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@ -4167,7 +4167,7 @@ cell neighborhood in $O(m)$ time."
, succeeds = "aarx-clgta-96"
, update = "98.07 rote, 98.03 mitchell, 97.03 rote"
, abstract = "Exploiting the concept of so-called light edges, we introduce
a new way of defining the greedy triangulation GT(S) of a point
a new way of defining the greedy triangulation GT(S) of a point
set S. It provides a decomposition of
GT(S) into levels, and the number of levels allows us to
bound the total edge length of GT(S). In particular, we show
@ -12292,9 +12292,9 @@ method that uses very different techniques."
, pages = "201--290"
, url = "http://wwwpi6.fernuni-hagen.de/Publikationen/tr198.pdf"
, succeeds = "ak-vd-96"
, cites = "ahknu-vdcfc-95, abms-claho-94, aesw-emstb-91, agss-ltacv-89, ahl-sqrpc-90, aiks-fkpmd-91, a-dppuv-82, aaag-ntsp-95, aa-skfgpf-95, aacktrx-tin-96, a-nemts-83, as-vdco-95, ay-aampt1-90, ay-aampt2-90, ar-cvddp-96, a-gvdps-89, a-lbvdc-98, abky-cabmm-88, abcw-cpdts-88, ab-rdt-85, a-sdcgp-85, a-pdpaa-87, a-rpccc-87, a-iadbu-88, a-lcpd-88, a-ndrcv-90, a-vdsfg-91, ae-oacwv-84, aha-mttls-92, ai-grvd-88, as-solri-92, abi-cvus-88, br-rasas-90, bo-arcgi-79, bs-dcms-76, bwy-oetac-80, be-mgot-92i, beg-pgmg-94, b-bnvd, bt-gatuf-85, bmt-dppbp-96, bcdt-osc3d-91i, bdsty-arsol-92, bg-tdrcs-93, bsty-vdhdc-95, bt-rcdt-93, bh-cpcoe-88, b-rgsdd-89, b-vdch-79, b-gtfga-80, bms-hcvdl-94, b-gog-55, c-vmpmp-85, cd-svd-88, crw-gc-91, csy-oscpf-95, cdns-nsrgs-95, c-ochaa-93, ce-iacko-87, cgt-ecabs-95, cx-alesw-96, c-bvdcp-86, c-cdt-89, c-tapga-89, c-gqmgc-93,
cd-vdbcd-85, ckstw-vdl3s-95, csw-fmasp-95, c-wcanh-76, cms-mfdca-94, c-narsc-87, c-agmst-89, cms-frric-93, cs-arscg-89, cmrs-tspvd-93, cw-pspp-86, dj-wtacg-89, ds-oiti-89, bms-plpco-93, dfnp-stdt-91, d-eaclm-77, dk-savd-87, dk-bspwa-97, dn-cgacp-85, d-slsv-34, d-pp-44, d-rysoa-92, dv-cprac-77, dds-saeid-92, de-apphd-96, d-nhndt-87, d-tdt-90, d-fhcdt-92, d-udrdp-50, dl-cvdrp-91, ddg-fsp-83, dfs-dgaag-90, dl-pmtds-89, dl-msp-76, d-ec-83, d-fdcac-87, d-hdvdl-91, e-acg-87, e-atccd-90, e-ubids-95, egs-ueplf-89, egs-oplms-86, eks-sspp-83, em-tdas-94, eos-calha-86, es-vda-86, ess-ztha-93, es-itfwr-96, es-otatd-91, et-qtaml-93, et-ubcdt-93, etw-otama-92, ei-drmwc-79, f-sodt-90, f-iwatd-86, f-nsa2d-92, f-vddt-92, f-savd-87, gs-nsagv-69, gj-cigtn-79, g-3dmud-95, g-agt-85, grr-vdlsm-95, grss-sracp-95,
