An \omega-wedge is the (closed ) set of all points contained between two rays that are emanating from a single point (the apex), and are separated by an angle \omega < \pi. Given a convex polygon P, we place the \omega-wedge such that P is inside the wedge and both rays are tangent to P. The \omega-cloud of P is the curve traced by the apex of the \omega-wedge as it rotates around Pwhile maintaining tangency in both rays . We investigate reconstructing a polygon P from its \omega-cloud. Previous work on reconstructing P from probes with the \omega-wedge required knowledge of the points of tangency between P  the two rays of the \omega-wedge  . Here we show that if \omega is known, the \omega-cloud alone uniquely determines P, and we give a linear-time reconstruction algorithm. Furthermore, even if we only know that \omega < \pi/ 2, we can still reconstruct P , albeit in cubic time in the number of vertices. This reduces to quadratic time if in addition we are given the location of one of the vertices of P .
An \omega-wedge is the closed set of points contained between two rays that are emanating from a single point (the apex), and are separated by an angle \omega < \pi. Given a convex polygon P, we place the \omega-wedge such that P is inside the wedge and both rays are tangent to P. The set of apex positions of all such placements of the \omega-wedge is called the \omega-cloud of P . We investigate reconstructing a polygon P from its \omega-cloud. Previous work on reconstructing P from probes with the \omega-wedge required knowledge of the points of tangency between P and the two rays of the \omega-wedge in addition to the location of the apex . Here we consider the setting where the maximal \omega-cloud alone is given. We give two conditions under which it uniquely defines P: (i) when \omega < \pi is fixed/given, or (ii) when what is known is that \omega < \pi/ 2. We show that if neither of these two conditions hold, then P may not be unique. We show that, when the uniqueness conditions hold, the polygon P can be reconstructed in O(n) time with O(1) working space in addition to the input, where n is the number of arcs in the input \omega-cloud .
R
(closed ) set of all
closed set of
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\omega-cloud of P is the curve traced by the apex of
set of apex positions of all such placements of
R
as it rotates around Pwhile maintaining tangency in both rays
is called the \omega-cloud of P
A
and
A
in addition to the location of the apex
R
show that if \omega is known, the
consider the setting where the maximal
R
uniquely determines P, and we give a linear-time reconstruction algorithm. Furthermore, even if we only know
is given. We give two conditions under which it uniquely defines P: (i) when \omega < \pi is fixed/given, or (ii) when what is known is
R
2, we can still reconstruct P , albeit in cubic time in
2. We show that if neither of these two conditions hold, then P may not be unique. We show that, when the uniqueness conditions hold, the polygon P can be reconstructed in O(n) time with O(1) working space in addition to the input, where n is the number of arcs in
R
number of vertices. This reduces to quadratic time if in addition we are given the location of one of the vertices of P
input \omega-cloud