DSpace Collection:
http://hdl.handle.net/2117/3199
Thu, 29 Jan 2015 08:18:33 GMT2015-01-29T08:18:33Zwebmaster.bupc@upc.eduUniversitat Politècnica de Catalunya. Servei de Biblioteques i DocumentaciónoGeometric Biplane Graphs II: Graph Augmentation
http://hdl.handle.net/2117/26085
Title: Geometric Biplane Graphs II: Graph Augmentation
Authors: Hurtado Díaz, Fernando Alfredo; Garcia Olaverri, Alfredo Martin; Korman Cozzetti, Matias; Matos, Inés P.; Saumell, Maria; Silveira, Rodrigo Ignacio; Tejel Altarriba, Francisco Javier; Tóth, Csaba D.
Abstract: We study biplane graphs drawn on a nite point set
S
in the plane in general position.
This is the family of geometric graphs whose vertex set is
S
and which can be decomposed
into two plane graphs. We show that every su ciently large point set admits a 5-connected
biplane graph and that there are arbitrarily large point sets that do not admit any 6-
connected biplane graph. Furthermore, we show that every plane graph (other than a
wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are
arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph
by adding pairwise noncrossing edges.Mon, 26 Jan 2015 19:01:27 GMThttp://hdl.handle.net/2117/260852015-01-26T19:01:27ZHurtado Díaz, Fernando Alfredo; Garcia Olaverri, Alfredo Martin; Korman Cozzetti, Matias; Matos, Inés P.; Saumell, Maria; Silveira, Rodrigo Ignacio; Tejel Altarriba, Francisco Javier; Tóth, Csaba D.noWe study biplane graphs drawn on a nite point set
S
in the plane in general position.
This is the family of geometric graphs whose vertex set is
S
and which can be decomposed
into two plane graphs. We show that every su ciently large point set admits a 5-connected
biplane graph and that there are arbitrarily large point sets that do not admit any 6-
connected biplane graph. Furthermore, we show that every plane graph (other than a
wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are
arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph
by adding pairwise noncrossing edges.Terrain visibility with multiple viewpoints
http://hdl.handle.net/2117/25140
Title: Terrain visibility with multiple viewpoints
Authors: Hurtado Díaz, Fernando Alfredo; Löffler, Maarten; Matos, Inés P.; Sacristán Adinolfi, Vera; Saumell, Maria; Silveira, Rodrigo Ignacio; Staals, Frank
Abstract: We study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider three fundamental visibility structures: the visibility map, the colored visibility map, and the Voronoi visibility map. We study the complexity of each structure for both 1.5D and 2.5D terrains, and provide efficient algorithms to construct them. Our algorithm for the visibility map in 2.5D terrains improves on the only existing algorithm in this setting.Tue, 23 Dec 2014 18:23:25 GMThttp://hdl.handle.net/2117/251402014-12-23T18:23:25ZHurtado Díaz, Fernando Alfredo; Löffler, Maarten; Matos, Inés P.; Sacristán Adinolfi, Vera; Saumell, Maria; Silveira, Rodrigo Ignacio; Staals, FranknoWe study the problem of visibility in polyhedral terrains in the presence of multiple viewpoints. We consider three fundamental visibility structures: the visibility map, the colored visibility map, and the Voronoi visibility map. We study the complexity of each structure for both 1.5D and 2.5D terrains, and provide efficient algorithms to construct them. Our algorithm for the visibility map in 2.5D terrains improves on the only existing algorithm in this setting.On perfect and quasiperfect dominations in graphs
http://hdl.handle.net/2117/24831
Title: On perfect and quasiperfect dominations in graphs
Authors: Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. LuzTue, 25 Nov 2014 10:36:52 GMThttp://hdl.handle.net/2117/248312014-11-25T10:36:52ZCáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. LuznoLD-graphs and global location-domination in bipartite graphs
http://hdl.handle.net/2117/24830
Title: LD-graphs and global location-domination in bipartite graphs
Authors: Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel
Abstract: A dominating setS of a graph G is a locating-dominating-set, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD - codes
and the cardinality of an LD-code is the
location-domination number
,
¿
(
G
).
