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Journal articles on the topic 'Triangulation'

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Published: 4 June 2021

Last updated: 30 July 2024

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1

Mitchell, Scott A. "Finding a Covering Triangulation Whose Maximum Angle is Provably Small." International Journal of Computational Geometry & Applications 07, no. 01n02 (February 1997): 5–20. http://dx.doi.org/10.1142/s021819599700003x.

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We consider the following problem: given a planar straight-line graph, find a covering triangulation whose maximum angle is as small as possible. A covering triangulation is a triangulation whose vertex set contains the input vertex set and whose edge set contains the input edge set. The covering triangulation problem differs from the usual Steiner triangulation problem in that we may not add a vertex on any input edge. Covering triangulations provide a convenient method for triangulating multiple regions sharing a common boundary, as each region can be triangulated independently. We give an explicit lower bound γopt on the maximum angle in any covering triangulation of a particular input graph in terms of its local geometry. Our algorithm produces a covering triangulation whose maximum angle γ is provably close to γopt. Specifically, we show that [Formula: see text], i.e., our γ is not much closer to π than is γopt. To our knowledge, this result represents the first nontrivial bound on a covering triangulation's maximum angle. Our algorithm adds O(n) Steiner points and runs in time O(n log n), where n is the number of vertices of the input. We have implemented an O(n2) time version of our algorithm.

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2

Wu, Bai Chao, Ai Ping Tang, and Lian Fa Wang. "A Constrained Delaunay Triangulation Algorithm Based on Incremental Points." Applied Mechanics and Materials 90-93 (September 2011): 3277–82. http://dx.doi.org/10.4028/www.scientific.net/amm.90-93.3277.

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The foundation ofdelaunay triangulationandconstrained delaunay triangulationis the basis of three dimensional geographical information system which is one of hot issues of the contemporary era; moreover it is widely applied in finite element methods, terrain modeling and object reconstruction, euclidean minimum spanning tree and other applications. An algorithm for generatingconstrained delaunay triangulationin two dimensional planes is presented. The algorithm permits constrained edges and polygons (possibly with holes) to be specified in the triangulations, and describes some data structures related to constrained edges and polygons. In order to maintain the delaunay criterion largely，some new incremental points are added onto the constrained ones. After the data set is preprocessed, the foundation ofconstrained delaunay triangulationis showed as follows: firstly, the constrained edges and polygons generate initial triangulations，then the remained points completes the triangulation . Some pseudo-codes involved in the algorithm are provided. Finally, some conclusions and further studies are given.

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3

Keil, J. Mark, and Tzvetalin S. Vassilev. "Approximating the MaxMin and MinMax Area Triangulations using Angular Constraints." Serdica Journal of Computing 4, no. 3 (October 21, 2010): 321–34. http://dx.doi.org/10.55630/sjc.2010.4.321-334.

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We consider sets of points in the two-dimensional Euclidean plane. For a planar point set in general position, i.e. no three points collinear, a triangulation is a maximal set of non-intersecting straight line segments with vertices in the given points. These segments, called edges, subdivide the convex hull of the set into triangular regions called faces or simply triangles. We study two triangulations that optimize the area of the individual triangles: MaxMin and MinMax area triangulation. MaxMin area triangulation is the triangulation that maximizes the area of the smallest area triangle in the triangulation over all possible triangulations of the given point set. Similarly, MinMax area triangulation is the one that minimizes the area of the largest area triangle over all possible triangulations of the point set. For a point set in convex position there are O(n^2 log n) time and O(n^2) space algorithms that compute these two optimal area triangulations. No polynomial time algorithm is known for the general case. In this paper we present an approach to approximation of the MaxMin and MinMax area triangulations of a general point set. The algorithm, based on angular constraints and perfect matchings between triangulations, runs in O(n^3) time and O(n^2) space. We determine the approximation factors as functions of the minimal angles inthe optimal (unknown) triangulation and the approximating one.

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4

Levcopoulos, Christos, and Drago Krznaric. "Quasi-Greedy Triangulations Approximating the Minimum Weight Triangulation." Journal of Algorithms 27, no. 2 (May 1998): 303–38. http://dx.doi.org/10.1006/jagm.1997.0918.

