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Erschienen in:
2006 | OriginalPaper | Buchkapitel
Throwing Stones Inside Simple Polygons
verfasst von : Otfried Cheong, Hazel Everett, Hyo-Sil Kim, Sylvain Lazard, René Schott
Erschienen in: Algorithmic Aspects in Information and Management
Verlag: Springer Berlin Heidelberg
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Given two sets
A
and
B
of
m
non-intersecting line segments in the plane, we show how to compute in
O
(
m
log
m
) time a data structure that uses
O
(
m
) space and allows to answer the following query in
O
(log
m
) time: Given a parabola
γ
:
y
=
ax
2
+
bx
+
c
, does
γ
separate
A
and
B
? This structure can be used to build a data structure that stores a simple polygon and allows ray-shooting queries along parabolic trajectories with vertical main axis. For a polygon with complexity
n
, we can answer such “stone throwing” queries in
O
(log
2
n
) time, using
O
(
n
log
n
) space and
O
(
n
log
2
n
) preprocessing time. This matches the best known bound for circular ray shooting in simple polygons.