The distance between shape centroids is less than a quarter of the shape perimeter

In this paper we consider the distance between the shape centroid computed from the shape interior points and the shape centroid computed from the shape boundary points. We show that the distance between those centroids is upper bounded by the quarter of the perimeter of the shape considered. The obtained upper bound is sharp and cannot be improved.Next, we introduce the shape centredness as a new shape descriptor which, informally speaking, should indicate to which degree a shape has a uniquely defined centre. By exploiting the result mentioned above, we give a formula for the computation of the shape centredness. Such a computed centredness is invariant with respect to translation, rotation and scaling transformations.

► The distance between the six shape centroids is less than the shape perimeter.
► The shape centredness C(S) of S is computed asC(S)=1−4Shape‐perimeter·∥Distance‐between‐shape‐centroids∥.
► C(S) is invariant w.r.t. isometric and similarity transformations.
► C(S) ranges over (0, 1].