goy-cvdsl-93, gs-cdtp-78, gks-ricdv-92, gmr-vdmpp-92, gs-pmgsc-85, h-ca-75, h-rvdlp-92, h-gbitp-91, hkp-itnc-91, hn-sc-89, hns-psscp-88, h-pcprn-91, hks-uevsi-93, h-oarms-79, iklm-cdf3s-93, iki-awvdr-94, iss-nriac-92, j-3dtlt-89, j-ctddt-91, j-gspgm-91, jrz-ccdt-91, km-accvd-91, km-icdmc-92, kk-cdtmb-88, kg-cgwac-92, k-csmwt-94, ks-tpscv-93, k-eccs-79, k-ndot-80, k-osps-83, kr-fcm-85, k-cddvd-80, k-cavd-89, kl-ltrab-93, kl-fsc-95, kl-mpsp-95, kmm-ricav-93a, kw-vdbgm-88, k-sssgt-56, kl-fsacl-95, l-dtmmi-94, l-esta-94, l-scsi-77, l-vdlmh-94, l-ricsa-95, l-tdvdl-80, l-matps-82, l-knnvd-82, ld-gvdp-81, ll-gdtpg-86, lw-vdllm-80, lk-qgtam-96, l-ltcrn-94, msw-sblp-92, ms-pggrg-80, m-dtchn-84, mr-vdcdg-90, mks-cplvd-96, m-tdird-76, m-mnfcp-70, m-lpltw-84, mmo-cavd-91, m-zkav-93,
, cites = "ahknu-vdcfc-95, abms-claho-94, aesw-emstb-91, agss-ltacv-89, ahl-sqrpc-90, aiks-fkpmd-91, a-dppuv-82, aaag-ntsp-95, aa-skfgpf-95, aacktrx-tin-96, a-nemts-83, as-vdco-95, ay-aampt1-90, ay-aampt2-90, ar-cvddp-96, a-gvdps-89, a-lbvdc-98, abky-cabmm-88, abcw-cpdts-88, ab-rdt-85, a-sdcgp-85, a-pdpaa-87, a-rpccc-87, a-iadbu-88, a-lcpd-88, a-ndrcv-90, a-vdsfg-91, ae-oacwv-84, aha-mttls-92, ai-grvd-88, as-solri-92, abi-cvus-88, br-rasas-90, bo-arcgi-79, bs-dcms-76, bwy-oetac-80, be-mgot-92i, beg-pgmg-94, b-bnvd, bt-gatuf-85, bmt-dppbp-96, bcdt-osc3d-91i, bdsty-arsol-92, bg-tdrcs-93, bsty-vdhdc-95, bt-rcdt-93, bh-cpcoe-88, b-rgsdd-89, b-vdch-79, b-gtfga-80, bms-hcvdl-94, b-gog-55, c-vmpmp-85, cd-svd-88, crw-gc-91, csy-oscpf-95, cdns-nsrgs-95, c-ochaa-93, ce-iacko-87, cgt-ecabs-95, cx-alesw-96, c-bvdcp-86, c-cdt-89, c-tapga-89, c-gqmgc-93,
cd-vdbcd-85, ckstw-vdl3s-95, csw-fmasp-95, c-wcanh-76, cms-mfdca-94, c-narsc-87, c-agmst-89, cms-frric-93, cs-arscg-89, cmrs-tspvd-93, cw-pspp-86, dj-wtacg-89, ds-oiti-89, bms-plpco-93, dfnp-stdt-91, d-eaclm-77, dk-savd-87, dk-bspwa-97, dn-cgacp-85, d-slsv-34, d-pp-44, d-rysoa-92, dv-cprac-77, dds-saeid-92, de-apphd-96, d-nhndt-87, d-tdt-90, d-fhcdt-92, d-udrdp-50, dl-cvdrp-91, ddg-fsp-83, dfs-dgaag-90, dl-pmtds-89, dl-msp-76, d-ec-83, d-fdcac-87, d-hdvdl-91, e-acg-87, e-atccd-90, e-ubids-95, egs-ueplf-89, egs-oplms-86, eks-sspp-83, em-tdas-94, eos-calha-86, es-vda-86, ess-ztha-93, es-itfwr-96, es-otatd-91, et-qtaml-93, et-ubcdt-93, etw-otama-92, ei-drmwc-79, f-sodt-90, f-iwatd-86, f-nsa2d-92, f-vddt-92, f-savd-87, gs-nsagv-69, gj-cigtn-79, g-3dmud-95, g-agt-85, grr-vdlsm-95, grss-sracp-95,
