Shape elongation from optimal encasing rectangles

Let SS be a shape with a polygonal boundary. We show that the boundary of the maximally elongated rectangle R(S)R(S) which encases the shape SS contains at least one edge of the convex hull of SS. Such a nice property enables a computationally efficient construction of R(S)R(S).In addition, we define the elongation of a given shape SS as the ratio of the length of R(S)R(S) (determined by the longer edge of R(S)R(S)) and the width of R(S)R(S) (determined by the shorter edge of R(S)R(S)) and show that a so defined shape elongation measure has several desirable properties. Several examples are given in order to illustrate the behavior of the new elongation measure. As a by-product, of the method developed here, we obtain a new method for the computation of the shape orientation, where the orientation of a given shape SS is defined by the direction of the longer edge of R(S)R(S).