Related papers: The Deltoid Curve and Triangle Transformations
We develop a technique to study curves in a variety which has a degeneration into some union of varieties. The class of such varieties is very broad, but the theory becomes particularly useful when the variety has a degeneration into a…
We provide algorithms to reconstruct rational ruled surfaces in three-dimensional projective space from the `apparent contour' of a single projection to the projective plane. We deal with the case of tangent developables and of general…
A variational approach to the reconstruction of a shape (2D simple manifolds) as triangulated surface from given level set using shape gradients is presented. It involves an energy functional that depends on the local shape characteristics…
A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical…
In this paper, we give a simple description of the deformations of a map between two smooth curves with partially prescribed branching, in the cases that both curves are fixed, and that the source is allowed to vary. Both descriptions work…
We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves…
The geometry of the Heisenberg group acting on the plane arises naturally in geometric topology as a degeneration of the familiar spaces $\mathbb{S}^2,\mathbb{H}^2$ and $\mathbb{E}^2$ via conjugacy limit as defined by Cooper, Danciger, and…
Simple drawings are drawings of graphs in which the edges are Jordan arcs and each pair of edges share at most one point (a proper crossing or a common endpoint). We introduce a special kind of simple drawings that we call generalized…
An approach to defining quadratic implicit curves is to prescribe two tangent lines and a secant line going through the points of tangency. This paper will show that this method can be generalized to a higher number of tangents, resulting…
We investigate when the tangent bundle of a projective manifold has a non-trivial first order (or positive-dimensional) deformation. This leads to a new conjectural characterization of the complex projective space.
We give an example of a projective manifold with dense entire curves such that every Brody curve is degenerate.
The Centre Symmetry Set of a planar curve $M$ is the envelope of affine chords of $M$, i.e. the lines joining points on $M$ with parallel tangent lines. In this paper we study global geometrical properties of this set including the number…
The goal of this paper is to describe all local diffeomorphisms mapping a family of circles, in an open subset of $\r^3$, into straight lines. This paper contains two main results. The first is a complete description of the rectifiable…
The field of multiple view geometry has seen tremendous progress in reconstruction and calibration due to methods for extracting reliable point features and key developments in projective geometry. Point features, however, are not available…
We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister…
We introduce the Frenet theory of curves in dual space $\d^3$. After defining the curvature and the torsion of a curve, we classify all curves in dual plane with constant curvature. We also establish the fundamental theorem of existence in…
We describe some theoretical results on triangulations of surfaces and we develop a theory on roots, decompositions and genus-surfaces. We apply this theory to describe an algorithm to list all triangulations of closed surfaces with at most…
A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp…
Let $K$ be a local field of residue characteristic $p$. Let $C$ be a curve over $K$ whose minimal proper regular model has totally degenerate semi-stable reduction. Under certain hypotheses, we compute the prime-to-$p$ rational torsion…
An elementary geometric construction known as Napoleon's theorem produces an equilateral triangle built on the sides of any initial triangle: the centroids of each equilateral triangle meeting the original sides, all outward or all inward,…