相关论文: Cusp singularities of plane envelopes

In this paper, we extend the method developed in [17, 18] to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in…

In the paper, we investigate properties of the nine-dimensional variety of the inflection points of the plane cubic curves. The description of local monodromy groups of the set of inflection points near singular cubic curves is given. Also,…

If the fundamental group of the complement of a smooth embedding f: S^2 \subset R^4 is a cyclic group, the map can be deformed to the standard embedding by a generic one-parameter family with at most cusp singularities. If two smooth…

We construct a family of plane curves as pull-backs of a conic for abelian coverings of P^2. If the conic is tangent to the ramification lines one obtains a family of curves of degree 2n with 3n singularities of type A_{n-1}. We calculate…

We study surfaces of constant positive Gauss curvature in Euclidean 3-space via the harmonicity of the Gauss map. Using the loop group representation, we solve the regular and the singular geometric Cauchy problems for these surfaces, and…

We show that the normal points of a cubic hypersurface in projective space have canonical singularities unless the hypersurface is an iterated cone over an elliptic curve. As an application, we give a simple linear algebraic description of…

A horospherical torus about a cusp of a hyperbolic manifold inherits a Euclidean similarity structure, called a cusp shape. We bound the change in cusp shape when the hyperbolic structure of the manifold is deformed via cone deformation…

In this paper, we consider rational cuspidal plane curves having exactly two cusps whose complements have logarithmic Kodaira dimension two. We classify such curves with the property that the strict transforms of them via the minimal…

We consider in this paper the $FRS$-deformations of a family of space curves with codimension $\leq 3$. Some geometric aspects of a space curve such as flattenings, vertices and twistings points has been studied.

We investigate singularities of all parallel surfaces to a given regular surface. In generic context, the types of singularities of parallel surfaces are cuspidal edge, swallowtail, cuspidal lips, cuspidal beaks, cuspidal butterfly and…

We study equisingular deformation problems for curves and surfaces in algebraic families, with particular emphasis on situations where nodal behavior is no longer generic. Extending classical Severi theory, we develop deformation--theoretic…

We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or…

Considering the tangent plane at a point to a surface in the four-dimensional Euclidean space, we find an invariant of a pair of two tangents in this plane. If this invariant is zero, the two tangents are said to be conjugate. When the two…

We completely classify all plane curves of degree at most 30 with a unique cuspidal (locally unibranch) singular point and rational normalization in terms of the Newton pairs parameterizing the cusp. We distinguish between prime and…

Let f = 0 be an implicit singular plane curve. When deforming f = 0, inflections and vertex emerge from the singularities. In this papper, we classify the deformations of f = 0 with respect to the inflections and the vertices in the cases…

A wide range of equations related to free surface motion in two dimensions exhibit the formation of cusp singularities either in time, or as function of a parameter. We review a number of specific examples, relating in particular to fluid…

We study singularities of constant positive Gaussian curvature surfaces and determine the way they bifurcate in generic 1-parameter families of such surfaces. We construct the bifurcations explicitly using loop group methods. Constant…

We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also…

We report on the problem of the existence of complex and real algebraic curves in the plane with prescribed singularities up to analytic and topological equivalence. The question is whether, for a given positive integer $d$ and a finite…

We consider classical curvature flows: 1-parameter families of convex embeddings of the 2-sphere into Euclidean 3-space which evolve by an arbitrary (non-homogeneous) function of the radii of curvature. The associated flow of the radii of…