Related papers: Where is the cone?

This is a survey article on real algebra and geometry, and in particular on its recent applications in optimization and convexity. We first introduce basic notions and results from the classical theory. We then explain how these relate to…

This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half takes up and develops…

We discuss how the shape of a special Cosserat rod can be represented as a path in the special Euclidean algebra. By shape we mean all those geometric features that are invariant under isometries of the three-dimensional ambient space. The…

Since the end of the 19th century, and after the works of F. Klein and H. Poincar\'e, it is well known that models of elliptic geometry and hyperbolic geometry can be given using projective geometry, and that Euclidean geometry can be seen…

A rational triangle is a triangle with rational side lengths. We consider three different families of rational triangles having a fixed side and whose vertices are rational points in the plane. We display a one-to-one correspondence between…

We describe the curves of constant (geodesic) curvature and torsion in the three-dimensional round sphere. These curves are the trajectory of a point whose motion is the superposition of two circular motions in orthogonal planes. The global…

In a recent paper by Cook, et al., which introduced the concept of unexpected plane curves, the focus was on understanding the geometry of the curves themselves. Here we expand the definition to hypersurfaces of any dimension and, using…

Progress on the problem whether the Hilbert schemes of locally Cohen-Macaulay curves in projective 3 space are connected has been hampered by the lack of an answer to a question that was raised by Robin Hartshorne in his paper "On the…

We find all analytic surfaces in space $\mathbb{R}^3$ such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The problem of finding such surfaces traces back to the works of…

We provide a complete classification of the critical sets and their images for quadratic maps of the real plane. Critical sets are always conic sections, which provides a starting point for the classification. The generic cases, maps whose…

Eliminating the arbitrary coefficients in the equation of a generic plane curve of order $n$ by computing sufficiently many derivatives, one obtains a differential equation. This is a projective invariant. The first one, corresponding to…

We classify maximal quartic generalised projective special real curves up to equivalence. A maximal quartic generalised projective special real curve consists of connected components of the intersection of the hyperbolic points of a quartic…

The Universe is a physical object. Physical objects have shapes and sizes. General relativity is insufficient to describe the global shape and size of the Universe: the Hilbert-Einstein equations only treat limiting quantities towards an…

In this paper we consider planar sections and visual contours of co-dimension one affine immersions. The main theorem says that the third order Taylor expansion of the difference between the visual contour and planar section functions is…

In this paper we provide a computational approach to the shape of curves which are rational in polar coordinates, i.e. which are defined by means of a parametrization (r(t),\theta(t)) where both r(t),\theta(t) are rational functions. Our…

The Newton polygon of the implicit equation of a rational plane curve is explicitly determined by the multiplicities of any of its parametrizations. We give an intersection-theoretical proof of this fact based on a refinement of the…

The purpose of this article is to find a family of curves parametrized by arc length and that depend on an angular function and an intrinsic fraction function, which is defined as the quotient between torsion and curvature. We find for this…

We study three families of polyhedral cones whose sections are regular simplices, cubes, and crosspolytopes. We compute solid angles and conic intrinsic volumes of these cones. We show that several quantities appearing in stochastic…

The solution of the cubic equation has a century-long history; however, the usual presentation is geared towards applications in algebra and is somewhat inconvenient to use in optimization where frequently the main interest lies in real…

Two perpendicular segments which divide a given triangle into 4 regions of equal area is called a quadrisection of the triangle. Leonhard Euler proved in 1779 that every scalene triangle has a quadrisection with its triangular part on the…