mirror of https://github.com/CGAL/cgal
Enhanced. Introduced nop strategy, minkowski_sum_by_decomposition(), and hole filtration
This commit is contained in:
Efi Fogel
parent
e006eaa6c6
commit
34a36aa2ca
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@ -4167,7 +4167,7 @@ cell neighborhood in $O(m)$ time."
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, succeeds = "aarx-clgta-96"
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, update = "98.07 rote, 98.03 mitchell, 97.03 rote"
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, abstract = "Exploiting the concept of so-called light edges, we introduce
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a new way of defining the greedy triangulation GT(S) of a point
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a new way of defining the greedy triangulation GT(S) of a point
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set S. It provides a decomposition of
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GT(S) into levels, and the number of levels allows us to
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bound the total edge length of GT(S). In particular, we show
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@ -12292,9 +12292,9 @@ method that uses very different techniques."
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, pages = "201--290"
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, url = "http://wwwpi6.fernuni-hagen.de/Publikationen/tr198.pdf"
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, succeeds = "ak-vd-96"
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, 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,
|
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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,
|
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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,
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, 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,
|
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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,
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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"
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, update = "00.11 smid, 00.03 bibrelex, 99.03 bibrelex, 98.11 bibrelex, 98.07 mitchell, 98.03 bibrelex, 97.03 icking"
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, annote = "Chapter 5 of su-hcg-00"
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@ -12847,9 +12847,9 @@ method that uses very different techniques."
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, url = "http://www.ifor.math.ethz.ch/staff/fukuda/fukuda.html"
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, update = "98.03 houle, 97.03 pocchiola, 96.05 fukuda"
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, annote = "Reverse search is a general exhaustive search technique
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which came out of the new convex hull algorithm by the authors.
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which came out of the new convex hull algorithm by the authors.
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This technique can be applied to many enumeration problems in computer
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science, operations research and geometry.
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science, operations research and geometry.
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It is highly suitable for parallelization."
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}
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@ -16291,7 +16291,7 @@ rendering. Contains pseudocode."
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, number = 4
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, year = 1997
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, note = "Special issue on parallel I/O. An earlier version
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appears in Proc. of the 8th Annual
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appears in Proc. of the 8th Annual
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ACM Symposium on Parallel Algorithms and
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Architectures (SPAA~'96), Padua, Italy, June 1996, 109--118"
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, update = "97.03 murali"
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@ -18449,12 +18449,12 @@ the interior. Contains pseudocode."
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, succeeds = "d-rld-89"
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, update = "98.07 bibrelex, 98.03 mitchell, 93.09 held"
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, annote = "He considers rectilinear paths in a rectilinear simple polygon. In $O(n\log n)$
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preprocessing time and space, he builds a data structure that supports
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$O(\log n)$ time queries for distance between two points ($O(1)$ time between
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two polygon vertices). He is actually searching for paths that are``smallest'' in
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that they are shortest simultaneously in rectilinear link distance and
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$L_1$ length (which is always possible). See improvements to $O(n)$ time and
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space by Lingas, Maheshwari, and Sack~\cite{lms-parld-95} and
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preprocessing time and space, he builds a data structure that supports
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$O(\log n)$ time queries for distance between two points ($O(1)$ time between
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two polygon vertices). He is actually searching for paths that are``smallest'' in
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that they are shortest simultaneously in rectilinear link distance and
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$L_1$ length (which is always possible). See improvements to $O(n)$ time and
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space by Lingas, Maheshwari, and Sack~\cite{lms-parld-95} and
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Schuierer~\cite{s-odssr-96}."
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}
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@ -20044,6 +20044,16 @@ $O(n^2)$ in the plane."
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, keywords = "Delaunay triangulation, probabilistic analysis"
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}
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@article{bfhhm-epsph-15
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, author = "A. Baram and E. Fogel and D. Halperin and M. Hemmer and S. Morr"
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, title = "Exact {M}inkowski sums of Polygons With Holes"
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, journal = "Internat. J. Comput. Geom. Appl."