An LD-set
S
of a graph
G
is
global
if it is an LD-set for both
G
and its complement,
G
. One of the main contributions of this work is the definition of the
LD-graph
,an
edge-labeled graph associated to an LD-set, that will be very helpful to deduce some
properties of location-domination in graphs. Concretely, we use LD-graphs to study
the relation between the location-domination number in a bipartite graph and its
complementTue, 25 Nov 2014 10:34:30 GMThttp://hdl.handle.net/2117/248302014-11-25T10:34:30ZHernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuelnodomination, location, complement graph, bipartite graphA dominating setS of a graph G is a locating-dominating-set, LD-set for short, if every vertex v not in S is uniquely determined by the set of neighbors of v belonging to S. Locating-dominating sets of minimum cardinality are called LD - codes
and the cardinality of an LD-code is the
location-domination number
,
¿
(
G
).
An LD-set
S
of a graph
G
is
global
if it is an LD-set for both
G
and its complement,
G
. One of the main contributions of this work is the definition of the
LD-graph
,an
edge-labeled graph associated to an LD-set, that will be very helpful to deduce some
properties of location-domination in graphs. Concretely, we use LD-graphs to study
the relation between the location-domination number in a bipartite graph and its
complementThe graph distance game and some graph operations
http://hdl.handle.net/2117/24796
Title: The graph distance game and some graph operations
Authors: Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. Luz
Abstract: In the graph distance game, two players alternate in constructing a max-
imal path. The objective function is the distance between the two endpoints of the
path, which one player tries to maximize and the other tries to minimize. In this paper
we examine the distance game for various graph operations: the join, the corona and
the lexicographic product of graphs. We provide general bounds and exact results for
special graphsFri, 21 Nov 2014 12:24:49 GMThttp://hdl.handle.net/2117/247962014-11-21T12:24:49ZCáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. LuznoDistance game, graph operationsIn the graph distance game, two players alternate in constructing a max-
imal path. The objective function is the distance between the two endpoints of the
path, which one player tries to maximize and the other tries to minimize. In this paper
we examine the distance game for various graph operations: the join, the corona and
the lexicographic product of graphs. We provide general bounds and exact results for
special graphsOn k-enclosing objects in a coloured point set
http://hdl.handle.net/2117/24559
Title: On k-enclosing objects in a coloured point set
Authors: Barba, Luis; Durocher, Stephane; Fraser, Robert; Hurtado Díaz, Fernando Alfredo; Mehrabi, Saeed; Mondal, Debajyoti; Morrison, Jason Morrison; Skala, Matthew; Wahid, Mohammad Abdul
Abstract: We introduce the exact coloured
k
-enclosing object
problem: given a set
P
of
n
points in
R
2
, each of
which has an associated colour in
f
1
;:::;t
g
, and a vec-
tor
c
= (
c
1
;:::;c
t
), where
c
i
2
Z
+
for each 1
i
t
,
nd a region that contains exactly
c
i
points of
P
of
colour
i
for each
i
. We examine the problems of nd-
ing exact coloured
k
-enclosing axis-aligned rectangles,
squares, discs, and two-sided dominating regions in a
t
-coloured point setTue, 04 Nov 2014 19:07:20 GMThttp://hdl.handle.net/2117/245592014-11-04T19:07:20ZBarba, Luis; Durocher, Stephane; Fraser, Robert; Hurtado Díaz, Fernando Alfredo; Mehrabi, Saeed; Mondal, Debajyoti; Morrison, Jason Morrison; Skala, Matthew; Wahid, Mohammad AbdulnoWe introduce the exact coloured
k
-enclosing object
problem: given a set
P
of
n
points in
R
2
, each of
which has an associated colour in
f
1
;:::;t
g
, and a vec-
tor
c
= (
c
1
;:::;c
t
), where
c
i
2
Z
+
for each 1
i
t
,
nd a region that contains exactly
c
i
points of
P
of
colour
i
for each
i
. We examine the problems of nd-
ing exact coloured
k