goy-cvdsl-93, gs-cdtp-78, gks-ricdv-92, gmr-vdmpp-92, gs-pmgsc-85, h-ca-75, h-rvdlp-92, h-gbitp-91, hkp-itnc-91, hn-sc-89, hns-psscp-88, h-pcprn-91, hks-uevsi-93, h-oarms-79, iklm-cdf3s-93, iki-awvdr-94, iss-nriac-92, j-3dtlt-89, j-ctddt-91, j-gspgm-91, jrz-ccdt-91, km-accvd-91, km-icdmc-92, kk-cdtmb-88, kg-cgwac-92, k-csmwt-94, ks-tpscv-93, k-eccs-79, k-ndot-80, k-osps-83, kr-fcm-85, k-cddvd-80, k-cavd-89, kl-ltrab-93, kl-fsc-95, kl-mpsp-95, kmm-ricav-93a, kw-vdbgm-88, k-sssgt-56, kl-fsacl-95, l-dtmmi-94, l-esta-94, l-scsi-77, l-vdlmh-94, l-ricsa-95, l-tdvdl-80, l-matps-82, l-knnvd-82, ld-gvdp-81, ll-gdtpg-86, lw-vdllm-80, lk-qgtam-96, l-ltcrn-94, msw-sblp-92, ms-pggrg-80, m-dtchn-84, mr-vdcdg-90, mks-cplvd-96, m-tdird-76, m-mnfcp-70, m-lpltw-84, mmo-cavd-91, m-zkav-93,
m-uuam-28, m-rcvdp-93, m-spop-93, mmp-dgp-87, ms-getcc-88, mp-fitcp-78, m-osclv-90, m-lavd-91, osy-gvdl-86, osy-gvdl-87, oy-rmpmd-85, oim-iimvd-84, obs-stcav-92, p-etspi-77, pl-ecgvd-95, p-kpudz-82, p-mrpdt-92, ps-cgi-85, p-scnsg-57, r-odtr-91, rr-oprav-94, r-aiv-94, r-mrpdt-90, r-tbvdm-93, rsl-ashts-77, st-pwvt-88, s-cvdhd-82, s-mplbc-85, s-chdch-86, s-nfhdv-87, s-cdtvd-88, s-sdlpc-91, s-barga-93, s-cg-78, sh-cpp-75, s-icpps-85, s-atubl-94, s-let-78, s-vidt-80, s-sagdt-91, s-facdt-87, s-mmdpsl-91a, sd-csdta-95, s-smane-92, si-cvdom-92, soi-toari-90, s-rngam-83, si-atpvd-86, t-otdt-93, too-natdv-83, t-gcvdm-86, t-rngfp-80, v-mstkd-88, v-sgagc-91, v-dmrsl-09, v-nadpc-08, w-eucdt-93, ws-oacdt-87, w-cnddt-81, w-sedbe-91a, www-sdpfo-87, y-cmstk-82, y-amp-87, y-oavds-87, zm-sdnah-91, ZZZ"
, update = "00.11 smid, 00.03 bibrelex, 99.03 bibrelex, 98.11 bibrelex, 98.07 mitchell, 98.03 bibrelex, 97.03 icking"
, annote = "Chapter 5 of su-hcg-00"
@ -12847,9 +12847,9 @@ method that uses very different techniques."
, url = "http://www.ifor.math.ethz.ch/staff/fukuda/fukuda.html"
, update = "98.03 houle, 97.03 pocchiola, 96.05 fukuda"
, annote = "Reverse search is a general exhaustive search technique
which came out of the new convex hull algorithm by the authors.
which came out of the new convex hull algorithm by the authors.
This technique can be applied to many enumeration problems in computer
science, operations research and geometry.
science, operations research and geometry.
It is highly suitable for parallelization."
}
@ -16291,7 +16291,7 @@ rendering. Contains pseudocode."
, number = 4
, year = 1997
, note = "Special issue on parallel I/O. An earlier version
appears in Proc. of the 8th Annual
appears in Proc. of the 8th Annual
ACM Symposium on Parallel Algorithms and
Architectures (SPAA~'96), Padua, Italy, June 1996, 109--118"
, update = "97.03 murali"
@ -18449,12 +18449,12 @@ the interior. Contains pseudocode."