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, volume = 1
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, year = 2015
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, pages = "01--01"
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, note = "To appear"
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}
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@techreport{bg-dpd-91
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, author = "M. Bern and J. Gilbert"
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, title = "Drawing the planar dual"
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@ -21011,10 +21021,10 @@ cubes with side-lengths not exceeding 1 in the $3$-dimensional
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euclidean space. Let $S$ and $T$ be two points lying outside
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the open cubes. Assume one needs to find a short path emanating
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from $S$ and terminating at $T$ avoiding the cubes of $\cal P$
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under the restriction that the cubes are not known prior to the search.
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under the restriction that the cubes are not known prior to the search.
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In fact the positions and the side-lengths of the cubes become known
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to the searcher as the cubes are contacted. We give an algorithm to
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construct a path of length less than
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construct a path of length less than
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$\frac 32 d + 3 \sqrt 3 \log d + 5$,
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where $d > 3 \sqrt 3$ denotes the distance between S and T."
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}
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@ -24325,7 +24335,7 @@ experimental results are given."
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, abstract = "This paper
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presents the main algorithmic and design choices that have been
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made
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to implement triangulations in
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to implement triangulations in
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the computational geometry algorithms library CGAL."
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}
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@ -25614,9 +25624,9 @@ present a polynomial-time exact algorithm to solve this problem."
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rectangles floating in 3-space, with edges represented by
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vertical lines of sight. We apply an extension of the
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{Erd\H os}-Szekeres Theorem in a geometric setting to obtain an
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upper bound of 56 for size of the largest complete graph which
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is representable. On the other hand, we construct a
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representation of the complete graph with 22 vertices.
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upper bound of 56 for size of the largest complete graph which
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is representable. On the other hand, we construct a
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representation of the complete graph with 22 vertices.
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These are the best existing bounds."
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}
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@ -28630,7 +28640,7 @@ determinants."
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, year = 1993
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, update = "98.03 mitchell"
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, abstract = "We calculate the partition
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of the configuration space $I\!\!R^2 x S^1$
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of the configuration space $I\!\!R^2 x S^1$
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of a car-like robot, only moving forwards, with respect to the
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type of the length optimal paths. This kind of robot is subject to
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kinematic constraints on its path curvature and its orientation.
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@ -33579,35 +33589,35 @@ determinants."
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Given a set of polygonal obstacles of $n$ vertices in the plane,
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the problem of processing the all-pairs Euclidean {\em short} path
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queries is that of reporting an obstacle-avoiding path $P$ (or
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its length) between two arbitrary query points $p$ and $q$ in the
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plane, such that the length of $P$ is within a small factor of the
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its length) between two arbitrary query points $p$ and $q$ in the
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plane, such that the length of $P$ is within a small factor of the
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length of a Euclidean {\em shortest} obstacle-avoiding path between
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$p$ and $q$. The goal is to answer each short path query quickly
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by constructing data structures that capture path information in
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the obstacle-scattered plane. For the related all-pairs Euclidean
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{\em shortest} path problem, the best known algorithms for even
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very simple cases (e.g., {\em rectilinear} shortest paths among
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by constructing data structures that capture path information in
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the obstacle-scattered plane. For the related all-pairs Euclidean
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{\em shortest} path problem, the best known algorithms for even
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very simple cases (e.g., {\em rectilinear} shortest paths among
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disjoint {\em rectangular} obstacles in the plane) require
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at least quadratic space and time to construct a data structure,
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so that a length query can be answered in polylogarithmic time.
