Related papers: On tangents to curves

We compute the limit of tangents of an arbitrary surface. We obtain as a byproduct an embedded version of Jung's desingularization theorem for surface singularities with finite limits of tangents.

In this paper, tangent-, principal normal-, and binormal-wise associated curves are defined such that each of these vectors of any given curve lies on the osculating, normal, and rectifying plane of its mate, respectively. For each…

The aim of this paper is to give some characterizations for N-Legendre and N-slant curves in the unit tangent bundles of surfaces endowed with natural diagonal lifted structures.

It is well known that the area $U$ of the triangle formed by three tangents to a parabola $X$ is half of the area $T$ of the triangle formed by joining their points of contact. In this article, we study some properties of $U$ and $T$ for…

We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense.…

The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this…

The present paper deals with a study of curves on a smooth surface whose position vector always lies in the tangent plane of the surface and it is proved that such curves remain invariant under isometry of surfaces. It is also shown that…

This article establishes several remarkably simple identities relating certain metric invariants of level curves of real and complex functions. In particular, we relate lengths of level curves to their curvature and to the gradient field of…

We investigate a tangent space at a point of a general metric space and metric space valued derivatives. The conditions under which two different subspace of a metric space have isometric tangent spaces in a common point of these subspaces…

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…

In this note is we exhibit an elementary method to construct explicitly curves over finite fields with many points. Despite its elementary character the method is very efficient and can be regarded as a partial substitute for the use of…

A quadrisecant line is one which intersects a curve in at least four points, while an essential secant captures something about the knottedness of a knot. This survey article gives a brief history of these ideas, and shows how they may be…

There is an abstract notion of connection in any tangent category. In this paper, we show that when applied to the tangent category of affine schemes, this recreates the classical notion of a connection on a module (and similarly, in the…

The pedal of a curve in the Euclidean plane is a classical subject which has a singular point at the inflection point of the original curve or the pedal point. The primitive of a curve is a curve given by the inverse construction for making…

A problem involving a square in the curvilinear triangle made by two touching congruent circles and their common tangent is generalized.

Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived.

In this part of the series five-dimensional tangent vectors are introduced first as equivalence classes of parametrized curves and then as differential-algebraic operators that act on scalar functions. I then examine their basic algebraic…

We define and study trivial points on towers of curves over number fields, and we show their finiteness in some cases. We relate these to the unboundeness of the gonality of the curves, which we show under some hypothesis. The problem is…

We develop a theory for quotients of geometries and obtain sufficient conditions for the quotient of a geometry to be a geometry. These conditions are compared with earlier work on quotients, in particular by Pasini and Tits. We also…

This paper describes how to define and work with differential equations in the abstract setting of tangent categories. The key notion is that of a curve object which is, for differential geometry, the structural analogue of a natural number…