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Expected asymptotically optimal planar point location

2004
*
International Journal of Production Economics
*

Given

doi:10.1016/s0925-7721(04)00047-1
fatcat:gefgbn33tvaaxldoq3arazq4n4
*a*fixed distribution*of**point**location*queries among the triangles*in**a*triangulation*of*the plane,*a*data structure is presented that achieves, within constant*multiplicative*factors, the entropy ... The data structure is*a*simple variation*of*Kirkpatrick's classic*planar**point**location*structure [D.G. Kirkpatrick, SIAM J. ...*In**a**point**location*problem,*a**subdivision**of*space is provided, and*a*data structure is constructed such that, for each query*point*,*a*pointer to the triangle*in*the*subdivision**in*which the query*point*...##
###
Expected asymptotically optimal planar point location

2004
*
Computational geometry
*

Given

doi:10.1016/j.comgeo.2004.03.010
fatcat:vnia3qlevvh6pbplsq63uk3fjm
*a*fixed distribution*of**point**location*queries among the triangles*in**a*triangulation*of*the plane,*a*data structure is presented that achieves, within constant*multiplicative*factors, the entropy ... The data structure is*a*simple variation*of*Kirkpatrick's classic*planar**point**location*structure [D.G. Kirkpatrick, SIAM J. ...*In**a**point**location*problem,*a**subdivision**of*space is provided, and*a*data structure is constructed such that, for each query*point*,*a*pointer to the triangle*in*the*subdivision**in*which the query*point*...##
###
Point-Location in The Arrangement of Curves
[article]

2020
*
arXiv
*
pre-print

We design

arXiv:2007.11451v2
fatcat:udqcs4ot7rdovjtf2mxqxnfohm
*a*data structure for solving the*point**location*problem queries*in*O(log C(n)+log S(n)) time using O(T(n)+S(n)log(S(n))) preprocessing time, if*a*polygonal*subdivision**of*total size S(n), with ...*A*data-structure for*point*-*location*, preprocesses the curves into*a*data structure*of*polynomial size*in*n, such that the queries can be answered*in*time polylogarithmic*in*n. ... Finding*a**point**in**a**planar**subdivision*and returning the face that contain that*point*is called*point**location*. ...##
###
Subdivision Surfaces for Scattered-data Approximation
[chapter]

2001
*
Eurographics
*

The novelty

doi:10.1007/978-3-7091-6215-6_7
fatcat:b5rreyfhjfdjnizamchmtdum2e
*of*our scheme is that it applies*subdivision*only to the ordinates*of*control*points*, whereas the triangulated mesh*in*the plane is fixed. ... We propose*a*modified Loop*subdivision*surface scheme for the approximation*of*scattered data*in*the plane. ... We start with*a*set*of**planar**points*Figure 4 . ...##
###
Subdivision Surfaces for Scattered-data Approximation
[article]

2001
*
EUROVIS 2005: Eurographics / IEEE VGTC Symposium on Visualization
*

The novelty

doi:10.2312/vissym/vissym01/055-064
fatcat:soq2wf2uobbrxixf4itvvg7wau
*of*our scheme is that it applies*subdivision*only to the ordinates*of*control*points*, whereas the triangulated mesh*in*the plane is fixed. ... We propose*a*modified Loop*subdivision*surface scheme for the approximation*of*scattered data*in*the plane. ... We start with*a*set*of**planar**points*¢ C¤ v § % 4 ¤ ¤ @ (¦ § © ) with associated function values ¢ Q ¤ and*a*triangulation d*of*these*points*. ...##
###
Deterministic Rectangle Enclosure and Offline Dominance Reporting on the RAM
[chapter]

2014
*
Lecture Notes in Computer Science
*

Many additional ideas are then incorporated, including

doi:10.1007/978-3-662-43948-7_7
fatcat:t2psiqxxnnblpcw5p5yeyo4mjq
*a*linear-time algorithm for merging shallow cuttings and an algorithm for an offline tree*point**location*problem. ... We revisit*a*classical problem*in*computational geometry that has been studied since the 1980s:*in*the rectangle enclosure problem we want to report all k enclosing pairs*of*n input rectangles*in*2D. ... We thank Christos Makris from the University*of*Patras, Greece for preliminary work on this problem. ...##
###
2D Fractional Cascading on Axis-aligned Planar Subdivisions
[article]

2020
*
arXiv
*
pre-print

*In*2D fractional cascading, the input includes

*a*

*planar*

*subdivision*for every vertex

*of*G and the query is

*a*

*point*q and

*a*subgraph π and the goal is to

*locate*the cell containing q

*in*all the

*subdivisions*...

*In*this paper, we show that it is possible to circumvent the lower bound

*of*Chazelle and Liu for axis-aligned

*planar*

*subdivisions*. ... Given

*a*

*planar*

*subdivision*S

*of*complexity n,

*in*the

*planar*

*point*

*location*problem, we are asked to preprocess S such that given any query

*point*q

*in*the plane, we can find the face f

*in*S containing q ...

