Spaces of Geodesic Triangulations of Surfaces

Abstract

We give a short proof of the contractibility of the space of geodesic triangulations with fixed combinatorial type of a convex polygon in the Euclidean plane. Moreover, for any $$n>0$$ n > 0 , we show that there exists a space of geodesic triangulations of a polygon with a triangulation, whose n-th homotopy group is not trivial.

The Deformation Space of Delaunay Triangulations of the Sphere

“…It was conjectured that for constant curvature surfaces they are homotopy equivalent to their smooth counterparts by Connelly et al [5]. This conjecture has been confirmed by Bloch-Connelly-Henderson [2] for convex polygons, and a new proof based on Tuttes' embedding theorem was provided by Luo [13]. Recently, this conjecture was proved for the cases of flat tori and closed surfaces of negative curvature (see Erickson-Lin [10] and Luo-Wu-Zhu [14,15]).…”

Spaces of Geodesic Triangulations Are Cells

“…Tutte's embedding theorem is a fundamental result on the construction of straight-line embeddings of a 3-connected planar graph in the plane, which has profound applications in computational geometry. The connection between Tutte's embedding theorem and spaces of geodesic triangulations of surfaces was first established in [16] if the surface is a convex polygon in the plane. This connection is further generalized to the cases of surfaces of negative curvature in [18] to prove Theorem 1.2.…”

A note on graph drawings with star-shaped boundaries in the plane

“…We recall the following definitions from [20]. An embedded n-sided polygon Ω in the plane is determined by a map φ from the set…”

“…They constructed an example showing that GT (Ω, T ) was not necessarily path-connected if we didn't assume star-shaped boundary. More complicated examples given in [20] show that the spaces of geodesic triangulations of general polygons can be complicated, in the sense that their homotopy groups can have large ranks. We answer the contractibility problem in a special case where the boundary polygon is a non-convex quadrilateral.…”

“…We can vary the coefficients w ij to construct families of geodesic triangulations of a convex polygon. This idea will lead to a simple proof of the contractibility of the space of geodesic triangulations of convex polygons [20].…”