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### Shortest paths in simple polygons with polygon-meet constraints

2004 Information Processing Letters
We study a constrained version of the shortest path problem in simple polygons, in which the path must visit a given target polygon.  ...  The algorithms are linear, both in time and space, in terms of the complexity of the two polygons.  ...  Convex target polygon In this case, the problem can be considered as a special case of the general touring polygons problem (general TPP) studied by Dror et al. [3] .  ...

### Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane

Mikkel Abrahamsen, Panos Giannopoulos, Maarten Löffler, Günter Rote
2020 Discrete & Computational Geometry
We first give an O(n 4 log 3 n)-time algorithm that computes an optimal fence for the case where the input consists of polygons of two colours with n corners in total.  ...  We study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane with k different colours, compute a shortest "fence" F, i.e., a union of curves of minimum  ...  This work was initiated at the workshop on Fixed-Parameter Computational Geometry at the Lorentz Center in Leiden in May 2018.  ...

### Intégrale, Longueur, Aire

H. Lebesgue
1902 Annali di Matematica Pura ed Applicata
ongueurs des segments ou des aires des polygones.  ...  Dc BoIs R~YMO~D (*) est donc un cas particulier de la d6finition classique~ m~me en dtendant comme je l'ai fait le sens du mot in-t~grale.  ...  Consid6rons un ligne polygonale P inscrite dans cette courbe.  ...

### Thiessen Polygons [chapter]

2017 Encyclopedia of GIS
When there are multiple convex polygons in the same constraint relation, the complement of one convex polygon can be affected by the complements of all the other convex polygons.  ...  In constraint databases, any concave polygon or polygon with holes has to be divided and represented by multiple convex polygons.  ...  The LTIS database contains the following information for use in our analysis: • Accidents ı In general, it records the total duration (measured in [Event-Hrs] ) affected by all accidents on a given day  ...

### Optimal patrolling of fragmented boundaries

Andrew Collins, Jurek Czyzowicz, Leszek Gasieniec, Adrian Kosowski, Evangelos Kranakis, Danny Krizanc, Russell Martin, Oscar Morales Ponce
2013 Proceedings of the 25th ACM symposium on Parallelism in algorithms and architectures - SPAA '13
These two fundamental types of strategies were studied in the past in the robotics community in different theoretical and experimental settings.  ...  Our main contribution is a proof that the optimal solution to the patrolling problem is attained either by the cyclic strategy, in which all the robots move in one direction around the curve, or by the  ...  Unsurprisingly, many general watchman route problems are NP-hard (e.g. watchman tours of simple polygon with holes, suggested in [7] and corrected in [13] ), touring a sequence of non-convex polygons  ...

### Foundations [chapter]

2011 Methods of Geometry
The image f[D] of the domain is called the range off. Preserved and invariant properties In many apphcations, a function maps its domain D into itself.  ...  The intersection of any family of convex point sets is convex.  ...  For example, 2 is an upper bound for the set S of values f(x) = x 2 /(l + x 2 ), indicated by the dotted hne in figure B.l.  ...

### The Genesis of Architectural Space

Gyula J. Hajnóczi
2021 Építés - Építészettudomány
This valuable writing – although published in Hungarian as a series of journal articles or in copied version – is hardly known to the architectural profession.2 The English version published here helps  ...  Space" is a concise summary of Gyula Hajnóczi's spatial theory research that offers a thorough overview also for less experienced researchers in architectural theory.  ...  but convex, i.e. turn in a full-bodied, "articulated" way.  ...

### Topology building and random polygon generation

Chongjian Zhu
1994
This problem translates directly to polygon nesting in computational geometry: given a set of polygons, find their nesting structure.  ...  In order to test and verify the time complexity of our polygon nesting algorithm, we present an algorithm that generates x-monotone polygons uniformly at random over a vertex set of n points.  ...  Then they partitioned the convex polygonal line into convex chains and further partitioned the convex chains into the x-monotone polygonal lines.  ...

