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A popular classroom demonstration is to draw a cycloid on a blackboard with a piece of chalk inserted through a hole at a point P with radius r = R from the center of a wood disk of radius R that is rolling without slipping along the chalk…
An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right,…
Let $P\subset\mathbb{R}^{2}$ be a set of $n$ points. In this paper we show two new algorithms, one to compute the number of triangulations of $P$, and one to compute the number of pseudo-triangulations of $P$. We show that our algorithms…
Suppose that a polygon $P$ is given as an array containing the vertices in counterclockwise order. We analyze how many vertices (including the index of each of these vertices) we need to know before we can bound $P$, i.e., report a bounded…
In the classic circle packing problem, one asks whether a given set of circles can be packed into a given container. Packing problems like this have been shown to be $\mathsf{NP}$-hard. In this paper, we present new sufficient conditions…
We consider the problem of deciding, given a sequence of regions, if there is a choice of points, one for each region, such that the induced polyline is simple or weakly simple, meaning that it can touch but not cross itself. Specifically,…
In a containment problem, the goal is to preprocess a set of geometric objects so that, given a geometric query object, we can report all the objects containing the query object. We consider the containment problem where input objects are…
This paper provides (in french) a framework for an alternative demonstration of result of Khimshiashvili and Panina on the characterization of critical points of the area on the manifold of polygons with fixed sidelengths as being the…
Correlation clustering is a central topic in unsupervised learning, with many applications in ML and data mining. In correlation clustering, one receives as input a signed graph and the goal is to partition it to minimize the number of…
In this paper we consider the classic problems of scattering of waves from perfectly conducting cylinders with piecewise smooth boundaries. The scattering problems are formulated as integral equations and solved using a Nystr\"om scheme…
The manuscript presents a theoretical proof in conglomeration with new definitions on Inaccessibility and Inside for a point S related to a simple or self intersecting polygon P. The proposed analytical solution depicts a novel way of…
The problem deals with an exact calculation of the intersection area of a circle arbitrary placed on a grid of square shaped elements with gaps between them (finite fill factor). Usually an approximation is used for the calculation of the…
Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two…
A clique transversal in a graph is a set of vertices intersecting all maximal cliques. The problem of determining the minimum size of a clique transversal has received considerable attention in the literature. In this paper, we initiate the…
We address the problem of computing the minimum number of triangles to separate a set of blue points from a set of red points in $\mathbb{R}^2$. A set of triangles is a \emph{separator} of one color from the other if every point of that…
We call a continuous path of polygons decreasing if the convex hulls of the polygons form a decreasing family of sets. For an arbitrary polygon of more than three vertices, we characterize the polygons contained in it that can be reached by…
Let $\s^1$ be a circle in Euclidean plane. We consider the problem of finding the shape of a planar curve which is an extremal of the potential energy that measures the distance to $\s^1$. We describe the shape of these curves…
We classify the set of quadrilaterals that can be inscribed in convex Jordan curves, in the continuous as well as in the smooth case. This answers a question of Makeev in the special case of convex curves. The difficulty of this problem…
The aim in packing problems is to decide if a given set of pieces can be placed inside a given container. A packing problem is defined by the types of pieces and containers to be handled, and the motions that are allowed to move the pieces.…
We study Snipperclips, a computer puzzle game whose objective is to create a target shape with two tools. The tools start as constant-complexity shapes, and each tool can snip (i.e., subtract its current shape from) the other tool. We study…