The Jordan curve theorem in the Khalimsky plane

[EN] The connectivity in Alexandroff topological spaces is equivalent to the path connectivity. This fact gets some specific properties to Z2, equipped with the Khalimsky topology. This allows a sufficiently precise description of the curves in Z2 and permit to prove a digital Jordan curve theorem in Z2.Bouassida, E. (2008). The Jordan curve theorem in the Khalimsky plane. Applied General Topology. 9(2):253-262. doi:10.4995/agt.2008.1805.SWORD2532629

A digital analogue of the Jordan curve theorem

AbstractWe study certain closure operations on Z2, with the aim of showing that they can provide a suitable framework for solving problems of digital topology. The Khalimsky topology on Z2, which is commonly used as a basic structure in digital topology nowadays, can be obtained as a special case of the closure operations studied. By proving an analogy of the Jordan curve theorem for these closure operations, we show that they provide a convenient model of the real plane and can therefore be used for studying topological and geometric properties of digital images. We also discuss some advantages of the closure operations investigated over the Khalimsky topology

Relative Convex Hull Determination from Convex Hulls in the Plane

A new algorithm for the determination of the relative convex hull in the plane of a simple polygon A with respect to another simple polygon B which contains A, is proposed. The relative convex hull is also known as geodesic convex hull, and the problem of its determination in the plane is equivalent to find the shortest curve among all Jordan curves lying in the difference set of B and A and encircling A. Algorithms solving this problem known from Computational Geometry are based on the triangulation or similar decomposition of that difference set. The algorithm presented here does not use such decomposition, but it supposes that A and B are given as ordered sequences of vertices. The algorithm is based on convex hull calculations of A and B and of smaller polygons and polylines, it produces the output list of vertices of the relative convex hull from the sequence of vertices of the convex hull of A.Comment: 15 pages, 4 figures, Conference paper published. We corrected two typing errors in Definition 2: $I_S$ has to be defined based on $O_S$, and $I_E$ has to be defined based on $O_E$ (not just using $O$). These errors appeared in the text of the original conference paper, which also contained the pseudocode of an algorithm where $I_S$ and $I_E$ appeared as correctly define