2 research outputs found
Visibility-Monotonic Polygon Deflation
A deflated polygon is a polygon with no visibility crossings. We answer a
question posed by Devadoss et al. (2012) by presenting a polygon that cannot be
deformed via continuous visibility-decreasing motion into a deflated polygon.
We show that the least n for which there exists such an n-gon is seven. In
order to demonstrate non-deflatability, we use a new combinatorial structure
for polygons, the directed dual, which encodes the visibility properties of
deflated polygons. We also show that any two deflated polygons with the same
directed dual can be deformed, one into the other, through a
visibility-preserving deformation.Comment: 19 pages, 139 figures, abridged version submitted to CCCG 201
Visibility-Preserving Convexifications Using Single-Vertex Moves
Devadoss asked: (1) can every polygon be convexified so that no internal visibility (between vertices) is lost in the process? Moreover, (2) does such a convexification exist, in which exactly one vertex is moved at a time (that is, using single-vertex moves)? We prove the redundancy of the “single-vertex moves” condition: an affirmative answer to (1) implies an affirmative answer to (2). Since Aichholzer et al. recently proved (1), this settles (2)