-enclosing axis-aligned rectangles,
squares, discs, and two-sided dominating regions in a
t
-coloured point setCell-paths in mono- and bichromatic line arrangements in the plane
http://hdl.handle.net/2117/24536
Title: Cell-paths in mono- and bichromatic line arrangements in the plane
Authors: Aichholzer, Oswin; Cardinal, Jean; Hackl, Thomas; Hurtado Díaz, Fernando Alfredo; Korman Cozzetti, Matias; Pilz, Alexander; Silveira, Rodrigo Ignacio; Uehara, Ryuhei; Vogtenhuber, Birgit; Welzl, Emo
Abstract: We show that in every arrangement of n red and blue lines | in general position and not all of the same color | there is a path through a linear number of cells where red and blue lines are crossed alternatingly (and
no cell is revisited). When all lines have the same color, and hence the preceding alternating constraint is dropped, we prove that the dual graph of the arrangement always contains a path of length (n2).Fri, 31 Oct 2014 18:38:08 GMThttp://hdl.handle.net/2117/245362014-10-31T18:38:08ZAichholzer, Oswin; Cardinal, Jean; Hackl, Thomas; Hurtado Díaz, Fernando Alfredo; Korman Cozzetti, Matias; Pilz, Alexander; Silveira, Rodrigo Ignacio; Uehara, Ryuhei; Vogtenhuber, Birgit; Welzl, EmonoWe show that in every arrangement of n red and blue lines | in general position and not all of the same color | there is a path through a linear number of cells where red and blue lines are crossed alternatingly (and
no cell is revisited). When all lines have the same color, and hence the preceding alternating constraint is dropped, we prove that the dual graph of the arrangement always contains a path of length (n2).Software for discussing parametric polynomial systems : the Gröbner cover
http://hdl.handle.net/2117/24349
Title: Software for discussing parametric polynomial systems : the Gröbner cover
Authors: Montes Lozano, Antonio; Wibmer, Michael
Abstract: We present the canonical Gröbner Cover method for discussing parametric polynomial systems of equations. Its objective is to decompose the parameter space into subsets (segments) for which it exists a generalized reduced Gröbner basis in the whole segment with fixed set of leading power products on it. Wibmer's Theorem guarantees its existence. The Gröbner Cover is designed in a joint paper of the authors, and the Singular grobcov.lib library [15] implementing it, is developed by Montes. The algorithm is canonic and groups the solutions having the same kind of properties into different disjoint segments. Even if the algorithms involved have high complexity, we show how in practice it is effective in many applications of medium difficulty. An interesting application to automatic deduction of geometric theorems is roughly described here, and another one to provide a taxonomy for exact geometrical loci computations, that is experimentally implemented in a web based application using the dynamic geometry software Geogebra, is explained in another session.Sat, 11 Oct 2014 15:23:46 GMThttp://hdl.handle.net/2117/243492014-10-11T15:23:46ZMontes Lozano, Antonio; Wibmer, Michaelnoautomatic theorem discovering, canonical algorithm, Groebner cover, parametric polynomialWe present the canonical Gröbner Cover method for discussing parametric polynomial systems of equations. Its objective is to decompose the parameter space into subsets (segments) for which it exists a generalized reduced Gröbner basis in the whole segment with fixed set of leading power products on it. Wibmer's Theorem guarantees its existence. The Gröbner Cover is designed in a joint paper of the authors, and the Singular grobcov.lib library [15] implementing it, is developed by Montes. The algorithm is canonic and groups the solutions having the same kind of properties into different disjoint segments. Even if the algorithms involved have high complexity, we show how in practice it is effective in many applications of medium difficulty. An interesting application to automatic deduction of geometric theorems is roughly described here, and another one to provide a taxonomy for exact geometrical loci computations, that is experimentally implemented in a web based application using the dynamic geometry software Geogebra, is explained in another session.Software using the Gröbner cover for geometrical loci computation and classification