##
###
Succinct Geometric Indexes Supporting Point Location Queries
[article]

2008
*
arXiv
*
pre-print

Our first and main result is

arXiv:0805.4147v1
fatcat:o27rcrfifjej5hs7l56mjkq7je
*a*succinct geometric index that can answer*point**location*queries,*a*fundamental problem*in*computational geometry, on*planar*triangulations*in*O(lg n) time. ... We then generalize our succinct geometric index to*planar**subdivisions*, and design indexes for other types*of*queries. ... . ⊓ ⊔*Point**Location**in**Planar**Subdivisions*We now generalize the techniques*in*Section 3 to design succinct geometric indexes supporting*point**location*queries*in*general*planar**subdivisions*. ...##
###
Page 2368 of Mathematical Reviews Vol. , Issue 81F
[page]

1981
*
Mathematical Reviews
*

P. 81f:68056

*A*note on*locating**a*set*of**points**in**a**planar**subdivision*. SIAM J. Comput. 8 (1979), no. 4, 542-545. ...*Locating**a**point*p*in**a**subdivision*means finding the*subdivision*region which contains p. ...##
###
A CASE STUDY IN ALGORITHM ENGINEERING FOR GEOMETRIC COMPUTING

2001
*
International journal of computational geometry and applications
*

We present the criteria that have inspired the preliminary design

doi:10.1142/s0218195901000390
fatcat:bpb6wyvko5ey7fayv7xbzkoh2m
*of*GeomLib and discuss the guidelines that we h*a*ve followed*in*the initial implementation. ... The goal*of*this paper is to prove the applicability*of*some advanced software design concepts to geometric computing through*a*vertical case study. ... Acknowledgments We w ould like to thank Franco Preparata and Mike Goodrich for their encouragement and support, Jim Baker for his collaboration*in*writing*a*preliminary version*of*this paper, Natasha Gelfand ...##
###
A higher dimensional formulation for robust and interactive distance queries

2006
*
Proceedings of the 2006 ACM symposium on Solid and physical modeling - SPM '06
*

*In*this framework, minimum distance between

*a*

*point*and

*planar*curve is solved using

*a*single polynomial equation

*in*three variables (two variables for

*a*position

*of*the

*point*and one for the curve). ... Given

*a*particular query

*point*, the solution space's remaining degrees-offreedom are fixed and we can numerically compute the minimum distance

*in*

*a*very efficient way. ... All opinions, findings, conclusions or recommendations expressed

*in*this document are those

*of*the author and do not necessarily reflect the views

*of*the sponsoring agencies. ...

##
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Window Queries On Planar Subdivisions Arising From Overlapping Polygons

2014
*
Canadian Conference on Computational Geometry
*

*A*dynamic data structure is presented for storing the partitioning that arises from m polygons with at most k sides each

*in*space O(km 2 ) if the polygons are assumed to be convex, and space O(k 2 m 2 ... We present efficient algorithms for partitioning 2dimensional space into faces arising from the boundaries

*of*overlapping polygons. ... The problem reduces to

*planar*

*point*

*location*;

*a*query that intersects no segments can be effectively reduced to

*a*single

*point*, as we need only

*locate*the face that contains some

*point*from the query region ...

##
###
I/O-Efficient Map Overlay and Point Location in Low-Density Subdivisions
[chapter]

2007
*
Lecture Notes in Computer Science
*

*in*O(scan (n)) i/o's, where n is the total number

*of*edges

*in*the two

*subdivisions*, (ii) answer

*a*single

*point*

*location*query

*in*O(log B n) i/o's and k batched

*point*

*location*queries

*in*O(scan (n) + sort ... We present improved and simplified i/o-efficient algorithms for two problems on

*planar*low-density

*subdivisions*, namely map overlay and

*point*

*location*. ... An optimal static structure for

*point*

*location*

*in*general

*planar*

*subdivisions*was already given by Goodrich et al. [18] for the standard i/o-model and by Bender et al. ...

##
###
Comparison of three bounding regions with cubic convergence to planar freeform curves

2015
*
The Visual Computer
*

*In*other comparison criteria, the BCA approach is more efficient and stable than the other two methods

*in*producing

*a*hierarchy

*of*bounding regions that approximate

*a*family

*of*freeform

*planar*curves within ...

*a*given error bound. ... four junction

*points*(each by adding to the center the result

*of*

*a*scalarvector

*multiplication*). ...

##
###
I/O-Efficient Flow Modeling on Fat Terrains
[chapter]

2007
*
Lecture Notes in Computer Science
*

*in*O(scan (n)) i/o's, where n is the total number

*of*edges

*in*the two

*subdivisions*, (ii) answer

*a*single

*point*

*location*query

*in*O(log B n) i/o's and k batched

*point*

*location*queries

*in*O(scan (n) + sort ... We present improved and simplified i/o-efficient algorithms for two problems on

*planar*low-density

*subdivisions*, namely map overlay and

*point*

*location*. ... An optimal static structure for

*point*

*location*

*in*general

*planar*

*subdivisions*was already given by Goodrich et al. [18] for the standard i/o-model and by Bender et al. ...

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