### Subject Index Volumes 1–200

2001 Discrete Mathematics
, 2215, 2955, 5032, 5222, 5322, see also Eulerian tour trails, 4690, 5064 Euler (cont.)  ...  , see n-convex d-convexity, see n-convexity d-covered, see n-covered d-critical, see n-critical d-cube, see n-cube D-cycle, 1442 d-dimensional, see n-dimensional d-disjunct, see n-disjunct d-distance,  ...  convex bodies, 1950 convex bodies, Wilson, 88, 255, 514, 601, 1376 convex bodies, , 1490 convex bodies, , 1570 convex bodies, , 2526 convex bodies, , 2568 convex bodies, , 2691 857, 1137, 1200, 1305  ...

### Full Issue PDF

Dwayne Lepitzki
We thank the Ontario Reptile and Amphibian Atlas for making a vast amount of data available in the public domain and the hundreds of volunteers who have taken the time to submit their observations to those  ...  MacDonald, whose expertise and knowledge of all things amphibious was instrumental to the success of this project in the early days.  ...  We determined the home range of individuals by calculating the 100% minimum convex polygon (MCP) using the genmcp function in GME, which provides the smallest possible convex polygon that encompasses all  ...

### Parts of visual form: computational aspects

K. Siddiqi, B.B. Kimia
1995 IEEE Transactions on Pattern Analysis and Machine Intelligence
A representation of parts for recognition requires that they be invariant to rigid transformations, robust in the presence of occlusions, stable with changes in viewing geometry, and be arranged in a hierarchy  ...  Neckbased parts arise from narrowings in shape, or the local minima in distance between two p o i n ts on the boundary, while limb-based parts arise from a pair of negative curvature minima which h a ve  ...  fence.  ...

### Proceedings of the IX International Workshop on Locational Analysis and Related Problems [article]

Marta Baldomero-Naranjo, Inmaculada Espejo-Miranda, Luisa I. Martínez-Merino, Juan Manuel Muñoz-Ocaña, Antonio M. Rodríguez-Chía
2020 arXiv   pre-print
The International Workshop on Locational Analysis and Related Problems will take place during January 30-February 1, 2019 in C\'adiz (Spain).  ...  Previous meetings took place in Segovia (September 27-29, 2017), M\'alaga (September 14-16, 2016), Barcelona (November 25-28, 2015), Sevilla (October 1-3, 2014), Torremolinos (M\'alaga, June 19-21, 2013  ...  polygonal chain.  ...

### Elements of Mathematics: A Problem-Centered Approach to History and Foundations [article]

Gabor Toth
2022 Zenodo
In general, naïve theory in mathematics is a term referring to a mathematical theory that employs natural language to describe its objects of study.  ...  Many terms in a naïve theory are not defined with mathematical rigor, and thus the theory is prone to "excesses," possibly leading to inconsistencies.  ...  If Q participates in a polygonal path for C, then this path can be split into the union of two polygonal paths, one for C 1 and the other for C 2 .  ...

### Non-parametric Spatial-Domain Algorithm for Analysis and Mapping of Urban Scenes Using 3D Point Clouds [article]

Hussein Al-Gurrani, University Of Calgary, University Of Calgary, Ayman Habib, Naser El-Sheimy
2016
With half of the current seven billion people living in urban areas, urbanization most certainly will continue in the future.  ...  Such a steep urbanization growth curve requires and justifies the growing interest not only in urban sustainability but also in sustainable development in general, which explains the growing interest as  ...  Figure 5 - 5 Figure 5-16 Detected polygons and recognized windows in a facade. (a) Detected polygons.  ...

### An Invitation to Tensor-Triangular Geometry

Paul Balmer
2017 Notices of the American Mathematical Society
Embark on a playful mathematical tour, aided by Lisl Gaal's illustrations of familiar scenes and whimsical triggers for the imagination.  ...  Along the way, find fruit stands arranged using polynomial multiplication, checkerboard tablecloths sewn with patterns of primes in a two-dimensional number system, and deceptive cats revealing that simple  ...  C y n t h i a D w o r k , H a r v a r d U n i v e r s i t y , Privavy in the Land of Plenty (AMS Josiah Willard Gibbs Lecture).  ...
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