, succeeds = "d-rld-89"
, update = "98.07 bibrelex, 98.03 mitchell, 93.09 held"
, annote = "He considers rectilinear paths in a rectilinear simple polygon. In $O(n\log n)$
preprocessing time and space, he builds a data structure that supports
$O(\log n)$ time queries for distance between two points ($O(1)$ time between
two polygon vertices). He is actually searching for paths that are``smallest'' in
that they are shortest simultaneously in rectilinear link distance and
$L_1$ length (which is always possible). See improvements to $O(n)$ time and
space by Lingas, Maheshwari, and Sack~\cite{lms-parld-95} and
preprocessing time and space, he builds a data structure that supports
$O(\log n)$ time queries for distance between two points ($O(1)$ time between
two polygon vertices). He is actually searching for paths that are``smallest'' in
that they are shortest simultaneously in rectilinear link distance and
$L_1$ length (which is always possible). See improvements to $O(n)$ time and
space by Lingas, Maheshwari, and Sack~\cite{lms-parld-95} and
Schuierer~\cite{s-odssr-96}."
}
@ -21011,10 +21011,10 @@ cubes with side-lengths not exceeding 1 in the $3$-dimensional
euclidean space. Let $S$ and $T$ be two points lying outside
the open cubes. Assume one needs to find a short path emanating
from $S$ and terminating at $T$ avoiding the cubes of $\cal P$
under the restriction that the cubes are not known prior to the search.
under the restriction that the cubes are not known prior to the search.
In fact the positions and the side-lengths of the cubes become known
to the searcher as the cubes are contacted. We give an algorithm to
construct a path of length less than
construct a path of length less than
$\frac 32 d + 3 \sqrt 3 \log d + 5$,
where $d > 3 \sqrt 3$ denotes the distance between S and T."
}
@ -24325,7 +24325,7 @@ experimental results are given."
, abstract = "This paper
presents the main algorithmic and design choices that have been
made
to implement triangulations in
to implement triangulations in
the computational geometry algorithms library CGAL."
}
@ -25614,9 +25614,9 @@ present a polynomial-time exact algorithm to solve this problem."
rectangles floating in 3-space, with edges represented by
vertical lines of sight. We apply an extension of the
{Erd\H os}-Szekeres Theorem in a geometric setting to obtain an
upper bound of 56 for size of the largest complete graph which
is representable. On the other hand, we construct a
representation of the complete graph with 22 vertices.
upper bound of 56 for size of the largest complete graph which
is representable. On the other hand, we construct a
representation of the complete graph with 22 vertices.
These are the best existing bounds."
}
@ -28630,7 +28630,7 @@ determinants."
, year = 1993
, update = "98.03 mitchell"
, abstract = "We calculate the partition
of the configuration space $I\!\!R^2 x S^1$
of the configuration space $I\!\!R^2 x S^1$
of a car-like robot, only moving forwards, with respect to the
type of the length optimal paths. This kind of robot is subject to
kinematic constraints on its path curvature and its orientation.
@ -33590,35 +33590,35 @@ determinants."
Given a set of polygonal obstacles of $n$ vertices in the plane,
the problem of processing the all-pairs Euclidean {\em short} path
queries is that of reporting an obstacle-avoiding path $P$ (or
its length) between two arbitrary query points $p$ and $q$ in the
plane, such that the length of $P$ is within a small factor of the
its length) between two arbitrary query points $p$ and $q$ in the
plane, such that the length of $P$ is within a small factor of the
length of a Euclidean {\em shortest} obstacle-avoiding path between
$p$ and $q$. The goal is to answer each short path query quickly
by constructing data structures that capture path information in
the obstacle-scattered plane. For the related all-pairs Euclidean
{\em shortest} path problem, the best known algorithms for even
very simple cases (e.g., {\em rectilinear} shortest paths among
by constructing data structures that capture path information in
the obstacle-scattered plane. For the related all-pairs Euclidean
{\em shortest} path problem, the best known algorithms for even
very simple cases (e.g., {\em rectilinear} shortest paths among
disjoint {\em rectangular} obstacles in the plane) require
at least quadratic space and time to construct a data structure,
so that a length query can be answered in polylogarithmic time.
The previously best known solution to the all-pairs Euclidean
{\em short} path problem also uses a data structure of quadratic
space and superquadratic construction time, in order to answer a
length query in polylogarithmic time. In this paper, we present a
data structure that requires nearly linear space and takes subquadratic
time to construct. Precisely, for any given $\epsilon$ satisfying
so that a length query can be answered in polylogarithmic time.