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The previously best known solution to the all-pairs Euclidean
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{\em short} path problem also uses a data structure of quadratic
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space and superquadratic construction time, in order to answer a
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length query in polylogarithmic time. In this paper, we present a
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data structure that requires nearly linear space and takes subquadratic
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time to construct. Precisely, for any given $\epsilon$ satisfying
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so that a length query can be answered in polylogarithmic time.
|
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The previously best known solution to the all-pairs Euclidean
|
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{\em short} path problem also uses a data structure of quadratic
|
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space and superquadratic construction time, in order to answer a
|
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length query in polylogarithmic time. In this paper, we present a
|
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data structure that requires nearly linear space and takes subquadratic
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time to construct. Precisely, for any given $\epsilon$ satisfying
|
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$0$ $<$ $\epsilon$ $\leq$ $1$, our data structure can be built
|
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in $o(q^{3/2})$ $+$ $O((n\log n)/\epsilon)$ time and
|
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$O(n\log n+n/\epsilon)$ space, where $q$, $1$ $\leq$ $q$ $\leq$ $n$,
|
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is the minimum number of faces needed to cover all the vertices of
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a certain planar graph we use. This data structure enables us to
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in $o(q^{3/2})$ $+$ $O((n\log n)/\epsilon)$ time and
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$O(n\log n+n/\epsilon)$ space, where $q$, $1$ $\leq$ $q$ $\leq$ $n$,
|
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is the minimum number of faces needed to cover all the vertices of
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a certain planar graph we use. This data structure enables us to
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report the length of a short path between two arbitrary query points
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in $O((\log n)/\epsilon+1/\epsilon^2)$ time and the actual path
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in $O((\log n)/\epsilon+1/\epsilon^2+L)$ time, where $L$ is the
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number of edges of the output path. The constant approximation
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factor, $6+\epsilon$, for the short paths that we compute is quite
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small. Our techniques are parallelizable and can also be used
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to improve the previously best known results on several related
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in $O((\log n)/\epsilon+1/\epsilon^2+L)$ time, where $L$ is the
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number of edges of the output path. The constant approximation
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factor, $6+\epsilon$, for the short paths that we compute is quite
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small. Our techniques are parallelizable and can also be used
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to improve the previously best known results on several related
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graphic and geometric problems."
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}
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@ -36877,7 +36887,7 @@ avoids overlap. This is useful in cartography."
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This paper shows that the $i$-level of an arrangement of hyperplanes in
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$E^d$ has at most ${{i+d-1}\choose {d-1}}$ local minima.
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This bound follows from ideas previously used to prove bounds on $(\leq k)$-sets.
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Using linear programming duality,
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Using linear programming duality,
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the Upper Bound Theorem is obtained as a corollary,
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giving yet another proof of this
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celebrated bound on the number of vertices of a simple polytope
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@ -42595,10 +42605,10 @@ Contains C code."
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, succeeds = "dp-olacd-91"
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, update = "98.11 bibrelex, 98.07 bibrelex, 95.09 mitchell"
|
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, annote = "In this paper you will find the definition of a Constrained
|
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Delaunay Triangulation, some theorems and the pseudocode of
|
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the algorithms to program it. On-Line means that you can
|
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insert points and required edges in any order. With this
|
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algorithm you can update an old CDT without retriangulating
|
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Delaunay Triangulation, some theorems and the pseudocode of
|
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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."
|
||||
}
|
||||
|
||||
|
|
@ -44319,7 +44329,7 @@ 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"
|
||||
|
|
@ -53238,17 +53248,17 @@ 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."
|
||||
}
|
||||
|
||||
|
|
@ -57337,18 +57347,18 @@ 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
|
||||
|
|
@ -57456,7 +57466,7 @@ 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."
|
||||
|
|
@ -93025,7 +93035,7 @@ 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"
|
||||
|
|
@ -97136,7 +97146,7 @@ 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
|
||||
|
|
@ -105727,22 +105737,22 @@ 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
|
||||
|
|
@ -106538,9 +106548,9 @@ 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 $k=3,\ldots,m$."
