Related papers: The normal map as a vector field
Legendre curves are smooth plane curves which may have singular points, but still have a well defined smooth normal (and corresponding tangent) vector field. Because of the existence of singular points, the usual curvature concept for…
We introduce circular evolutes and involutes of framed curves in the Euclidean space. Circular evolutes of framed curves stem from the curvature circles of Bishop directions and singular value sets of normal surfaces of Bishop directions.…
For any chord diagram on a circle there exists a complete graph on sufficiently many vertices such that any generic immersion of it to the plane contains a plane closed curve whose chord diagram contains the given chord diagram as a…
In this paper, we use Conley index theory to examine the Poincare index of an isolated invariant set. We obtain some limiting conditions on a critical point of a planar vector field to be an isolated invariant set. As a result we show the…
The ruled surfaces, i.e., surfaces generated by one parametric set of lines, are widely used in the~field of applied geometry. An~isophote on a surface is a curve consisting of surface points whose normals form a constant angle with some…
Geometric constructions are widely used in computer graphics and engineering drawing. A right generalized cylinder is a ruled surface whose base curve is a plane curve perpendicular to the rulings. The paper discusses relations between the…
In the present paper, we address the problem how to get a map from knots in the cylinder and on the thickened torus to some (generalisation of) virtual knots called virtual-flat knots. The main construction takes a diagram on a cylinder…
The aim of this paper is to investigate the sufficient condition for the invariance of a normal curve on a smooth immersed surface under isometry. We also find the the deviations of the tangential and normal components of the curve with…
We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map,…
The normal form theory for polynomial vector fields is extended to those for $C^\infty$ vector fields vanishing at the origin. Explicit formulas for the $C^\infty$ normal form and the near identity transformation which brings a vector field…
Let Y and X denote C^k vector fields on a possibly noncompact surface with empty boundary, k >0. Say that Y tracks X if the dynamical system it generates locally permutes integral curves of X. Let K be a locally maximal compact set of…
We examine pairs of closed plane curves that have the same closing property as two conic sections in Poncelet's porism. We show how the vertex curve can be computed for a given envelope and vice versa. Our formulas are universal in the…
The aim of the present paper is to investigate intrinsically the notion of a concircular $\pi$-vector field in Finsler geometry. This generalizes the concept of a concircular vector field in Riemannian geometry and the concept of a…
We establish normal forms for conformal vector fields on pseudo-Riemannian manifolds in the neighborhood of a singularity. For real-analytic Lorentzian manifolds, we show that the vector field is analytically linearizable or the manifold is…
We review and extend a technique for recovering a smooth function from its averages over a wide class of curves in a general region of Euclidean space. The method is based on complexification of the underlying vector fields defining the…
The main objective of the present paper is to investigate a sufficient condition for which a rectifying curve on a smooth surface remains invariant under isometry of surfaces, and also it is shown that under such an isometry the component…
Indices of vector fields on (complex analytic) singular varieties have been considered by various authors from several different viewpoints. All these indices coincide with the classical local index of Poincar\'e-Hopf when the ambient…
A non-autonomous version of the standard map with a periodic variation of the parameter is introduced and studied. Symmetry properties in the variables and parameters of the map are found and used to find relations between rotation numbers…
This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth…
The theory of frames normal for general connections on differentiable bundles is developed. Links with the existing theory of frames normal for covariant derivative operators (linear connections) in vector bundles are revealed. The…