Maximum-Area Rectangles in a Simple Polygon

We study the problem of finding maximum-area rectangles contained in a polygon in the plane. There has been a fair amount of work for this problem when the rectangles have to be axis-aligned or when the polygon is convex. We consider this problem in a simple polygon with n vertices, possibly with holes, and with no restriction on the orientation of the rectangles. We present an algorithm that computes a maximum-area rectangle in O(n^3 log n) time using O(kn^2) space, where k is the number of reflex vertices of P. Our algorithm can report all maximum-area rectangles in the same time using O(n^3) space. We also present a simple algorithm that finds a maximum-area rectangle contained in a convex polygon with n vertices in O(n^3) time using O(n) space

Finding planar regions

Preiskovali smo problem iskanja ravnin na trianguliranem terenu. Za množico točk v prostoru zgradimo Delaunayjevo triangulacijo in z dvema različnima metodama poiščemo ravnino na terenu. V prvi metodi uporabimo algoritem za iskanje največjega konveksnega poligona. Algoritem se dobro obnese na manjši množici točk, na večji množici točk, pa zaradi svoje kvadratične časovne zahtevnosti ne pride v poštev. V drugi metodi uporabimo aproksimacijski algoritem. Ta se bolje obnese tudi na večji množici podatkov, kot tudi na realnih geografskih podatkih, ki jih lahko dobimo na spletnem portalu LIDAR. Implementiran vmesnik nam pomaga, da na enostaven način testiramo obe metodi in vizualiziramo rezultate.We studied the problem of finding planar regions in a triangulated terrain. For a set of points in 3-space, we construct the Delaunay triangulation. Then, with two different methods we look for a region which is flat. The first method uses an algorithm to find the largest convex polygon. The algorithm works well on smaller sets of points, but on larger sets of points it performs poorly due to its quadratic time complexity. In the second method, we use an approximation algorithm. It performs better on larger data sets, as well as on real geographical data, which can be obtained on the web portal LIDAR. The implemented interface helps us to test both methods in an easy way and visualize the results