相关论文: A focus on focal surfaces
We prove that the focal set generated by the reflection of a point source off a translation invariant surface consists of two sets: a curve and a surface. The focal curve lies in the plane orthogonal to the symmetry direction containing the…
In this paper we describe projective curves and surfaces such that almost all their hyperplane sections are projectively equivalent. Our description is complete for curves and close to being complete for smooth surfaces. In the appendix we…
We study surfaces with a constant ratio of principal curvatures in Euclidean and simply isotropic geometries and characterize rotational, channel, ruled, helical, and translational surfaces of this kind under some technical restrictions…
In this paper we classify certain special ruled surfaces in $\R^3$ under the general theorem of characterization of constant angle surfaces. We study the tangent developable and conical surfaces from the point of view the constant angle…
We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of…
We investigate plane curves intersecting in at most two unibranched points to study the algebraic exceptional set appearing in standard conjectures of diophantine and hyperbolic geometry. Our first result compares the local geometry of two…
In this article we obtain the classification of the congruences of lines with one-dimensional focal locus. It turns out that one can restrict to study the case of $\mathbb{P}^3$.
We develop an essentially algebraic method to study biharmonic curves into an implicit surface. Although our method is rather general, it is especially suitable to study curves into surfaces defined by a polynomial equation: in particular,…
A hyperbolic framed curve is a smooth curve with a moving frame in hyperbolic 3-space. It may have singularities. By using this moving frame, we can investigate the differential geometry properties of curves, even at singular points. In…
Let $\calP$ be a general pencil of curves of degree $d$ in the projective plane. In this paper we review the computation of the number of curves in $\calP$ that have a hyperflex line, a flex bitangent line or a tritangent line. Then we…
This paper surveys and gives a uniform exposition of results contained in papers published by the team of authors. The subject is degenerations of surfaces, especially to unions of planes. More specifically, we deduce some properties of the…
We take the fundamental group of the complement of the branch curve of a generic projection induced from canonical embedding of a surface. This group is stable on connected components of moduli spaces of surfaces. Since for many classes of…
We exploit a key result from visual psychophysics---that individuals perceive shape qualitatively---to develop the use of a geometrical/topological "invariant'' (the Morse--Smale complex) relating image structure with surface structure.…
We survey determinantal singularities, their deformations, and their topology. This class of singularities generalizes the well studied case of complete intersections in several different aspects, but exhibits a plethora of new phenomena…
Near a singular point of a surface or a curve, geometric invariants diverge in general, and the orders of diverge, in particular the boundedness about these invariants represent geometry of the surface and the curve. In this paper, we study…
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…
A directed curve is a possibly singular curve with well-defined tangent lines along the curve. Then the tangent surface to a directed curve is naturally defined as the ruled surface by tangent geodesics to the curve, whenever any affine…
We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…
We investigate combinatorial aspects of exceptional sequences in the derived category of coherent sheaves on certain smooth and complete algebraic surfaces. We show that to any such sequence there is canonically associated a complete toric…
We study discrete curvatures computed from nets of curvature lines on a given smooth surface, and prove their uniform convergence to smooth principal curvatures. We provide explicit error bounds, with constants depending only on properties…