a vertex. Finally, we show explicitly the relations of the Frenet frame and the Euclidean curvatures of $γ$ with the Frenet frame and the Euclidean curvatures of its focal curve $C_γ$.

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