2 research outputs found
Probing Convex Polygons with a Wedge
Minimizing the number of probes is one of the main challenges in
reconstructing geometric objects with probing devices. In this paper, we
investigate the problem of using an -wedge probing tool to determine
the exact shape and orientation of a convex polygon. An -wedge consists
of two rays emanating from a point called the apex of the wedge and the two
rays forming an angle . To probe with an -wedge, we set the
direction that the apex of the probe has to follow, the line , and the initial orientation of the two rays. A valid -probe of a
convex polygon contains within the -wedge and its outcome
consists of the coordinates of the apex, the orientation of both rays and the
coordinates of the closest (to the apex) points of contact between and each
of the rays.
We present algorithms minimizing the number of probes and prove their
optimality. In particular, we show how to reconstruct a convex -gon (with
all internal angles of size larger than ) using -probes;
if , the reconstruction uses -probes. We show
that both results are optimal. Let be the number of vertices of whose
internal angle is at most , (we show that ). We
determine the shape and orientation of a general convex -gon with
(respectively , ) using (respectively , )
-probes. We prove optimality for the first case. Assuming the algorithm
knows the value of in advance, the reconstruction of with or
can be achieved with probes,- which is optimal.Comment: 31 pages, 27 figure
Minimum Enclosing Area Triangle with a Fixed Angle
Given a set S of n points in the plane and a fixed angle 0 < Ο < Ο, we show how to find all triangles of minimum area with angle Ο that enclose S in O(n log n) time