The previously best known solution to the all-pairs Euclidean
{\em short} path problem also uses a data structure of quadratic
space and superquadratic construction time, in order to answer a
length query in polylogarithmic time. In this paper, we present a
data structure that requires nearly linear space and takes subquadratic
time to construct. Precisely, for any given $\epsilon$ satisfying
$0$ $<$ $\epsilon$ $\leq$ $1$, our data structure can be built
in $o(q^{3/2})$ $+$ $O((n\log n)/\epsilon)$ time and
$O(n\log n+n/\epsilon)$ space, where $q$, $1$ $\leq$ $q$ $\leq$ $n$,
is the minimum number of faces needed to cover all the vertices of
a certain planar graph we use. This data structure enables us to
in $o(q^{3/2})$ $+$ $O((n\log n)/\epsilon)$ time and
$O(n\log n+n/\epsilon)$ space, where $q$, $1$ $\leq$ $q$ $\leq$ $n$,
is the minimum number of faces needed to cover all the vertices of
a certain planar graph we use. This data structure enables us to
report the length of a short path between two arbitrary query points
in $O((\log n)/\epsilon+1/\epsilon^2)$ time and the actual path
in $O((\log n)/\epsilon+1/\epsilon^2+L)$ time, where $L$ is the
number of edges of the output path. The constant approximation
factor, $6+\epsilon$, for the short paths that we compute is quite
small. Our techniques are parallelizable and can also be used
to improve the previously best known results on several related
in $O((\log n)/\epsilon+1/\epsilon^2+L)$ time, where $L$ is the
number of edges of the output path. The constant approximation
factor, $6+\epsilon$, for the short paths that we compute is quite
small. Our techniques are parallelizable and can also be used
to improve the previously best known results on several related
graphic and geometric problems."
}
@ -36888,7 +36888,7 @@ avoids overlap. This is useful in cartography."
This paper shows that the $i$-level of an arrangement of hyperplanes in
$E^d$ has at most ${{i+d-1}\choose {d-1}}$ local minima.
This bound follows from ideas previously used to prove bounds on $(\leq k)$-sets.
Using linear programming duality,
Using linear programming duality,
the Upper Bound Theorem is obtained as a corollary,
giving yet another proof of this
celebrated bound on the number of vertices of a simple polytope
@ -42606,10 +42606,10 @@ Contains C code."
, succeeds = "dp-olacd-91"
, update = "98.11 bibrelex, 98.07 bibrelex, 95.09 mitchell"
, annote = "In this paper you will find the definition of a Constrained
Delaunay Triangulation, some theorems and the pseudocode of
the algorithms to program it. On-Line means that you can
insert points and required edges in any order. With this
algorithm you can update an old CDT without retriangulating
Delaunay Triangulation, some theorems and the pseudocode of
the algorithms to program it. On-Line means that you can
insert points and required edges in any order. With this
algorithm you can update an old CDT without retriangulating
the old data."
}
@ -44330,7 +44330,7 @@ Contains C code."
@techreport{d-vrtdd-09
, author = "Olivier Devillers"
, title = "Vertex Removal in Two Dimensional {Delaunay} Triangulation:
, title = "Vertex Removal in Two Dimensional {Delaunay} Triangulation:
Asymptotic Complexity is Pointless"
, thanks = "triangles"
, institution = "INRIA"
@ -53249,17 +53249,17 @@ library."
, update = "98.11 bibrelex, 98.03 mitchell, 97.11 bibrelex, 97.03 rote"
, abstract = "We call a line $l$ a separator for a set $S$ of objects in
the plane if $l$ avoids all the objects and
partitions $S$ into two nonempty subsets, one consisting
partitions $S$ into two nonempty subsets, one consisting
of objects lying above $l$ and the
other of objects lying below $l$. We present an
other of objects lying below $l$. We present an
$O(n log n)$-time algorithm for
finding a separator line for a set of $n$ segments, provided
the ratio between the diameter of the set of segments and
finding a separator line for a set of $n$ segments, provided
the ratio between the diameter of the set of segments and
the length of the smallest segment is bounded.
No subquadratic algorithms are known for the general case.
No subquadratic algorithms are known for the general case.
