We present an algorithm for a pair of pursuers, each with one flashlight, searching for an unpredictable, moving target in a 2D environment (simple polygon). Given a polygon with n edges and m concave regions, the algorithm decides in time O(n + nm + m) whether the polygon can be cleared by the two 1-searchers, and if so, constructs a search schedule. The pursuers are allowed to move on the bou...

There are many di-erent kinds of guards in a simple polygon that have been proposed and discussed. In this paper, we consider a new type of guard, boundary guard, which is a guard capable of moving along a boundary of a polygon and every interior point of the polygon can be seen by the mobile guard. We propose an algorithm to 1nd the shortest boundary guard of a simple polygon P in O(n log n) t...

The perimeter and area generating functions of exactly solvable polygon models satisfy q-functional equations, where q is the area variable. The behaviour in the vicinity of the point where the perimeter generating function diverges can often be described by a scaling function. We develop the method of qlinear approximants in order to extract the approximate scaling behaviour of polygon models ...

We establish the relation between the ISO(2, 1) homotopy invariants, and the polygon representation of (2+1)-dimensional gravity. The polygon closure conditions, together with the SO(2, 1) cycle conditions, are equivalent to the ISO(2, 1) cycle conditions for the representations ρ : π 1 (Σ g,N) → ISO(2, 1). Also, the symplectic structure on the space of invariants is closely related to that of ...

We show how to represent a simple polygon P by a grid (pixel-based) polygon Q that is simple and whose Hausdorff or Fréchet distance to P is small. For any simple polygon P , a grid polygon exists with constant Hausdorff distance between their boundaries and their interiors. Moreover, we show that with a realistic input assumption we can also realize constant Fréchet distance between the bounda...

In differential geometry the study of smooth submanifolds with distinguished curvature properties has a long history and belongs to the central themes of this Þeld. Modern work on smooth submanifolds, and on surfaces in particular, relies heavily on geometric and analytic machinery which has evolved over hundreds of years. However, non-smooth surfaces are also natural mathematical objects, even...