http://hdl.handle.net/2117/24171
Title: Software using the Gröbner cover for geometrical loci computation and classification
Authors: Abanades, Miguel A.; Botan, Francisco; Montes Lozano, Antonio; Recio Muñiz, Tomás
Abstract: We describe here a properly recent application of the Gröbner Cover algorithm (GC) providing an algebraic support to Dynamic Geometry computations of geometrical loci. It provides a complete algebraic solution of locus computation as well as a suitable taxonomy allowing to distinguish the nature of the different components. We included a new algorithm Locus into the Singular grobcov.lib library for this purpose. A web prototype has been implemented using it in Geogebra.Fri, 26 Sep 2014 08:32:28 GMThttp://hdl.handle.net/2117/241712014-09-26T08:32:28ZAbanades, Miguel A.; Botan, Francisco; Montes Lozano, Antonio; Recio Muñiz, TomásnoDynamical geometry, Groebner cover, Locus, TaxonomyWe describe here a properly recent application of the Gröbner Cover algorithm (GC) providing an algebraic support to Dynamic Geometry computations of geometrical loci. It provides a complete algebraic solution of locus computation as well as a suitable taxonomy allowing to distinguish the nature of the different components. We included a new algorithm Locus into the Singular grobcov.lib library for this purpose. A web prototype has been implemented using it in Geogebra.The degree-diameter problem in maximal bipartite planar graphs
http://hdl.handle.net/2117/24097
Title: The degree-diameter problem in maximal bipartite planar graphs
Authors: Dalfó Simó, Cristina; Huemer, Clemens; Salas, Julian
Abstract: The (A ,D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree and diameter D. We consider the (A ,D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the ( , 2) problem, the number of vertices is n = + 2; and for the ( , 3) problem, n = 3 -1 if is odd and n = 3 -2 if is even. Then, we study the general case ( A ,D) and obtain that an upper bound on n is approximately 3(2D+1)( -2)bD/2c,
and another one is C( - 2)bD/2c if D and C is a sufficiently large constant.
Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately ( - 2)k if D = 2k, and 3( - 3)k if D = 2k + 1, for and D sufficiently large in both cases.Thu, 18 Sep 2014 10:55:22 GMThttp://hdl.handle.net/2117/240972014-09-18T10:55:22ZDalfó Simó, Cristina; Huemer, Clemens; Salas, JuliannoMaximal planar bipartite graphsThe (A ,D) (degree/diameter) problem consists of finding the largest possible number of vertices n among all the graphs with maximum degree and diameter D. We consider the (A ,D) problem for maximal planar bipartite graphs, that are simple planar graphs in which every face is a quadrangle. We obtain that for the ( , 2) problem, the number of vertices is n = + 2; and for the ( , 3) problem, n = 3 -1 if is odd and n = 3 -2 if is even. Then, we study the general case ( A ,D) and obtain that an upper bound on n is approximately 3(2D+1)( -2)bD/2c,
and another one is C( - 2)bD/2c if D and C is a sufficiently large constant.
Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on n for maximal planar bipartite graphs, which is approximately ( - 2)k if D = 2k, and 3( - 3)k if D = 2k + 1, for and D sufficiently large in both cases.Colored spanning graphs for set visualization
http://hdl.handle.net/2117/23314
Title: Colored spanning graphs for set visualization
Authors: Hurtado Díaz, Fernando Alfredo; Korman Cozzetti, Matias; Van Kreveld, Matias; Löffler, Maarten; Sacristán Adinolfi, Vera; Silveira, Rodrigo Ignacio; Speckmann, Bettina
Abstract: We study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red
points belong exclusively to the red set, and purple points belong to both sets.
A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected.