|
||||
}
|
||||
|
||||
|
|
@ -113817,7 +113827,7 @@ 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."
|
||||
}
|
||||
|
|
@ -113845,7 +113855,7 @@ 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."
|
||||
}
|
||||
|
||||
|
|
@ -124672,7 +124682,7 @@ 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."
|
||||
}
|
||||
|
|
@ -129982,12 +129992,12 @@ 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 $O(log^2 n)$."
|
||||
}
|
||||
|
|
@ -130870,7 +130880,7 @@ 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"
|
||||
}
|
||||
|
|
@ -151783,13 +151793,13 @@ 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.}
|
||||
|
|
@ -151867,7 +151877,7 @@ pages = {179--189}
|
|||
acmid = {939190},
|
||||
publisher = {IEEE Computer Society},
|
||||
address = {Washington, DC, USA},
|
||||
}
|
||||
}
|
||||
|
||||
@inproceedings{Arthur2007Kmeans,
|
||||
author = {Arthur, David and Vassilvitskii, Sergei},
|
||||
|
|
@ -151883,7 +151893,7 @@ pages = {179--189}
|
|||
acmid = {1283494},
|
||||
publisher = {Society for Industrial and Applied Mathematics},
|
||||
address = {Philadelphia, PA, USA},
|
||||
}
|
||||
}
|
||||
|
||||
@book{botsch2010polygon,
|
||||
title={Polygon mesh processing},
|
||||
|
|
@ -151922,5 +151932,4 @@ pages = {179--189}
|
|||
publisher = {ACM},
|
||||
address = {New York, NY, USA},
|
||||
keywords = {Design and analysis of algorithms, computational geometry, shortest path problems},
|
||||
}
|
||||
|
||||
}
|
||||
|
|
|
|||
|
|
@ -18,11 +18,11 @@ used in many applications, such as motion planning and computer-aided
|
|||
design and manufacturing. This package contains functions that compute
|
||||
the planar Minkowski sums of two polygons. (Here, \f$ A\f$ and \f$ B\f$
|
||||
are two closed polygons in \f$ \mathbb{R}^2\f$, which may have holes; see
|
||||
Chapter \ref Chapter_2D_Regularized_Boolean_Set-Operations "2D
|
||||
Regularized Boolean Set-Operations" for the precise definition of valid
|
||||
Chapter \ref Chapter_2D_Regularized_Boolean_Set-Operations
|
||||
"2D Regularized Boolean Set-Operations" for the precise definition of valid
|
||||
polygons), and the planar Minkowski sum of a simple polygon and a
|
||||
disc---an operation also referred to as <I>offsetting</I> or <I>dilating</I>
|
||||
a polygon). \cgalFootnote{The family of valid types of summands is slightly
|
||||
a polygon.\cgalFootnote{The family of valid types of summands is slightly
|
||||
broader for certain operations, e.g., a degenerate polygon consisting of
|
||||
line segments is a valid operand for the approximate-offsetting operation.}
|
||||
This package, like the \ref Chapter_2D_Regularized_Boolean_Set-Operations
|
||||
|
|
@ -32,7 +32,7 @@ by the \ref chapterArrangement_on_surface_2 "2D Arrangements" package.
|
|||
The two packages are integrated well to
|
||||
allow mixed operations. For example, it is possible to apply Boolean set
|
||||
operations on objects that are the result of Minkowski sum
|
||||
computations. \cgalFootnote{The operands of the Minkowski sum operations
|
||||
computations.\cgalFootnote{The operands of the Minkowski sum operations
|
||||
supported by this package must be (linear) polygons, as opposed to the
|
||||
operands of the Boolean set operations supported by the
|
||||
\ref Chapter_2D_Regularized_Boolean_Set-Operations
|
||||
|
|
@ -51,11 +51,11 @@ edges ordered according to the angle they form with the \f$ x\f$-axis. As the
|
|||
two input polygons are convex, their edges are already sorted by the angle
|
||||
they form with the \f$ x\f$-axis; see the figure above.
