We consider the problem of nding a strip separating between two polygons, whose intersection with a third (convex) polygon is of maximum area. We present an optimal linear-time algorithm for computing the optimum strip. When the third polygon is not convex, the running time of the algorithm is quadratic in the size of the input. The application in mind is the piecewise-linear surface interpolat...

We derive a formula for an n-th order divided difference of the inverse of a function. The formula has a simple and surprising structure: it is a sum over partitions of a convex polygon with n+1 vertices. The formula provides a numerically stable method of computing divided differences of k-th roots. It also provides a new way of enumerating all partitions of a convex polygon of a certain type,...

We consider the geometric optimization problem of nding the maximum area axis-parallel rectangle (MAAPR) in an n-vertex general polygon. We characterize the MAAPR for general polygons by considering di erent cases based on the types of contacts between the rectangle and the polygon. We present a general framework for solving a key subcase of the MAAPR problem which dominates the running time fo...

We study the problem of constructing convex polygons and convex polyhedra given the number of visible edges and visible faces from some orthogonal projections. In 2D, we find necessary and sufficient conditions for the existence of a feasible polygon of size N and give an algorithm to construct one, if it exists. When N is not known, we give an algorithm to find the maximum and minimum size of ...

Viviani’s theorem states that the sum of distances from any point inside an equilateral triangle to its sides is constant. We consider extensions of the theorem and show that any convex polygon can be divided into parallel segments such that the sum of the distances of the points to the sides on each segment is constant. A polygon possesses the CVS property if the sum of the distances from any ...

We give a short proof of the following geometric inequality: for any two triangular meshes A and B of the same polygon C, if the number of vertices in A is at most the number of vertices in B, then the maximum length of an edge in A is at least the minimum distance between two vertices in B. Here the vertices in each triangular mesh include the vertices of the polygon and possibly additional St...

Let II be a convex lattice polygon with b boundary points and c (5 1) interior points. We show that for any given a , the number b satisfies b 5 2e + 7 , and identify the polygons for which equality holds. A lattice polygon II is a simple polygon whose vertices are points of the integral lattice. We let A = 4(11) denote the area of II , b{U) the number of lattice points on the boundary of II , ...

In this paper, an O ( n log log n ) algorithm is proposed for minimally rectangular partitioning a simple rectilinear polygon. For any simple rectilinear polygon P , a vertex-edge visible pair is a vertex and an edge that can be connected by a horizontal or vertical line segment that lies entirely inside P . We show that, if the vertex-edge visible pairs are found, the maximum matching and the ...