Authors: Vincent Pilaud ¹; Francisco Santos ^2,3

• 1 École normale supérieure - Paris
• 2 Universidad de Cantabria [Santander] [UC / UniCan]
• 3 Universidad de Cantabria [Santander] = University of Cantabria [Spain] = Université de Cantabrie [Espagne] [UC / UniCan]

A $k$-triangulation of a convex polygon is a maximal set of diagonals so that no $k+1$ of them mutually cross. $k$-triangulations have received attention in recent literature, with motivation coming from several interpretations of them. We present a new way of looking at $k$-triangulations, where certain star polygons naturally generalize triangles for $k$-triangulations. With this tool we give new, direct proofs of the fundamental properties of $k$-triangulations (number of edges, definition of flip). This interpretation also opens up new avenues of research that we briefly explore in the last section.