Abstract
We consider whether any two triangulations of a polygon or a point set on a nonplanar surface with a given metric can be transformed into each other by a sequence of edge flips. The answer is negative in general with some remarkable exceptions, such as polygons on the cylinder, and on the flat torus, and certain configurations of points on the cylinder.
Highlights
Most of the problems considered so far in computational geometry are restricted to the plane, or to the Euclidean 3-space
It is interesting to emphasize the different behavior that these surfaces show when we study the graph of triangulations of a polygon: while the graph of triangulations is connected both in the cylinder and in the flat torus, polygons with nonconnected graphs can be constructed in the two nonorientable surfaces
We introduce some definitions and preliminary results that will lead to a proof of the connectivity of the graph of triangulations of a triangulable polygon on the flat torus
Summary
Most of the problems considered so far in computational geometry are restricted to the plane, or to the Euclidean 3-space. Any closed connected surface S admits a metric that allows polygons and point sets whose graphs of (metrical) triangulations are nonconnected.
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