We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem is NP-hard. Hence we give an (1/2¿+1)-approximation, where ¿ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets.Thu, 26 Jun 2014 17:02:39 GMThttp://hdl.handle.net/2117/233142014-06-26T17:02:39ZHurtado Díaz, Fernando Alfredo; Korman Cozzetti, Matias; Van Kreveld, Matias; Löffler, Maarten; Sacristán Adinolfi, Vera; Silveira, Rodrigo Ignacio; Speckmann, BettinanoWe study an algorithmic problem that is motivated by ink minimization for sparse set visualizations. Our input is a set of points in the plane which are either blue, red, or purple. Blue points belong exclusively to the blue set, red
points belong exclusively to the red set, and purple points belong to both sets.
A red-blue-purple spanning graph (RBP spanning graph) is a set of edges connecting the points such that the subgraph induced by the red and purple points is connected, and the subgraph induced by the blue and purple points is connected.
We study the geometric properties of minimum RBP spanning graphs and the algorithmic problems associated with computing them. Specifically, we show that the general problem is NP-hard. Hence we give an (1/2¿+1)-approximation, where ¿ is the Steiner ratio. We also present efficient exact solutions if the points are located on a line or a circle. Finally we consider extensions to more than two sets.An open-source toolbox for motion analysis of closed-chain mechanisms
http://hdl.handle.net/2117/23201
Title: An open-source toolbox for motion analysis of closed-chain mechanisms
Authors: Porta Pleite, Josep Maria; Ros Giralt, Lluís; Bohigas Nadal, Oriol; Manubens Ferriol, Montserrat; Rosales Gallegos, Carlos; Jaillet, Leonard Georges
Abstract: Many situations in Robotics require an effective analysis of the motions of a closed-chain mechanism. Despite appearing very often in practice (e.g. in parallel manipulators, reconfigurable robots, or molecular compounds), there is a lack of general tools to effectively analyze the complex configuration spaces of such systems. This paper describes the CUIK suite, an open-source toolbox for motion analysis of general closed-chain mechanisms. The package can determine the motion range of the whole mechanism or of some of its parts, detect singular configurations leading to control or dexterity issues, or find collision- and singularity-free paths between given configurations. The toolbox is the result of several years of research and development within the Kinematics and Robot Design group at IRI, Barcelona, and is available under GPLv3 license from http://www.iri.upc.edu/cuik.Wed, 11 Jun 2014 15:57:37 GMThttp://hdl.handle.net/2117/232012014-06-11T15:57:37ZPorta Pleite, Josep Maria; Ros Giralt, Lluís; Bohigas Nadal, Oriol; Manubens Ferriol, Montserrat; Rosales Gallegos, Carlos; Jaillet, Leonard GeorgesnoRobots
PARAULES AUTOR: kinematic constraints, motion analysis and planning, branch-and prune, higher-dimensional continuationMany situations in Robotics require an effective analysis of the motions of a closed-chain mechanism. Despite appearing very often in practice (e.g. in parallel manipulators, reconfigurable robots, or molecular compounds), there is a lack of general tools to effectively analyze the complex configuration spaces of such systems. This paper describes the CUIK suite, an open-source toolbox for motion analysis of general closed-chain mechanisms. The package can determine the motion range of the whole mechanism or of some of its parts, detect singular configurations leading to control or dexterity issues, or find collision- and singularity-free paths between given configurations. The toolbox is the result of several years of research and development within the Kinematics and Robot Design group at IRI, Barcelona, and is available under GPLv3 license from http://www.iri.upc.edu/cuik.Stabbing simplices of point sets with k-flats
http://hdl.handle.net/2117/22736
Title: Stabbing simplices of point sets with k-flats
Authors: Cano, Javier; Hurtado Díaz, Fernando Alfredo; Urrutia Galicia, Jorge
Abstract: Let S be a set of n points inRdin general position.A set H of k-flats is called an mk-stabber of S if the relative interior of anym-simplex with vertices in S is intersected by at least one element of H. In thispaper we give lower and upper bounds on the size of mínimum mk-stabbers of point sets in Rd. We study mainly mk-stabbers in the plane and in R3Mon, 28 Apr 2014 16:17:42 GMThttp://hdl.handle.net/2117/227362014-04-28T16:17:42ZCano, Javier; Hurtado Díaz, Fernando Alfredo; Urrutia Galicia, JorgenoLet S be a set of n points inRdin general position.A set H of k-flats is called an mk-stabber of S if the relative interior of anym-simplex with vertices in S is intersected by at least one element of H. In thispaper we give lower and upper bounds on the size of mínimum mk-stabbers of point sets in Rd. We study mainly mk-stabbers in the plane and in R3Witness bar visibility graphs
http://hdl.handle.net/2117/22337
Title: Witness bar visibility graphs
Authors: Cortés, Carmen; Hurtado Díaz, Fernando Alfredo; Márquez, Alberto; Valenzuela, Jesús
Abstract: Bar visibility graphs were introduced in the seventies as a model for some VLSI layout problems.