|
||||
The Minkowski sum can therefore be computed using an operation similar to the
|
||||
merge step of the merge-sort algorithm \cgalFootnote{See, for example,
|
||||
merge step of the merge-sort algorithm\cgalFootnote{See, for example,
|
||||
<a href="http://en.wikipedia.org/wiki/Merge_sort">
|
||||
http://en.wikipedia.org/wiki/Merge_sort</a>.} in \f$ O(m + n)\f$ time,
|
||||
starting from two bottommost vertices in \f$ P\f$ and in \f$ Q\f$ and merging
|
||||
the ordered list of edges.
|
||||
starting from the two bottommost vertices in \f$ P\f$ and in \f$ Q\f$ and
|
||||
merging the ordered list of edges.
|
||||
|
||||
\cgalFigureBegin{mink_figonecyc,ms_convex_polygon.png,ms_concave_polygon.png,ms_convolution.png}
|
||||
The convolution of a convex polygon and a non-convex polygon. The convolution
|
||||
|
|
@ -95,11 +95,11 @@ positive orientations (i.e., their boundaries wind in a
|
|||
counterclockwise order around their interiors). The
|
||||
<I>convolution</I> of these two polygons \cgalCite{grs-kfcg-83},
|
||||
denoted \f$ P * Q\f$, is a collection of line segments of the form
|
||||
\f$ [p_i + q_j, p_{i+1} + q_j]\f$, \cgalFootnote{Throughout this
|
||||
\f$ [p_i + q_j, p_{i+1} + q_j]\f$,\cgalFootnote{Throughout this
|
||||
chapter, we increment or decrement an index of a vertex modulo the
|
||||
size of the polygon.} where the vector \f$ {\mathbf{p_i p_{i+1}}}\f$
|
||||
lies between \f$ {\mathbf{q_{j-1} q_j}}\f$ and
|
||||
\f$ {\mathbf{q_j q_{j+1}}}\f$, \cgalFootnote{We say that a vector
|
||||
\f$ {\mathbf{q_j q_{j+1}}}\f$,\cgalFootnote{We say that a vector
|
||||
\f$ {\mathbf v}\f$ lies between two vectors \f$ {\mathbf u}\f$ and
|
||||
\f$ {\mathbf w}\f$ if we reach \f$ {\mathbf v}\f$ strictly before
|
||||
reaching \f$ {\mathbf w}\f$ if we move all three vectors to the
|
||||
|
|
@ -114,7 +114,7 @@ The segments of the convolution form a number of closed (not
|
|||
necessarily simple) polygonal curves called <I>convolution cycles</I>.
|
||||
The Minkowski sum \f$ P \oplus Q\f$ is the set of points
|
||||
having a non-zero winding number with respect to the cycles of
|
||||
\f$ P * Q\f$. \cgalFootnote{Informally speaking, the winding number
|
||||
\f$ P * Q\f$.\cgalFootnote{Informally speaking, the winding number
|
||||
of a point \f$ p \in\mathbb{R}^2\f$ with respect to some planar curve
|
||||
\f$ \gamma\f$ is an integer number counting how many times does
|
||||
\f$ \gamma\f$ wind in a counterclockwise direction around \f$ p\f$.}
|
||||
|
|
@ -157,14 +157,29 @@ the sub-sums \f$ S_{ij}\f$ when using the decomposition approach. As
|
|||
both approaches construct the arrangement of these segments and
|
||||
extract the sum from this arrangement, computing Minkowski sum using
|
||||
the convolution approach usually generates a smaller intermediate
|
||||
arrangement, hence it is faster and consumes less space. In most cases,
|
||||
arrangement; hence it is faster and consumes less space. In most cases,
|
||||
the reduced convolution method is faster than the full convolution
|
||||
method, as the respective induced arrangement is usually much smaller.