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5

Stanchev, Bozhidar Angelov, and Hristo Ivanov Paraskevov. "CONSTRAINING TRIANGULATION TO LINE SEGMENTS: A FAST METHOD FOR CONSTRUCTING CONSTRAINED DELAUNAY TRIANGULATION." Annals of the Academy of Romanian Scientists Series on Mathematics and Its Application 12, no. 1-2 (2020): 164–78. http://dx.doi.org/10.56082/annalsarscimath.2020.1-2.164.

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"In this paper we present an edge swapping approach for incorporating line segments into triangulation. If the initial triangulation is Delaunay, the algorithm tends to produce optimal Constrained Delaunay triangulation by improving the triangles’ aspect ratios from the local area being constrained. There are two types of methods for constructing Constrained Delaunay Triangulation: straight-forward ones which take both points and line segments as source data and produce constrained triangulation from them at once; and post-processing ones which take an already constructed triangulation and incorporate line segments into it. While most of the existing post-processing approaches clear the triangle’s edges intersected by the line segment being incorporated and fill the opened hole (cavity) by re-triangulating it, the only processing that our algorithm does is to change the triangulation connectivity and to improve the triangles’ aspect ratios through edge swapping. Hereof, it is less expensive in terms of both operating and memory costs. The motivation behind our approach is that most of the existing straight-forward triangulators are too slow and not stable enough. The idea is to use pure Delaunay triangulator to produce an initial Delaunay triangulation and later on to constrain it to the line segments (in other words, to split the processing into two steps, each of which is stable enough and the combination of them works much faster). The algorithm also minimizes the number of the newly introduced triangulation points - new points are added only if any of the line segment’s endpoints does not match an existing triangulation point."

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6

Cavicchioli, Maddalena. "Acute Triangulations of Trapezoids and Pentagons." Journal of Mathematics 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/747128.

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An acute triangulation of a polygon is a triangulation whose triangles have all their angles less than . The number of triangles in a triangulation is called the size of it. In this paper, we investigate acute triangulations of trapezoids and convex pentagons and prove new results about such triangulations with minimum size. This completes and improves in some cases the results obtained in two papers of Yuan (2010).

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7

Mirzoev, Tigran, and Tzvetalin S. Vassilev. "Quadratic Time Computable Instances of MaxMin and MinMax Area Triangulations of Convex Polygons." Serdica Journal of Computing 4, no. 3 (October 21, 2010): 335–48. http://dx.doi.org/10.55630/sjc.2010.4.335-348.

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We consider the problems of finding two optimal triangulations of a convex polygon: MaxMin area and MinMax area. These are the triangulations that maximize the area of the smallest area triangle in a triangulation, and respectively minimize the area of the largest area triangle in a triangulation, over all possible triangulations. The problem was originally solved by Klincsek by dynamic programming in cubic time [2]. Later, Keil and Vassilev devised an algorithm that runs in O(n^2 log n) time [1]. In this paper we describe new geometric findings on the structure of MaxMin and MinMax Area triangulations of convex polygons in two dimensions and their algorithmic implications. We improve the algorithm’s running time to quadratic for large classes of convex polygons. We also present experimental results on MaxMin area triangulation.

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8

BURTON, BENJAMIN A. "FACE PAIRING GRAPHS AND 3-MANIFOLD ENUMERATION." Journal of Knot Theory and Its Ramifications 13, no. 08 (December 2004): 1057–101. http://dx.doi.org/10.1142/s0218216504003627.

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The face pairing graph of a 3-manifold triangulation is a 4-valent graph denoting which tetrahedron faces are identified with which others. We present a series of properties that must be satisfied by the face pairing graph of a closed minimal ℙ2-irreducible triangulation. In addition we present constraints upon the combinatorial structure of such a triangulation that can be deduced from its face pairing graph. These results are then applied to the enumeration of closed minimal ℙ2-irreducible 3-manifold triangulations, leading to a significant improvement in the performance of the enumeration algorithm. Results are offered for both orientable and non-orientable triangulations.

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9

Ramprogus, Vince. "Triangulation." Nurse Researcher 12, no. 4 (April 2005): 4–6. http://dx.doi.org/10.7748/nr2005.04.12.4.4.c5954.