Our algorithm is based on the recent results of
Matousek, Pach, Sharir, Sifrony, and Welzl (1994) concerning
the union of fat triangles, but we also include an analysis
Matousek, Pach, Sharir, Sifrony, and Welzl (1994) concerning
the union of fat triangles, but we also include an analysis
which improves the bounds obtained by Matousek et al."
}
@ -57348,18 +57348,18 @@ a simple polygon with vertex set P. We prove that it is NP-complete
to find a minimum weight polygon or a maximum weight polygon for a
given vertex set, resulting in a proof of NP-completeness for the
corresponding area optimization problems. This answers a generalization
of a question stated by Suri in 1989.
of a question stated by Suri in 1989.
We give evidence that it is unlikely that the minimization
problem can be approximated.
problem can be approximated.
For the maximiation problem, we show that we can find in optimal
time O(n log n) a polygon
of more than half the area AR(conv(P)) of the convex hull conv(P)
of P, yielding a fast 1/2 approximation method for the problem.
We demonstrate that it is NP-complete to decide whether there
is a simple polygon of at least (2/3+eps)(AR(conv(P)).
is a simple polygon of at least (2/3+eps)(AR(conv(P)).
We also sketch an NP-hardness proof for the problem of finding a minimum-link
searating polygon for two finite point sets in the plane.
Finally, we turn to higher dimensions, where we prove that for
Finally, we turn to higher dimensions, where we prove that for
0<k<d+1, 1<d, it is
NP-hard to minimize the volume of the k-dimensional faces of a
d-dimensional simple nondegenerate polyhedron with a given vertex
@ -57467,7 +57467,7 @@ set, answering a generalization of a question stated by O'Rourke in 1980."
, update = "98.03 mitchell"
, abstract = "For a given set $A\subseteq\ (-\pi;+\pi]$ of angles, the problem ``Angle-Restricted Tour'' (ART) is to decide whether a set $P$ of $n$ points in the
Euclidean plane allows a closed directed tour consisting of straight line segments, such that all angles between consecutive line segments are from the
set $A$.
set $A$.
\par
We present a variety of combinatorial and algorithmic results on this problem. In particular, we show that any finite set of at least 5 points allows a
``pseudoconvex'' tour, where all angles are nonnegative."
@ -93036,7 +93036,7 @@ and implement some of them."
, nickname = "IWCIA '01"
, year = 2001
, pages = "139--151"
, comments = "Appears also in Electronic Notes in Theoretical
, comments = "Appears also in Electronic Notes in Theoretical
Computer Science, Volume 46,
www.elsevier.nl/locate/entcs/volume46.html"
, update = "01.11 smid"
@ -97147,7 +97147,7 @@ exclusive read exclusive write parallel random-access machine (EREW
PRAM). Let $P$ be a trapezoided rectilinear simple polygon with $n$
vertices. In $O(\log n)$ time using $O(n/{\log n})$ processors we can
optimally compute
\begin{enumerate}
\begin{enumerate}
\item minimum rectilinear link paths, or shortest paths in the $L_1$
metric from any point in $P$ to all vertices of $P$,
\item minimum rectilinear link paths from any segment inside $P$ to
@ -105738,22 +105738,22 @@ and robustness is significantly improved; the implementation has no numerical
tolerances and does not exhibit cycling problems. The algorithm also handles
penetrating polyhedra, making it useful for nonconvex polyhedral collision
detection. This paper presents the theoretical principles of V-clip, and gives
a pseudocode description of the algorithm. It also documents various tests
that compare V-clip, Lin-Canny, and the Enhanced Gilbert-Johnson-Keerthi
algorithm, a simplex-based algorithm that is widely used for the same
application. The results show V-clip to be a strong contender in this field,
comparing favorably with the other algorithms in most of the tests, in terms
a pseudocode description of the algorithm. It also documents various tests
that compare V-clip, Lin-Canny, and the Enhanced Gilbert-Johnson-Keerthi
algorithm, a simplex-based algorithm that is widely used for the same
application. The results show V-clip to be a strong contender in this field,
comparing favorably with the other algorithms in most of the tests, in terms
of both performance and robustness.
