On the Cover of the Digital Disk using concentric Digital Circles

This paper examines the minimum cover of a digital disk by concentric digital circles. If we take the existing definition of the digital circle, then we show a strict lower and upper bound on the size of the cover. We also generate a min-cover which achieves this lower bound. Now one of the minor boundary conditions of generating a digital circle is redefined. For this new definition, an asymptotic lower bound and a strict upper bound on the number of concentric circles required is provided. Also an optimal Greedy Algorithm for the construction of such a minimum cover using the new definition of boundary conditions is formulated. 1 A note about convention Since we are working with digital circles formed by circles with real radii, a tie between two points may occur while constructing the digital circle. If such a tie occurs, we break it by choosing the point nearer to the origin. 2 Notation • |CZ1 r(1)| denotes the number of points in octant one of the Type 1 digital circle of radius r. • |CZ r(1)| denotes the number of points of the Type 1 digital circle of radius r. We now give the two definitions of the digital circle with which we will be working in this paper. The first one is the traditional definition of the digital circle. The second definition is a slightly modified version of the first. In that, a certain boundary condition is relaxed. This will be useful because as we will see later, if the second definition is used, the number of circles required to cover the disk decreases quite a bit. For the two definitions of digital circles, there are two corresponding definition of digital disks. 3 Definitions Definition 1. CZ r(1) : The traditional definition of a digital circle. The one generated by Bresenham’s algorithm. We call this a Type 1 digital circle.