Related papers: Where is the cone?
Coarse geometry, the branch of topology that studies the global properties of spaces, was originally developed for metric spaces and then Roe introduced coarse structures as a large-scale counterpart of uniformities. In the literature,…
Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the…
If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…
Given two distinct reduced, irreducible curves of given degrees, contained in projective space but whose union is not contained in a hyperplane, what is the largest number of points of intersection they can have? When the projective space…
A space curve in a Euclidean 3-space $\mathbb E^3$ is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf…
From a topological viewpoint, a rational curve in the real projective plane is generically a smoothly immersed circle and a finite collection of isolated points. We give an isotopy classification of generic rational quintics in…
We prove that a surface in real 3-space containing a line and a circle through each point is a quadric. We also give some particular results on the classification of surfaces containing several circles through each point.
The author has been interested in regions surrounded by real algebraic curves of degree $1$ or $2$ in the plane. The author is mainly interested in their shapes and combinatorics. This is a fundamental and natural problem in mathematics…
Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard…
Boltzmann's work, ``Ueber die sogenannte H-Curve," discusses his demonstration of the essential characteristics of the H-curve in a clear, concise, and precise style, showcasing his efforts to persuade his peers. To make these findings more…
The notion of an unexpected curve in the plane was introduced in 2018, and was quickly generalized in several directions in a flurry of mathematical activity by many authors. In this expository paper we first describe some of the main…
It is a classical result that there are $12$ (irreducible) rational cubic curves through $8$ generic points in $\mathbb{P}_{\mathbb{C}}^2$, but little is known about the non-generic cases. The space of $8$-point configurations is…
The Hessian Topology is a subject having interesting relations with several areas, for instance, differential geometry, implicit differential equations, analysis and singularity theory. In this article we study the problem of realization of…
The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs.…
In recent years, several quantizations of real manifolds have been studied, in particular from the point of view of Connes' noncommutative geometry. Less is known for complex noncommutative spaces. In this paper, we review some recent…
The intersection of a quadric and a cubic surface in 3-space is a canonical curve of genus 4. It has 120 complex tritangent planes. We present algorithms for computing real tritangents, and we study the associated discriminants. We focus on…
We prove that the number of legendrian rational cubics in $\mathbb C P^3$ through three generic points and a line is three; also we classify all legendrian curves on a quadric surface. Several computations are additionally verified using…
A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are…
Contours may be viewed as the 2D outline of the image of an object. This type of data arises in medical imaging as well as in computer vision and can be modeled as data on a manifold and can be studied using statistical shape analysis.…
A very general surface of degree at least four in projective space of dimension three contains no curves other than intersections with surfaces. We find a formula for the degree of the locus of surfaces of degree at least five which contain…