One of the earliest and most well known problems in computational geometry is the socalled art gallery problem. The goal is to compute the minimum possible number guards placed on the vertices of a simple polygon in such a way that they cover the interior of the polygon. In this paper we consider the problem of guarding an art gallery which is modeled as a polygon with curvilinear walls. Our ma...

The art gallery problem asks for the smallest number of guards required to see every point of the interior of a polygon P . We introduce and study a similar problem called the chromatic art gallery problem. Suppose that two members of a finite point guard set S ⊂ P must be given different colors if their visible regions overlap. What is the minimum number of colors required to color any guard s...

The product of ratios that equals 1 in Ceva’s Theorem is analyzed in the case of non-concurrent Cevians, for triangles as well as arbitrary convex polygons. A general lemma on complementary systems of inequalities is proved, and used to classify the possible cases of non-concurrent Cevians. In the concurrent case, particular consideration is given to the Brocard configuration defined by equal a...

The solution of a linear reaction diffusion equation on a non-convex polygon is proved to be globally regular in a suitable weighted Sobolev space. This result is used to design an optimally convergent Fourier-Finite Element Method (FEM) where the mesh size is suitably refined. Furthermore, the coupled Non-Standard Finite Difference Method (NSFDM)-FEM is presented as a reliable scheme that repl...

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 KušnirenkoBernštein theorem. We apply this result to the determination of the Newton polygon of a curve parameterized by generic Laurent polynomials or by generic r...

The paper studies local convexity properties of digital curves. We locally define convex and concave parts from the slope of maximal digital straight segments and arithmetically characterize the smallest digital pattern required for checking convexity. Moreover, we introduce the concepts of digital edge and digital hull, a digital hull being a sequence of increasing or decreasing digital edges....

The concept of convex polygon-offset distance function was introduced in 2001 by Barequet, Dickerson, and Goodrich. Using this notion of point-to-point distance, they showed how to compute the corresponding nearestand farthest-site Voronoi diagram for a set of points. In this paper we generalize the polygon-offset distance function to be from a point to any convex object with respect to an m-si...

We propose a simple modi cation to the classical polygon rasterization pipeline that enables exact, e cient raycasting of bounded implicit surfaces without the use of a global spatial data structure or bounding hierarchy. Our algorithm requires two descriptions for each object: a (possibly non-convex) polyhedral bounding volume, and an implicit equation (including, optionally, a number of clipp...

We propose a simple modiication to the classical polygon rasterization pipeline that enables exact, eecient raycasting of bounded implicit surfaces without the use of a global spatial data structure or bounding hierarchy. Our algorithm requires two descriptions for each object: a (possibly non-convex) polyhedral bounding volume, and an implicit equation (including, optionally, a number of clipp...

We propose a simple modiication to the classical polygon rasterization pipeline that enables exact, eecient raycasting of bounded implicit surfaces without the use of a global spatial data structure or bounding hierarchy. Our algorithm requires two descriptions for each object: a (possibly non-convex) polyhedral bounding volume, and an implicit equation (including, optionally, a number of clipp...