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

Vahideh Keikha and Maarten Löffler and Ali Mohades and Jérôme Urhausen and Ivor van der Hoog
2017 arXiv   pre-print
We 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  ... 
arXiv:1705.11035v2 fatcat:2axjswqvujbhbhricckfxhtq6i

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

Vahideh Keikha, Maarten Löffler, Ali Mohades, Jérôme Urhausen, Ivor van der Hoog
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  ... 
arXiv:1708.00681v3 fatcat:juxboyefdvcozm7hblqsetlq5a

Oriented Convex Containers of Polygons [article]

R Nandakumar
2018 arXiv   pre-print
We consider the optimal containment of 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  ... 
arXiv:1802.10447v3 fatcat:ty4s6ysmdvbhtmqy2oay5cknt4

Laplacian smoothing revisited [article]

Dimitris Vartziotis, Benjamin Himpel
2014 arXiv   pre-print
Using 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.  ... 
arXiv:1406.4333v1 fatcat:sklpzswvwfaxnmpactik5en72m

Viviani's Theorem and its Extensions Revisited; Canghareeb, Sanghareeb and a New Definition of the Ellipse [article]

Elias Abboud
2014 arXiv   pre-print
Canghareeb and Sanghareeb are described to be "geometric creatures" living 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.  ... 
arXiv:1410.6442v1 fatcat:5ablq2nq5fdh5fjtpxyzjbvgjq

PolySimp: A Tool for Polygon Simplification based on the Underlying Scaling Hierarchy

Ding Ma, Zhigang Zhao, Ye Zheng, Renzhong Guo, Wei Zhu
2020 ISPRS International Journal of Geo-Information  
into 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.  ... 
doi:10.3390/ijgi9100594 fatcat:azm6avr6v5fghoykctgljoy5am

Convex Polygons in Geometric Triangulations

2017 Combinatorics, probability & computing  
We show that the 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.  ... 
doi:10.1017/s0963548317000141 fatcat:4yfjh3zprjaknf2mlhxlmzuwiu

On Calculating the Packing Efficiency for Embedding Hexagonal and Dodecagonal Sensors in a Circular Container

Marina Prvan, Julije Ožegović, Arijana Burazin Mišura
2019 Mathematical Problems in Engineering  
Hence, we construct an irregular 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).  ... 
doi:10.1155/2019/9624751 fatcat:qey5qwf7djavhcihxvl4t4ikda

Satellite Constellation Orbit Design Optimization with Combined Genetic Algorithm and Semianalytical Approach

Tania Savitri, Youngjoo Kim, Sujang Jo, Hyochoong Bang
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 .  ... 
doi:10.1155/2017/1235692 fatcat:mwpoksv6yjdo3lrx2kgnxqh3ya

Data association using empty convex polygonal regions in EKF-SLAM

Gururaj Kosuru, Satish Pedduri, K Madhava Krishna
2010 2010 IEEE International Conference on Robotics and Biomimetics  
The resulting data representation enables semantic reasoning on 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).  ... 
doi:10.1109/robio.2010.5723430 dblp:conf/robio/KosuruPK10 fatcat:6mhmktmkynbxrk67i2opo7zrju

Approximating the minimum weight steiner triangulation

David Eppstein
1994 Discrete & Computational Geometry  
As 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.  ... 
doi:10.1007/bf02574002 fatcat:wjk7pgqhljhvzdod7vc5dz32fe

A Bound on the Edge-Flipping Distance between Triangulations (Revisiting the Proof) [article]

Thomas Dagès, Alfred M. Bruckstein
2021 arXiv   pre-print
We provide 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 .  ... 
arXiv:2106.14408v1 fatcat:4tuvlxtwe5ctrf6i2v2idiwk2u

Affine invariant triangulations [article]

Prosenjit Bose, Pilar Cano, Rodrigo I. Silveira
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.  ... 
arXiv:2011.02197v1 fatcat:e3itntpbirbetdfxygpuevnvmy

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

Lisa Groiss, Bert Jüttler, Dominik Mokriš
2021 Computer Aided Geometric Design  
we correct 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.  ... 
doi:10.1016/j.cagd.2021.101975 fatcat:kfeqzjrjizdb7evx6vy3cyfgmq

Conforming polygonal finite elements

N. Sukumar, A. Tabarraei
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.  ... 
doi:10.1002/nme.1141 fatcat:yo3gvnl44jgbhkpmmcqypvwdvq
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