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Maximum-Area Triangle in a Convex Polygon, Revisited
[article]

2017
*
arXiv
*
pre-print

We

arXiv:1705.11035v2
fatcat:2axjswqvujbhbhricckfxhtq6i
*revisit*the following problem: Given*a**convex**polygon*P, find the largest-*area*inscribed*triangle*. ... Also we show by example that the algorithm presented*in*1979 by Dobkin and Snyder for finding the largest-*area*k-gon that is inscribed*in**a**convex**polygon*fails to find the optimal solution for k=4. ... Acknowledgments The authors would like to thank everybody who has discussed this discovery and its implications with them over the past months,*in*particular Bahareh Banyassady, Ahmad Biniaz, Prosenjit ...##
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Maximum-Area Quadrilateral in a Convex Polygon, Revisited
[article]

2017
*
arXiv
*
pre-print

*In*this note we show by example that the algorithm presented

*in*1979 by Dobkin and Snyder for finding the largest-

*area*k-gon that is inscribed

*in*

*a*

*convex*

*polygon*fails to find the optimal solution for ...

*In*this note, we

*revisit*the following problem: "Given

*a*

*convex*

*polygon*P , what is the largest-

*area*quadrilateral inscribed

*in*P ?" (see Figure 1 ). ... Introduction Surprisingly,

*in*[2] the authors show that the linear-time algorithm presented

*in*1979 by Dobkin and Snyder [1] for finding the largest-

*area*

*triangle*that is inscribed

*in*

*a*

*convex*

*polygon*...

##
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Oriented Convex Containers of Polygons
[article]

2018
*
arXiv
*
pre-print

We consider the optimal containment of

arXiv:1802.10447v3
fatcat:ty4s6ysmdvbhtmqy2oay5cknt4
*polygonal*regions within*convex*containers with the special property of 'orientedness' - an oriented region enables us to choose*a*preferred direction on the plane ... Rectangle containers of least*area*and perimeter for*a*given*convex*region For*a**convex**polygon*, let us call its rectangular container of minimum*area*R*A*and the rectangular container of minimum perimeter ... We now consider the least*area*and least perimeter right*triangles*that contain*a*given*convex*region (call these*triangles*RT*A*and RT P respectively) and look for*convex*regions for which these optimal ...##
###
Laplacian smoothing revisited
[article]

2014
*
arXiv
*
pre-print

Using

arXiv:1406.4333v1
fatcat:sklpzswvwfaxnmpactik5en72m
*a*simple backtracking line-search we compare several smoothing methods with respect to the resulting mesh quality. We also discuss their effectiveness*in*combination with topology modification. ... We use these insights to provide natural generalizations to*polygons*and polyhedra. ... Instead of λ 1 we can choose any combination of*area*and edge length associated to each*triangle*, which gives*a**convex*function. ...##
###
Viviani's Theorem and its Extensions Revisited; Canghareeb, Sanghareeb and a New Definition of the Ellipse
[article]

2014
*
arXiv
*
pre-print

Canghareeb and Sanghareeb are described to be "geometric creatures" living

arXiv:1410.6442v1
fatcat:5ablq2nq5fdh5fjtpxyzjbvgjq
*in*the plane under certain conditions relative to*a**triangle*. They have common and different features. ... The former lives on*a*line segment inside*a**triangle*and the latter lives along an ellipse. Their story ends with*a*new definition of the ellipse. ... Acknowledgement: This work is part of*a*research which was held and supported by Beit Berl College Research Fund. ...##
###
PolySimp: A Tool for Polygon Simplification based on the Underlying Scaling Hierarchy

2020
*
ISPRS International Journal of Geo-Information
*

into

doi:10.3390/ijgi9100594
fatcat:azm6avr6v5fghoykctgljoy5am
*a*set of bends,*triangles*, or*convex*hulls as basic geometric units for simplification. ... The pattern of far more small things than large ones is*a*de facto heavy tailed distribution.*In*this paper, we apply the scaling hierarchy for map generalization to*polygonal*features. ... The process stops at Iteration 2 as all the*polygon*components are*in*the shape of*triangles*, the*convex*degree of which is 1.*In*total, there were 67*convex*hulls. ...##
###
Convex Polygons in Geometric Triangulations

2017
*
Combinatorics, probability & computing
*

We show that the

doi:10.1017/s0963548317000141
fatcat:4yfjh3zprjaknf2mlhxlmzuwiu
*maximum*number of*convex**polygons**in**a*triangulation of n points*in*the plane is O(1.5029 n ). ... Given*a*planar straight-line graph G with n vertices, we also show how to compute efficiently the number of*convex**polygons**in*G. ... The authors are grateful to an anonymous reviewer for*a*very careful reading of the manuscript and for pertinent remarks. ...##
###
On Calculating the Packing Efficiency for Embedding Hexagonal and Dodecagonal Sensors in a Circular Container

2019
*
Mathematical Problems in Engineering
*

Hence, we construct an irregular

doi:10.1155/2019/9624751
fatcat:qey5qwf7djavhcihxvl4t4ikda
*convex*hexagon that is semiregularly tessellating the targeted*area*. ... Even though packing problems are common*in*many fields of research, not many authors concentrate on packing*polygons*of known dimensions into*a*circular shape to optimize*a*certain objective. ... 1 , 2 , 1 , 2 ∈ and 1 , 2 ≥ 3 (the smallest*polygon*is*a**triangle*). ...##
###
Satellite Constellation Orbit Design Optimization with Combined Genetic Algorithm and Semianalytical Approach

2017
*
International Journal of Aerospace Engineering
*

*A*target

*area*represented by

*a*

*polygon*defined by grid points is chosen instead of using

*a*target point only. The constellation consists of nonsymmetric and circular Low Earth Orbit (LEO) satellites. ...

