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We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.
We say that a metric graph is uniformly bounded if the degrees of all vertices are uniformly bounded and the lengths of edges are pinched between two positive constants; a metric space is approximable by a uniform graph if there is one…
We give some results on quadratic normality of reducible curves canonically embedded and partially extend this study to their projective normality.
This paper deals with relative normalizations of skew ruled surfaces in the Euclidean space $\mathbb{E}^{3}$. In section 2 we investigate some new formulae concerning the Pick invariant, the relative curvature, the relative mean curvature…
In this paper the problem of finding a normal form of triangles and plane quadrilaterals up to similarity is considered. Several normal forms for triangles and a normal form for quadrilaterals of special case are described. Normal forms of…
We introduce a large scale analogue of the classical fixed-point property for continuous maps, which shall apply to coarse maps. We also develop a coarse version of degree for coarse maps on Euclidean spaces. Then, applying a coarse…
A variational equation of the third order in three-dimensional space is proposed which describes autoparallel curves of some connection.
Any quasi-isometry of the complex of curves is bounded distance from a simplicial automorphism. As a consequence, the quasi-isometry type of the curve complex determines the homeomorphism type of the surface.
We carry out a survey on curves defined over finite fields that are Diophantine stable; that is, with the property that the set of points of the curve is not altered under a proper field extension. First, we derive some general results of…
We investigate the behaviour of vertices and inflexions on 1-parameter families of curves on smooth surfaces in the 3-space, which include a singular member. In particular, we discuss the context where the curves evolve as sections of a…
The equivalence problem of curves with values in a Riemannian manifold, is solved. The domain of validity of Frenet's theorem is shown to be the spaces of constant curvature. For a general Riemannian manifold new invariants must thus be…
In this paper, we consider the self-affinity of planar curves. It is regarded as an important property to characterize the log-aesthetic curves which have been studied as reference curves or guidelines for designing aesthetic shapes in CAD…
We investigate helicoidal (screw) surfaces generated not only by regular curves but also by curves with singular points. For curves with singular points, it is useful to use frontals in the Euclidean plane. The helicoidal surface of a…
The Cornu spirals on plane are the curves whose curvatures are linear. Generalized planar cornu spirals and Euler spirals in E^3, the curves whose curvatures are linear are defined in [1,5]. In this study, these curves are presented as the…
In this paper results from the differential geometry of curves are extended from normed planes to gauge planes which are obtained by neglecting the symmetry axiom. Based on the gauge analogue of the notion of Birkhoff orthogonality from…
In this paper, we investigate the differential geometry properties of curves of constant breadth according to Darboux frame in a given strict Walker 3-manifold. The considered curves are lying on a timelike surface in the Walker 3-manifold.
In this paper, we discuss some problems of elementary plane differential geometry and kinematics. Although the results are not new, the consistent use of complex-valued functions (plane curves) of a real variable (parameter) allows to…
We show that there are five types of planar curves such that arrangements of its translates are combinatorially equivalent to an arrangement of lines. These curves can be used to define norms giving constructions with many unit distances…
The three-dimensional linear regression problem is a problem of finding a spacial straight line best fitting a group of points in three-dimensional Euclidean space. This problem is considered in the present paper and a solution to it is…
We study some Riemannian metrics on the space of regular smooth curves in the plane, viewed as the orbit space of maps from $S^1$ to the plane modulo the group of diffeomorphisms of $S^1$, acting as reparameterizations. In particular we…