|
||||
However, in degenerate cases with many holes in the Minkowski sum, the
|
||||
full convolution method can be preferable, as it avoids costly
|
||||
intersection tests.
|
||||
|
||||
\subsection mink_ssecsum_conv Computing Minkowski Sum using Convolutions
|
||||
\subsection mink_ssec_hole_filter Filtering Out Holes
|
||||
|
||||
If a hole in one polygon is relatively small compared to the other
|
||||
polygon, the hole is irrelevant for the computation of
|
||||
\f$P\oplus Q \f$ \cgalCite{bfhhm-epsph-15}; It implies that the hole
|
||||
can be removed (that is, filled up) before the main computation starts.
|
||||
Theoretically, we can always fill up all the holes of at least one
|
||||
polygon, transforming it into a simple polygon, and still obtain
|
||||
exactly the same Minkowski sum. Practically, we remove all holesin one
|
||||
polygon whose bounding boxes are, in \f$x \f$- or \f$y \f$-direction,
|
||||
smaller than, or as large as, the bounding box of the other polygon.
|
||||
Obliterating holes in the input summands speeds up the computation of
|
||||
Minkowski sums, regardless of the approach used to compute the
|
||||
Minkowski sum.
|
||||
|
||||
\subsection mink_ssec_conv Computing Minkowski Sum using Convolutions
|
||||
|
||||
The function template \link minkowski_sum_2()
|
||||
`minkowski_sum_2(P, Q)`\endlink accepts two polygons
|
||||
|
|
@ -182,7 +197,8 @@ the function \link minkowski_sum_full_convolution_2()
|
|||
the \link Polygon_2 `Polygon_2`\endlink class template. The types of
|
||||
operands accepted by the function \link
|
||||
minkowski_sum_reduced_convolution_2()
|
||||
`minkowski_sum_reduced_convolution_2(P, Q)`\endlink
|
||||
`minkowski_sum_reduced_convolution_2(P, Q)`\endlink (and by the
|
||||
function \link minkowski_sum_2() `minkowski_sum_2(P, Q)`\endlink)
|
||||
are instances of either the \link Polygon_2 `Polygon_2`\endlink or
|
||||
\link Polygon_with_holes_2 `Polygon_with_holes_2`\endlink class templates.
|
||||
Even when the input polygons are restricted to be simple polygons, they
|
||||
|
|
@ -229,7 +245,7 @@ in both summands is large, the decomposition approach runs faster. In
|
|||
the following we describe how to employ the decomposition-based
|
||||
Minkowski sum procedure.
|
||||
|
||||
\subsection mink_ssecdecomp Decomposition Strategies
|
||||
\subsection mink_ssec_decomp_strategies Decomposition Strategies
|
||||
|
||||
In order to compute Minkowski sums of two polygon \f$ P \f$ and
|
||||
\f$ Q \f$ using the decomposition method, issue the call
|
||||
|
|
@ -242,7 +258,7 @@ of a type that models the concept
|
|||
`PolygonWithHolesConvexDecomposition_2`, which refines the concept
|
||||
`PolygonConvexDecomposition_2`. The same holds for \f$Q \f$.
|
||||
The two concepts `PolygonConvexDecomposition_2` and
|
||||
`PolygonWithHolesConvexDecomposition` refine a `Functor` concept
|
||||
`PolygonWithHolesConvexDecomposition_2` refine a `Functor` concept
|
||||
variant. Namely, they both require the provision of a function
|
||||
operator (`operator()`). The function operator of the model of the
|
||||
concept `PolygonConvexDecomposition_2` accepts a planar simple
|
||||
|
|
@ -251,14 +267,25 @@ polygon, while the function operator of the model of the concept
|
|||
with holes. Both return a range of convex polygons that represents
|
||||
the convex decomposition of the input polygon. If the decomposition
|
||||
strategy that decomposes \f$P \f$ is the same as the strategy that
|
||||
decompose \f$Q \f$, you can omit the forth argument, and
|
||||
issue the call `minkowski_sum_2(P, Q, decomp)`.