*A*multiobjective optimization study is conducted

*in*this study with percent coverage and

*revisit*time as the two main parameters to analyze the performance of the constellation. ... From these points,

*a*

*convex*hull

*polygon*is generated using

*a*computational geometry method, as depicted

*in*Figure 1 . ...

##
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Data association using empty convex polygonal regions in EKF-SLAM

2010
*
2010 IEEE International Conference on Robotics and Biomimetics
*

The resulting data representation enables semantic reasoning on

doi:10.1109/robio.2010.5723430
dblp:conf/robio/KosuruPK10
fatcat:6mhmktmkynbxrk67i2opo7zrju
*a*spatial level which reduces the misassociation of closely spaced data from different spatial configurations through the use of*convex**polygons*... The data representation is especially effective for association when*revisiting*previously mapped regions efficiently. ... We have also shown*a*method to calculate the largest*Maximum*Empty*Convex**Polygon*(MECP). ...##
###
Approximating the minimum weight steiner triangulation

1994
*
Discrete & Computational Geometry
*

As

doi:10.1007/bf02574002
fatcat:wjk7pgqhljhvzdod7vc5dz32fe
*in*[1],*a*nonpolynomial number of Steiner points may be needed. 9 We approximate the MWST of*a**convex**polygon*(not just point set) using ...*In*O(n log n) time we can compute*a*triangulation with O(n) new points, and no obtuse*triangles*, that approximates the MWST. ... Acknowledgments I would like to thank Mike Dillencourt for carefully reading*a*draft of this paper, and for many helpful discussions. ...##
###
A Bound on the Edge-Flipping Distance between Triangulations (Revisiting the Proof)
[article]

2021
*
arXiv
*
pre-print

We provide

arXiv:2106.14408v1
fatcat:4tuvlxtwe5ctrf6i2v2idiwk2u
*a*complete and detailed proof of this result*in**a*slightly generalised setting using*a*case-based analysis that fills several gaps left by previous proofs of the result. ... We*revisit*here*a*fundamental result on planar triangulations, namely that the flip distance between two triangulations is upper-bounded by the number of proper intersections between their straight-segment ... Furthermore, the*polygon*of successively concatenated*triangles*is*convex*when the*polygon*is entirely contained within*a*half-plane delimited by u i−1 u i . ...##
###
Affine invariant triangulations
[article]

2020
*
arXiv
*
pre-print

*In*addition, we provide different affine invariant sorting methods of

*a*point set S and of the vertices of

*a*

*polygon*P that can be combined with known algorithms to obtain other affine invariant triangulation ... We

*revisit*the A_S-norm from

*a*geometric perspective, and show that DT_A_S[S] can be seen as

*a*standard Delaunay triangulation of

*a*transformed point set based on S. ... Let be

*a*

*triangle*defined by the endpoints of an edge

*in*CH(S) and µ, that is different from B 1 . Since

*Area*( B 1 ) >

*Area*( ), it follows that

*Area*( B 1 )

*Area*( ) > 1. ...

##
###
27 variants of Tutte's theorem for plane near-triangulations and an application to periodic spline surface fitting

2021
*
Computer Aided Geometric Design
*

we correct

doi:10.1016/j.cagd.2021.101975
fatcat:kfeqzjrjizdb7evx6vy3cyfgmq
*a*minor inaccuracy*in**a*previous result concerning Floater-type parameterizations for genus-1 meshes. ... , which is an embedding of*a*planar graph with the property that all bounded faces are -possibly curved -*triangles*) of Tutte's Spring Embedding Theorem. ... It remains to analyze the existence of degenerate*triangles*, i.e.,*triangles*with zero*area*. These*triangles*form the degenerate sub-graph of Ǧ. ...##
###
Conforming polygonal finite elements

2004
*
International Journal for Numerical Methods in Engineering
*

*Polygonal*finite elements provide greater flexibility

*in*mesh generation and are better-suited for applications

*in*solid mechanics which involve

*a*significant change

*in*the topology of the material domain ...

*A*particular and notable contribution is the use of meshfree (natural-neighbour, nn) basis functions on

*a*canonical element combined with an affine map to construct conforming approximations on

*convex*... We also thank Professor John Bolander for providing the code for the Voronoi mesh generator, which was used to generate the

*polygonal*meshes considered

*in*this study. ...

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