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10

Singh, Surinder. "Triangulation." London Journal of Primary Care 2, no. 1 (June 2009): 86–87. http://dx.doi.org/10.1080/17571472.2009.11493257.

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11

KIMCHI, JUDITH, BARBARA POLIVKA, and JOANNE SABOL STEVENSON. "Triangulation." Nursing Research 40, no. 6 (November 1991): 364???366. http://dx.doi.org/10.1097/00006199-199111000-00009.

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12

Jackson, Emerson Abraham. "Triangulation." African Journal of Economic and Management Studies 9, no. 2 (June 11, 2018): 266–71. http://dx.doi.org/10.1108/ajems-01-2018-0034.

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Purpose The purpose of this paper is to provide forum for critical discourses in the area of retroduction thinking in the re-orientation of social sciences research at the Bank of Sierra Leone. It is acknowledged that economic research at the bank is more oriented towards the orthodox approach (Keynesian, etc.), while that of the un-orthodox (heterodox) view is being blighted by the need to prove economic theory application without much consideration given to the imperfect market economic conditions faced in the country. Design/methodology/approach The methodology of the paper is based on critical theoretical literature review of documents and also the assessment of current approach to research practices at the Bank of Sierra Leone. Findings There is actually a pool of professionals from different backgrounds, whose passion have been coordinated to embrace the idea of integrating variety of triangulation approaches (methodological, theoretical, data, and investigator) in their pursuit of exploring best practices for policy research at the bank. Research is an ontological journey and the process of its epistemological inquiry requires pool of experts, drawn from wide range of backgrounds (mainstream Social Sciences, Mathematical Science, etc.), whose approaches are pertinent in the pursuit of human discovery, and particularly for policy development, as in the case with the Bank of Sierra Leone. This work has highlighted several salient recommended points for future action in the lead towards energizing research at the bank. Originality/value The effort to provide this paper is very original, which is based on the author's experience of working in a research focused environment. This paper will be of value to future pursuance of work in promoting triangulation research at the Bank of Sierra Leone. It is hoped that the outcome of this will spearhead mixture of professional engagement of work activities that is geared towards addressing effective policy formulation at the bank in meeting its key objective (maintaining price stability).

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13

Manning, Louise. "Triangulation." Worldwide Hospitality and Tourism Themes 10, no. 3 (June 11, 2018): 297–312. http://dx.doi.org/10.1108/whatt-02-2018-0009.

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PurposeThe purpose of this paper is to critique the existing and emerging alternative approaches being used by regulators and industry to verify the presence and efficacy of food safety management systems (FSMS). It is the second paper in a theme issue of Worldwide Hospitality and Tourism Themes, discussing the importance of measuring food safety and quality culture.Design/methodology/approachThis paper, primarily focused on UK examples, examines academic and grey literature to consider the options for effective verification of FSMS with emphasis on the hospitality sector including the use of triangulation.FindingsThird-party certification (TPC) compliance audits alone will not deliver effective verification of the FSMS and the cultural context of how formal systems are implemented, monitored and internally verified. Triangulation needs to be undertaken during the FSMS verification process which at its simplest is a Question, Observe, Measure (QOM) triad and at its more complex involves TPC compliance audits and performance assessment using data analysis methodology and product and environmental testing.Originality/valueThe paper will be of value to practitioners, researchers and other stakeholders involved in the hospitality industry.

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14

Wilson, Chauncey E. "Triangulation." Interactions 13, no. 6 (November 2006): 46. http://dx.doi.org/10.1145/1167948.1167980.

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15

Hartley, Richard I., and Peter Sturm. "Triangulation." Computer Vision and Image Understanding 68, no. 2 (November 1997): 146–57. http://dx.doi.org/10.1006/cviu.1997.0547.

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16

Landman, Claude. "Triangulation." Journal français de psychiatrie 53, no. 2 (May 28, 2024): 53–56. http://dx.doi.org/10.3917/jfp.053.0053.

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17

NARASIMHAN, GIRI. "ON HAMILTONIAN TRIANGULATIONS IN SIMPLE POLYGONS." International Journal of Computational Geometry & Applications 09, no. 03 (June 1999): 261–75. http://dx.doi.org/10.1142/s0218195999000182.

