Computation of the Euler number using the contact perimeter

We present the computation of the Euler number of shapes using the contact perimeter. The contact perimeter was initially defined in [E. Bribiesca, Measuring 2D shape compactness using the contact perimeter, Comput. Math. Appl. 33 (1997) 1–9]. In this reference the contact perimeter was used to define a measure of compactness for 2D shapes. Now, in this paper we use the contact perimeter to compute the Euler number of unit-width objects composed of different side-connected cells and face-connected polyhedrons in two and three dimensions, respectively. Finally, we present some applications of this computation in knot and graph theory.