相关论文: Extremal areas of polygons with fixed perimeter
We describe the configuration space $\mathbf{S}$ of polygons with prescribed edge slopes, and study the perimeter $\mathcal{P}$ as a Morse function on $\mathbf{S}$. We characterize critical points of $\mathcal{P}$ (these are…
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in…
We study configuration spaces of linkages whose underlying graph are polygons with diagonal constrains, or more general, partial two-trees. We show that (with an appropriate definition) the oriented area is a Bott-Morse function on the…
We study polygon spaces arising from planar configurations of necklaces with some of the beads fixed and some of the beads sliding freely. These spaces include configuration spaces of flexible polygons and some other natural polygon spaces.…
In this paper we study the area function of polygons, where the vertices are sliding along curves. We give geometric criteria for the critical points and determine also the Hesse matrix at those points. This is the starting point for a…
The oriented area function $A$ is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the…
We determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the…
It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper, we announce an explicit formula of the Morse index for the signed area of a cyclic configuration. It depends…
We consider methods for finding a simple polygon of minimum (Min-Area) or maximum (Max-Area) possible area for a given set of points in the plane. Both problems are known to be NP-hard; at the center of the recent CG Challenge, practical…
We show that an appropriate generalization of the oriented area function is a perfect Morse function on the space of three-dimensional configurations of an equilateral polygonal linkage with odd number of edges. Therefore cyclic equilateral…
We compute the fixed point index of non-degenerate central configurations for the $n$-body problem in the euclidean space of dimension $d$, relating it to the Morse index of the gravitational potential function $\bar U$ induced on the…
Planar central configurations can be seen as critical points of the reduced potential or solutions of a system of equations. By the homogeneity and invariance of the potential with respect to SO(2), it is possible to see that the…
We study configuration spaces of hard spheres in a bounded region. We develop a general Morse-theoretic framework, and show that mechanically balanced configurations play the role of critical points. As an application, we find the precise…
Consider the Fulton-MacPherson configuration space of $n$ points on $\P^1$, which is isomorphic to a certain moduli space of stable maps to $\P^1$. We compute the cone of effective ${\mathfrak S}_n$-invariant divisors on this space. This…
Inspired in the theorem of Krein-Milamn, we investigate the existence of extreme points in compact convex subsets of asymmetric normed spaces. We focus our attention in the finite dimensional case, giving a geometric description of all…
Given an iterated function system of affine dilations with fixed points the vertices of a regular polygon, we characterize which points in the limit set lie on the boundary of its convex hull.
We investigate the enumerative geometry of point configurations in projective space. We define "projective configuration counts": these enumerate configurations of points in projective space such that certain specified subsets are in fixed…
We investigate the critical points of the basic (quasi-)modular forms $E_2$, $E_4$, and $E_6$. They occur where some associated polymorphic functions have poles. By an explicit description of these polymorphic functions as conformal maps,…
In this paper we study the concept of characteristic numbers and Chern slopes in the context of curve configurations in the real and complex projective plane. We show that some extremal line configurations inherit the same asymptotic…
We give an overview of theoretical and practical aspects of finding a simple polygon of minimum (Min-Area) or maximum (Max-Area) possible area for a given set of n points in the plane. Both problems are known to be NP-hard and were the…