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An n-vertex simple polygon P is said to have a Hamiltonian Triangulation if it has a triangulation whose dual graph contains a hamiltonian path. Such triangulations are useful in fast rendering engines in Computer Graphics. We give a new characterization of polygons with hamiltonian triangulations and use it to devise O(n log n)-time algorithms to recognize such polygons. We also give efficient algorithms for several related problems.

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18

Dyshkant, N. F. "About Some Localization Problems in Delaunay Triangulations." Modeling and Analysis of Information Systems 19, no. 6 (March 12, 2015): 112–26. http://dx.doi.org/10.18255/1818-1015-2012-6-112-126.

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We study some problems of nodes localization in a Delaunay triangulation and problem-solving procedures. For the problem of the set of nodes the computationally efficient approach that uses Euclidean minimum spanning tree of Delaunay triangulation is proposed. Efficient estimations for computational comlexity of the proposed methods in the average and in the worst cases are proved.computational geometry, geometric search, Delaunay triangulation, merging of overlapping triangulations, unregular discrete mesh, computational complexity

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19

Davies, David. "A Moderately Pessimistic Perspective on “Cooperative Naturalism”." Projections 12, no. 2 (December 1, 2018): 9–18. http://dx.doi.org/10.3167/proj.2018.120203.

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Murray Smith’s plea for a “cooperative naturalism” that adopts a “triangulational” approach to issues in film studies is both timely and well-defended. I raise three concerns, however: one is external, relating to this strategy’s limitations, and two are internal, relating to Smith’s application of the strategy. While triangulation seems appropriate when we ask about the nature of film experience, other philosophical questions about film have an ineliminable normative dimension that triangulation cannot address. Empirically informed philosophical reflection upon the arts must be “moderately pessimistic” in recognizing this fact. The internal concerns relate to Smith’s claims about the value and neurological basis of cinematic empathy. First, while empathy plays a central role in film experience, I argue that its neurological underpinnings fail to support the epistemic value he ascribes to it. Second, I question Smith’s reliance, in triangulating, upon the work of the Parma school on “mirror neurons.”

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20

Rakotosaona, Marie-Julie, Noam Aigerman, Niloy J. Mitra, Maks Ovsjanikov, and Paul Guerrero. "Differentiable surface triangulation." ACM Transactions on Graphics 40, no. 6 (December 2021): 1–13. http://dx.doi.org/10.1145/3478513.3480554.

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Triangle meshes remain the most popular data representation for surface geometry. This ubiquitous representation is essentially a hybrid one that decouples continuous vertex locations from the discrete topological triangulation. Unfortunately, the combinatorial nature of the triangulation prevents taking derivatives over the space of possible meshings of any given surface. As a result, to date, mesh processing and optimization techniques have been unable to truly take advantage of modular gradient descent components of modern optimization frameworks. In this work, we present a differentiable surface triangulation that enables optimization for any per-vertex or per-face differentiable objective function over the space of underlying surface triangulations. Our method builds on the result that any 2D triangulation can be achieved by a suitably perturbed weighted Delaunay triangulation. We translate this result into a computational algorithm by proposing a soft relaxation of the classical weighted Delaunay triangulation and optimizing over vertex weights and vertex locations. We extend the algorithm to 3D by decomposing shapes into developable sets and differentiably meshing each set with suitable boundary constraints. We demonstrate the efficacy of our method on various planar and surface meshes on a range of difficult-to-optimize objective functions. Our code can be found online: https://github.com/mrakotosaon/diff-surface-triangulation.

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21

Grigis, Alain. "Triangulation de Delaunay et Triangulation des Tores." Geometriae Dedicata 143, no. 1 (April 23, 2009): 81–88. http://dx.doi.org/10.1007/s10711-009-9374-1.

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22

GALTIER, JERÔME, FERRAN HURTADO, MARC NOY, STÉPHANE PÉRENNES, and JORGE URRUTIA. "SIMULTANEOUS EDGE FLIPPING IN TRIANGULATIONS." International Journal of Computational Geometry & Applications 13, no. 02 (April 2003): 113–33. http://dx.doi.org/10.1142/s0218195903001098.

