Inscribing Convex Polygons in Star-Shaped Objects

This study develops a new algorithm which automatically inscribes a convex polygon in a star shaped object \(\breve{O}\). Starting at \(\breve{O}\)’s mass center, our active contour (E-AC) expands until it encounters the boundary of \(\breve{O}\) (\(\partial \breve{O}\)). As a result it constructs a star-shaped polygon on \(\partial \breve{O}\). We measure the Euclidean distance from \(\partial \breve{O}\)’s mass center to each vertex of the star-shaped polygon defined by E-AC. The distances form a distance function, whose local minima construct star-shaped polygon inscribed in \(\partial \breve{O}\). Its consecutive convex triplets of vertices define a unique pair of convex polygons inscribed in \(\partial \breve{O}\). The Convex Core (CC) of \(\breve{O}\) is defined to be the polygon with the largest area (perimeter if the areas are equal). The CC is unique and invariant to rotation, translation and scaling. Experiments validate the new algorithm. The paper ends listing our contributions and comparing them with contemporary papers.