\par
From the V-Clip Collision Detection WWW page:
\par
V-Clip is a low-level collision detection algorithm. The basic operation
provided to the application is that of performing a collision check between
two objects. The decisions of when to perform the checks between which object
pairs are left to the application. In particular, the V-Clip library does not
include facilities for higher level collision check culling, using bounding
boxes or spheres, for example. If this is required, consider writing one
yourself, or consider using a complete collision detection package such as
provided to the application is that of performing a collision check between
two objects. The decisions of when to perform the checks between which object
pairs are left to the application. In particular, the V-Clip library does not
include facilities for higher level collision check culling, using bounding
boxes or spheres, for example. If this is required, consider writing one
yourself, or consider using a complete collision detection package such as
I-Collide or V-Collide.
\par
V-Clip is designed for objects that are bounded by closed surfaces, and not
@ -106549,9 +106549,9 @@ problems in computational geometry is presented."
, cites = "rwzw-cksck-91, rw-cckgp-92, eorw-fmakg-92"
, update = "98.03 mitchell, 97.11 bibrelex, 97.03 rote"
, abstract = "Given a set $S$ of $n$ points in the plane, we compute in time
$O(n^3)$ the total number of convex polygons whose vertices are
a subset of $S$. We give an $O(m n^3)$ algorithm for computing
the number of convex $k$-gons with vertices in $S$, for all
$O(n^3)$ the total number of convex polygons whose vertices are
a subset of $S$. We give an $O(m n^3)$ algorithm for computing
the number of convex $k$-gons with vertices in $S$, for all
values $k=3,\ldots,m$."
}
@ -113828,7 +113828,7 @@ small) triangulation of a convex polyhedron is NP-complete. Their 3SAT-reduction
, update = "98.07 bibrelex, 97.07 orourke"
, annote = "A new polygon visibility graph is introduced. It is
demonstrated that it encodes more geometric
information about the polygon than does the vertex
information about the polygon than does the vertex
visibility graph. For example, it determines the
shortest path tree for each vertex."
}
@ -113856,7 +113856,7 @@ small) triangulation of a convex polyhedron is NP-complete. Their 3SAT-reduction
, annote = "Vertex-edge visibility graphs of pseudo-polygons
are characterized combinatorially, showing that the
decision problem for them is in P. This also establishes
that the decision problem for vertex-vertex
that the decision problem for vertex-vertex
visibility graphs of pseudo-polygons is in NP."
}
@ -124693,7 +124693,7 @@ Previous title: On-Line Navigation Through Regions of Variable
points and we are able to report an approximation of the width
of this dynamic point set. Our data structure takes linear space
and allows for reporting the approximation with relative
accuracy $\epsilon$ in $O(sqrt(1/\epsilon)log n)$ time; and the
accuracy $\epsilon$ in $O(sqrt(1/\epsilon)log n)$ time; and the
update time is $O(\log^2 n)$. The method uses the tentative
prune-and-search strategy of Kirkpatrick and Snoeyink."
}
@ -130003,12 +130003,12 @@ Contains C code."
, update = "98.07 bibrelex+rote, 98.03 mitchell, 97.03 rote"
, abstract = "We consider the problem of approximating a convex figure
in the plane by a pair $(r,R)$ of homothetic (i. e., similar and
parallel) rectangles with $r$ contained in $C$ and $R$
containing $C$. We show the existence of such pairs where the
sides of the outer rectangle have length at most double the
length of the inner rectangle, thereby solving a problem posed
parallel) rectangles with $r$ contained in $C$ and $R$
containing $C$. We show the existence of such pairs where the
sides of the outer rectangle have length at most double the
length of the inner rectangle, thereby solving a problem posed
by P\'{o}lya and Szeg\H{o}.
If the $n$ vertices of a convex polygon $C$ are given as a
If the $n$ vertices of a convex polygon $C$ are given as a
sorted array, such an approximating pair of rectangles can be
computed in time $O(log^2 n)$."
}
@ -130891,7 +130891,7 @@ Contains C code."
, year = 1995
, pages = "1976--1982"
, comments = "also contains an interesting algorithm for the Fortune-Wilfong
problem, to find a minimal-length curvature-constrained
problem, to find a minimal-length curvature-constrained
path in a polygonal environment."
, update = "96.05 mitchell"
}
@ -145397,6 +145397,17 @@ of geometric optics."
any other by flips."