|
||||
decomposes \f$Q \f$, you can omit the forth argument, and
|
||||
issue the call `minkowski_sum_2(P, Q, decomp)`, where `decomp`
|
||||
is an object that represents the common strategy.
|
||||
The class template `Polygon_nop_decomposition_2`, which models the
|
||||
concept `PolygonConvexDecomposition_2`, is a trivial convex
|
||||
decomposition strategy referred to as the <I>nop<I/> strategy; it
|
||||
merely passes the input polygon to the next stage intact; use it in
|
||||
cases you know that the corresponding input polygon is convex to
|
||||
start with. If both \f$P \f$ and \f$Q \f$ are known to be convex,
|
||||
you can issue the call `minkowski_sum_2(P, Q, nop)`, where `nop`
|
||||
is an object that represents the nop strategy.
|
||||
|
||||
The Minkowski-sum package includes four models of the concept
|
||||
`PolygonConvexDecomposition_2` and two models of the refined concept
|
||||
`PolygonConvexDecomposition_2` (besides the trivial model
|
||||
`Polygon_nop_decomposition_2`) and two models of the refined concept
|
||||
`PolygonWithHolesConvexDecomposition_2` as described below. The first
|
||||
three are class templates that wrap the decomposition functions included
|
||||
in the \ref Chapter_2D_Polygon "Planar Polygon Partitioning" package.
|
||||
three are class templates that wrap the corresponding decomposition
|
||||
functions included in the
|
||||
\ref Chapter_2D_Polygon "Planar Polygon Partitioning" package.
|
||||
|
||||
<UL>
|
||||
<LI>The `Optimal_convex_decomposition_2<Kernel>` class template uses
|
||||
|
|
@ -310,10 +337,7 @@ vertex and having rational-coordinate endpoints on both sides.
|
|||
|
||||
The following are two models of the refined concept
|
||||
`PolygonWithHolesConvexDecomposition_2`. An instance of any one these
|
||||
two types can be used to decompose a polygon with holes. You can pass
|
||||
the instance as the third argument to call
|
||||
`minkowski_sum_2(P, Q, decomp)` to compute the Minkowski sum of two
|
||||
polygons with holes, \f$P \f$ and \f$Q \f$.
|
||||
two types can be used to decompose a polygon with holes.
|
||||
<UL>
|
||||
<LI>The `Polygon_vertical_decomposition_2<Kernel>` class template
|
||||
uses vertical decomposition to decompose the underlying arrangement;
|
||||
|
|
@ -332,6 +356,29 @@ angle-bisector decomposition strategy.
|
|||
|
||||
\cgalExample{Minkowski_sum_2/sum_by_decomposition.cpp}
|
||||
|
||||
\subsection mink_ssec_optimal_decomp Optimal Decomposition
|
||||
|
||||
Decomposition methods that handle polygons with holes are typically
|
||||
more costly than decomposition methods that handle only simple
|
||||
polygons. The hole filtration (see \ref mink_ssec_hole_filter) is
|
||||
applied before the actual construction starts (be it convolution
|
||||
based or decomposition based). The filteration may result with a
|
||||
polygon that does not have holes, or even a convex polygon, but this
|
||||
is unkown at the time of the call. To this end, we introduce the
|
||||
overloaded function template
|
||||
`minkowski_sum_by_decomposition_2(P, Q, no_holes_decomp, with_holes_decomp)`,
|
||||
where `no_holes_decomp` and `with_holes_decomp` are objects that model
|
||||
the concepts `PolygonConvexDecomposition_2` and
|
||||
`PolygonWithHolesConvexDecomposition_2`, respectively. If after the
|
||||
application of the hole filtration \f$P\f$ remains a polygon with holes,
|
||||
then the strategy represented by the object `with_holes_decomp` is
|
||||
applied to it. If, however, \f$P\f$ turns into a polygon without holes,
|
||||
then the strategy represented by the object `no_holes_decomp` is applied
|
||||
to it, unless the result is a convex polygon, in which case the nop
|
||||
strategy is applied. If \f$P\f$ is a polygon without holes to start with,
|
||||
then only convexity is checked. (Checking whether the result is convex
|
||||
inccurs a small overhead though.) The same holds for \f$Q\f$.