They have been also studied since then by the graph drawing community, and recently several
generalizations and restricted versions have been proposed.
We introduce a generalization, witness-bar visibility graphs, and we prove that this class encom-
passes all the bar-visibility variations considered so far. In addition, we show that many classes of graphs are contained in this family, including in particular all planar graphs, interval graphs, circular arc graphs and permutation graphsFri, 21 Mar 2014 16:03:37 GMThttp://hdl.handle.net/2117/223372014-03-21T16:03:37ZCortés, Carmen; Hurtado Díaz, Fernando Alfredo; Márquez, Alberto; Valenzuela, JesúsnoBar visibility graphs were introduced in the seventies as a model for some VLSI layout problems.
They have been also studied since then by the graph drawing community, and recently several
generalizations and restricted versions have been proposed.
We introduce a generalization, witness-bar visibility graphs, and we prove that this class encom-
passes all the bar-visibility variations considered so far. In addition, we show that many classes of graphs are contained in this family, including in particular all planar graphs, interval graphs, circular arc graphs and permutation graphsStabbing Segments with Rectilinear Objects
http://hdl.handle.net/2117/21700
Title: Stabbing Segments with Rectilinear Objects
Authors: Claverol Aguas, Mercè; Seara Ojea, Carlos; Garijo, Delia; Korman, M.; Silveira, Rodrigo Ignacio
Abstract: Given a set of n line segments in the plane, we say that a region R of the plane is a stabber if R contains exactly one end point of each segment of the set. In this paper we provide efficient algorithms for determining wheter or not a stabber exists for several shapes of stabbers. Specially, we consider the case in which the stabber can be described as the intersecction of isothetic halfplanes (thus the stabbers are halfplanes, strips, quadrants, 3-sided rectangles, or rectangles). We provided efficient algorithms reporting all combinatorially different stabbers of the shape. The algorithms run in O(n) time (for the halfplane case), O(n logn) time (for strips and quadrants), O(n^2) (for 3-sided rectangles), or O(n^3) time (for rectangles).Fri, 21 Feb 2014 17:24:50 GMThttp://hdl.handle.net/2117/217002014-02-21T17:24:50ZClaverol Aguas, Mercè; Seara Ojea, Carlos; Garijo, Delia; Korman, M.; Silveira, Rodrigo IgnacionoGiven a set of n line segments in the plane, we say that a region R of the plane is a stabber if R contains exactly one end point of each segment of the set. In this paper we provide efficient algorithms for determining wheter or not a stabber exists for several shapes of stabbers. Specially, we consider the case in which the stabber can be described as the intersecction of isothetic halfplanes (thus the stabbers are halfplanes, strips, quadrants, 3-sided rectangles, or rectangles). We provided efficient algorithms reporting all combinatorially different stabbers of the shape. The algorithms run in O(n) time (for the halfplane case), O(n logn) time (for strips and quadrants), O(n^2) (for 3-sided rectangles), or O(n^3) time (for rectangles).