An Efficient Algorithm to Calculate the Center of the Biggest Inscribed Circle in an Irregular Polygon

In this paper, an efficient algorithm to find the center of the biggest circle inscribed in a given polygon is described. This work was inspired by the publication of Daniel Garcia-Castellanos & Umberto Lombardo and their algorithm [1] used to find a landmass’ poles of inaccessibility. Two more efficient algorithms were found, one of them only applicable when the problem can be described as a linear problem, like in the case of a convex polygon. Introduction Given a polygon of N sides, how can we find the biggest inscribed circle? In other words, what is the furthest point from any vertex or edge? If the polygon has 3 sides, which would make it a triangle, this point is equivalent to the incircle and could be found by intersecting the bisection of any two of its angles. Likewise, if the polygon is a square, a pentagon, or any other regular shape, the same method works as illustrated in Figure 1. However, when that method is applied to an irregular polygon with more than 3 sides it might fail as shown in Figure 2. Figure 1. Center of the biggest inscribed circle in regular polygons using the intersection of any two bisections of the polygon’s angles. Figure 2. Intersection of two bisections does not necessarily yield the center of the biggest inscribed circle inside of an irregular polygon. When facing the problem with an irregular polygon, the solution can be found by taking the same approach used for the Poles of Inaccessibility (PIA) problem. A PIA is the most difficult point to reach within a landmass, commonly described as the point furthest from any coastline. If we look at a landmass as a polygon, the PIA of such landmass is identical to the center of the biggest circle inscribed in it. Summarizing the known algorithm’s methodology, to find the PIA we first describe a region R around a initial candidate for the PIA, then divide R into n by m cells and find the node (intersection between lines) furthest from any coastline to make it our next candidate PIA. Subsequently, R is centered around the new candidate PIA and shrunk by a factor of k, finding a new candidate and repeating the process until the desired precision is achieved. Figure 3 shows a visual demonstration of the methodology; for a detailed description of how the sequential algorithm used to find the PIA works refer to the original paper by Daniel Garcia-Castellanos & Umberto Lombardo. Figure 3. Visual demonstration of the sequential algorithm to find the PIA. Each point represents a node that has been compared against the candidate PIA in its iteration and the algorithm is recursively called until the density of nodes corresponds to the desired precision to be achieved. Monte Carlo’s approach The first proposed methodology is very similar to the known algorithm used to find the PIA. The key difference is in the choice of nodes, which would be done by the use of pseudo-randomly chosen coordinates within the region R, rather than sequentially by intersecting n and m lines. By choosing the nodes pseudo-randomly, the an iteration of the algorithm may finish at any arbitrary point rather than having to test n × m nodes in each of the regions. For this implementation, the region is shrunk when a new candidate PIA has not been found within the last k attempts. In Figure 4 we can visualize how the algorithm works and in Figure 5 the pseudo-code describing the algorithm is presented. Figure 4. Visual demonstration of the randomized algorithm to find the PIA. Each point represents a node that has been compared against the candidate PIA in its iteration and the algorithm is recursively called until the density of nodes corresponds to the desired precision to be achieved. while accuracy > minimum_accuracy do: # begin loop through nodes while consecutive_misses < k do: # select coordinates at random within bounds x := random(min_x, max_x) y := random(min_y, max_y) node := (x, y) # loop through edges and find shortest distance for edge := first_edge to last_edge do: if distance(node, edge) < smallest_distance smallest_distance := distance(node, edge) PIA := node end if end do # maximize the minimum distance through iterations # and keep track of the consecutive number of times # that a smallest distance has not been found if smallest_distance > maximin_distance maximin_distance := smallest_distance consecutive_misses := 0 else consecutive_misses := consecutive_misses + 1 end if end do # calculate current level of accuracy based on the # smallest distance between the upper and lower bound accuracy := min(max_x min_x, max_y min_y) # update the bounds of the region min_x := PIA(x) (max_x min_x) / (sqrt(2) * 2) max_x := PIA(x) + (max_x min_x) / (sqrt(2) * 2) min_y := PIA(y) (max_y min_y) / (sqrt(2) * 2) max_y := PIA(y) + (max_y min_y) / (sqrt(2) * 2)