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We generalize the operation of flipping an edge in a triangulation to that of flipping several edges simultaneously. Our main result is an optimal upper bound on the number of simultaneous flips that are needed to transform a triangulation into another. Our results hold for triangulations of point sets and for polygons.

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23

Geuenich, Jan, and Daniel Labardini-Fragoso. "Species with Potential Arising from Surfaces with Orbifold Points of Order 2, Part II: Arbitrary Weights." International Mathematics Research Notices 2020, no. 12 (June 12, 2018): 3649–752. http://dx.doi.org/10.1093/imrn/rny090.

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Abstract Let ${\boldsymbol{\Sigma }}=(\Sigma ,\mathbb{M},\mathbb{O})$ be either an unpunctured surface with marked points and order-2 orbifold points or a once-punctured closed surface with order-2 orbifold points. For each pair $(\tau ,\omega )$ consisting of a triangulation $\tau $ of ${\boldsymbol{\Sigma }}$ and a function $\omega :\mathbb{O}\rightarrow \{1,4\}$, we define a chain complex $C_\bullet (\tau , \omega )$ with coefficients in $\mathbb{F}_2=\mathbb{Z}/2\mathbb{Z}$. Given ${\boldsymbol{\Sigma }}$ and $\omega $, we define a colored triangulation of ${\boldsymbol{\Sigma }_\omega }=(\Sigma ,\mathbb{M},\mathbb{O},\omega )$ to be a pair $(\tau ,\xi )$ consisting of a triangulation of ${\boldsymbol{\Sigma }}$ and a 1-cocycle in the cochain complex that is dual to $C_\bullet (\tau , \omega )$; the combinatorial notion of colored flip of colored triangulations is then defined as a refinement of the notion of flip of triangulations. Our main construction associates to each colored triangulation a species and a potential, and our main result shows that colored triangulations related by a flip have species with potentials (SPs) related by the corresponding SP-mutation as defined in [25]. We define the flip graph of ${\boldsymbol{\Sigma }_\omega }$ as the graph whose vertices are the pairs $(\tau ,x)$ consisting of a triangulation $\tau $ and a cohomology class $x\in H^1(C^\bullet (\tau , \omega ))$, with an edge connecting two such pairs, $(\tau ,x)$ and $(\sigma ,z),$ if and only if there exist 1-cocycles $\xi \in x$ and $\zeta \in z$ such that $(\tau ,\xi )$ and $(\sigma ,\zeta )$ are colored triangulations related by a colored flip; then we prove that this flip graph is always disconnected provided the underlying surface $\Sigma $ is not contractible. In the absence of punctures, we show that the Jacobian algebras of the SPs constructed are finite-dimensional and that whenever two colored triangulations have the same underlying triangulation, the Jacobian algebras of their associated SPs are isomorphic if and only if the underlying 1-cocycles have the same cohomology class. We also give a full classification of the nondegenerate SPs one can associate to any given pair $(\tau ,\omega )$ over cyclic Galois extensions with certain roots of unity. The species constructed here are species realizations of the $2^{|\mathbb{O}|}$ skew-symmetrizable matrices that Felikson–Shapiro–Tumarkin associated in [17] to any given triangulation of ${\boldsymbol{\Sigma }}$. In the prequel [25] of this paper we constructed a species realization of only one of these matrices, but therein we allowed the presence of arbitrarily many punctures.

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24

FORTUNE, STEVEN. "NUMERICAL STABILITY OF ALGORITHMS FOR 2D DELAUNAY TRIANGULATIONS." International Journal of Computational Geometry & Applications 05, no. 01n02 (March 1995): 193–213. http://dx.doi.org/10.1142/s0218195995000118.

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We consider the correctness of 2-d Delaunay triangulation algorithms implemented using floating-point arithmetic. The α-pseudocircle through points a, b, c consists of three circular arcs connecting ab, bc, and ac, each arc inside the circumcircle of a, b, c and forming angle α with the circumcircle; a triangulation is α-empty if the α-pseudocircle through the vertices of each triangle is empty. We show that a simple Delaunay triangulation algorithm—the flipping algorithm—can be implemented to produce O(n∈)-empty triangulations, where n is the number of point sites and ∈ is the relative error of floating-point arithmetic; its worst-case running time is O(n2). We also discuss floating-point implementation of other 2-d Delaunay triangulation algorithms.