|
||||
|
||||
\section mink_secoffset Offsetting a Polygon
|
||||
|
||||
The operation of computing the Minkowski sum \f$ P \oplus B_r\f$ of a
|
||||
|
|
@ -388,7 +435,7 @@ sub-polygon, and finally calculating the union of these offsets
|
|||
sub-polygons; see \cgalFigureRef{mink_figpgn_offset} (b). However, as
|
||||
with the case of the Minkowski sum of a pair of polygons, it is also more
|
||||
efficient to compute the <I>convolution cycle</I> of the polygon and the
|
||||
disc \f$ B_r\f$, \cgalFootnote{As the disc is convex, it is guaranteed
|
||||
disc \f$ B_r\f$,\cgalFootnote{As the disc is convex, it is guaranteed
|
||||
that the convolution curve comprises a single cycle.} which can be
|
||||
constructed by applying the process described in the previous
|
||||
paragraph for convex polygons: The only difference is that a circular arc
|
||||
|
|
@ -399,7 +446,7 @@ the last step consists of computing the winding numbers of the faces
|
|||
of the arrangement induced by the convolution cycle and discarding the
|
||||
faces with zero winding numbers.
|
||||
|
||||
\subsection mink_ssecapprox_offset Approximating the Offset with a Guaranteed Error Bound
|
||||
\subsection mink_ssec_approx_offset Approximating the Offset with a Guaranteed Error Bound
|
||||
|
||||
Let \f$ P \f$ be a counterclockwise-oriented simple polygon all
|
||||
vertices of which \f$ p_0, \ldots, p_{n-1} \f$ have rational coordinates,
|
||||
|
|
@ -429,7 +476,7 @@ is supported by the line \f$ Ax + By + C' = 0 \f$, where
|
|||
rational number. Therefore, the line segments that compose the offset
|
||||
boundaries cannot be represented as segments of lines with rational
|
||||
coefficients.
|
||||
In Section \ref mink_ssecexact_offset we use the line-pair representation
|
||||
In Section \ref mink_ssec_exact_offset we use the line-pair representation
|
||||
to construct the offset polygonin an exact manner using the traits class
|
||||
for conic arcs.
|
||||
</UL>
|
||||
|
|
@ -504,7 +551,7 @@ the header file `bops_circular.h`, which defines the polygon types.
|
|||
|
||||
\cgalExample{Minkowski_sum_2/approx_offset.cpp}
|
||||
|
||||
\subsection mink_ssecexact_offset Computing the Exact Offset
|
||||
\subsection mink_ssec_exact_offset Computing the Exact Offset
|
||||
|
||||
As mentioned in the previous section, it is possible to represent
|
||||
offset polygons in an exact manner if the edges of the polygons are
|
||||
|
|
@ -548,7 +595,7 @@ handles polygons with holes, such as the
|
|||
`Polygon_vertical_decomposition_2<Kernel>` class template.
|
||||
\cgalAdvancedEnd
|
||||
|
||||
\subsection mink_ssecinner_offset Computing Inner Offsets
|
||||
\subsection mink_ssec_inner_offset Computing Inner Offsets
|
||||
|
||||
An operation closely related to the (outer) offset computation, is
|
||||
computing the <I>inner offset</I> of a polygon, or <I>insetting</I> it
|
||||
|
|
|
|||
Loading…
Reference in New Issue