}
@article{w-sf-76
, author = {Neal R. Wagner}
, title = {The Sofa Problem}
, year = {1976}
, journal = {The American Mathematical Monthly}
, volume = {83}
, number = {3}
, pages = {188--189}
, doi = {10.2307/2977022}
}
@inproceedings{ww-oedca-00
, author = "Uli Wagner and Emo Welzl"
, title = "Origin-Embracing Distributions, or A Continuous Analogue of the Upper Bound Theorem"
@ -151805,13 +151816,13 @@ amplification and suppression of local contrast. Contains C code."
@article{hh-esplp-08
, author = "I. Haran and D. Halperin"
, title = "An experimental study of point location in planar arrangements in CGAL"
, journal = "ACM Journal of Experimental Algorithmics"
, volume = "13"
, year = 2008
, pages = ""
}
, author = "I. Haran and D. Halperin"
, title = "An experimental study of point location in planar arrangements in CGAL"
, journal = "ACM Journal of Experimental Algorithmics"
, volume = "13"
, year = 2008
, pages = ""
}
@article{hkh-iiplgtds-12
, author = {{Hemmer}, M. and {Kleinbort}, M. and {Halperin}, D.}
@ -151900,7 +151911,7 @@ pages = {179--189}
acmid = {939190},
publisher = {IEEE Computer Society},
address = {Washington, DC, USA},
}
}
@inproceedings{Arthur2007Kmeans,
author = {Arthur, David and Vassilvitskii, Sergei},
@ -151916,7 +151927,7 @@ pages = {179--189}
acmid = {1283494},
publisher = {Society for Industrial and Applied Mathematics},
address = {Philadelphia, PA, USA},
}
}
@book{botsch2010polygon,
title={Polygon mesh processing},
@ -151955,7 +151966,7 @@ pages = {179--189}
publisher = {ACM},
address = {New York, NY, USA},
keywords = {Design and analysis of algorithms, computational geometry, shortest path problems},
}
}
@inproceedings{schnabel2007efficient,

37
Set_movable_separability_2/doc/Set_movable_separability_2/Set_movable_separability_2.txt
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@ -9,18 +9,39 @@ namespace CGAL {
\section sms_sec_intro Introduction
Problems of moving sets of objects, such as polygons in the plane and
polyhedra in three dimensions, without allowing collisions between the
objects are ubiquitous in many fields including motion planning,
computer graphics, VLSI layout, and manufacturing. One class of such
problems, referred to as <em>Movable Separability of Sets</em>
\cgalCite{t-mss-85}, considers the separability of sets of objects
under different kinds of motions and various definitions of
separation. The <em>moving sofa problem</em> or <em>sofa problem</em>
(see <a
href="https://en.wikipedia.org/wiki/Moving_sofa_problem">Moving sofa
problem</a> is a classic member of this class. It is a two-dimensional
idealisation of real-life furniture-moving problems; it asks for the
rigid two-dimensional shape of largest area \f$A\f$ that can be
maneuvered through an L-shaped planar region with legs of unit width
\cgalCite{w-sf-76}. The area \f$A\f$ thus obtained is referred to as
the sofa constant. The exact value of the sofa constant is an open
problem. These problems become progressively more challenging as the
allowable separation motions becomes more complex (have more degrees
of freedom), the number of objects involved grows, or the shape of the
objects becomes more complicated.
\cgalFigureBegin{sms_2_fig_sofa_problem,sofa_problem.png} The
Hammersley sofa has area 2.2074 but is not the largest solution.
\cgalFigureEnd
\section sms_sec_casting Casting
Casting is a manufacturing process where liquid material is poured
into a cavity inside a mold, which has the shape of a desired
product. After the material solidifies, the product is pulled out of
the mold. Typically a mold is used to manufacture numerous copies of a
product, in which case we need to make sure that the solidified
product can be separated from its mold without breaking it. The
challenge of designing a proper mold belongs to a larger topic termed
<em>Movable Separability of Sets</em>; see \cgalCite{t-mss-85}
These problems become progressively more challenging as the allowable
separation motions becomes more complex (have more degrees of
freedom), the number of objects involved grows, or the shape of the
objects becomes more complicated.
product. The challenge is design a proper mold, such that the solidified
product can be separated from its mold without breaking it.
This package provides a function called
`single_mold_translational_casting_2()` that given a simple closed

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