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25

Smith, Murray. "Triangulation Revisited." Projections 16, no. 1 (March 1, 2022): 11–24. http://dx.doi.org/10.3167/proj.2022.160102.

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What is the relationship between detailed critical analysis and the background assumptions made by a given theory of film spectatorship? In this article, I approach this question by looking at Vittorio Gallese and Michele Guerra’s The Empathic Screen in the light of the method of triangulation—the coordination and integration of phenomenological, psychological, and neuroscientific evidence, as set out in my Film, Art, and the Third Culture. In particular, I examine Gallese and Guerra’s arguments concerning the role of camera movement in prompting immersive, embodied simulation, as well as critiques of these arguments from David Bordwell and Malcolm Turvey. I focus on the special, irreducible role of critical analysis in these arguments. Detailed analysis of film form and style plays an essential role, I argue, in demonstrating the plausibility (or otherwise) of the thesis advanced by Gallese and Guerra. Such analysis is where the rubber of theoretical assumptions meets the road of the material work.

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26

Saudubray, Jérôme. "La triangulation." Humanisme N° 310, no. 1 (February 1, 2016): 92–95. http://dx.doi.org/10.3917/huma.310.0092.

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27

Beisecker, Dave. "Extending Triangulation." Southwest Philosophy Review 22, no. 2 (2006): 87–90. http://dx.doi.org/10.5840/swphilreview200622232.

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28

Gillies, David. "Aerial Triangulation." Photogrammetric Record 30, no. 149 (March 2015): 115–18. http://dx.doi.org/10.1111/phor.12095.

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29

Capon, L. B. "Triangulation adjustment." Australian Surveyor 35, no. 1 (March 1990): 77–86. http://dx.doi.org/10.1080/00050326.1990.10438644.

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30

MURPHY, SHIRLEY A. "Multiple Triangulation." Nursing Research 38, no. 5 (September 1989): 294???297. http://dx.doi.org/10.1097/00006199-198909000-00014.

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31

Worcester, Robert M., and Roger Mortimore. "Political Triangulation." Journal of Political Marketing 4, no. 2-3 (October 13, 2005): 45–72. http://dx.doi.org/10.1300/j199v04n02_03.

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32

Mitcbell, Ellen Sullivan. "Multiple triangulation." Advances in Nursing Science 8, no. 3 (April 1986): 18–26. http://dx.doi.org/10.1097/00012272-198604000-00004.

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33

Denzin, Norman K. "Triangulation 2.0." Journal of Mixed Methods Research 6, no. 2 (February 23, 2012): 80–88. http://dx.doi.org/10.1177/1558689812437186.

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34

Archibald, Mandy M. "Investigator Triangulation." Journal of Mixed Methods Research 10, no. 3 (February 17, 2015): 228–50. http://dx.doi.org/10.1177/1558689815570092.

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35

Saks, Michael J. "Methodological triangulation." Nature Human Behaviour 2, no. 11 (October 29, 2018): 806–7. http://dx.doi.org/10.1038/s41562-018-0458-5.

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36

Castelo, A., L. G. Nonato, M. F. Siqueira, R. Minghim, and G. Tavares. "The triangulation: An adaptive triangulation in any dimension." Computers & Graphics 30, no. 5 (October 2006): 737–53. http://dx.doi.org/10.1016/j.cag.2006.07.025.

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37

Noël, Raphaële, and Francine Cyr. "Comment penser la fonction du père ? Vers une vision systémique de la fonction de triangulation." Filigrane 19, no. 1 (July 26, 2010): 91–106. http://dx.doi.org/10.7202/044182ar.

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La conjugalité des parents impose à l’enfant un travail psychique d’aménagement des enjeux d’exclusion et d’inclusion constituant le paradoxe de la triangulation. Nous mettons en évidence un historique des triangulations : celles-ci se définissant comme contenants pour les contenus que sont les fonctions préoedipiennes et oedipiennes du père, ayant elles aussi un historique. Cela nous conduit à plonger dans la mécanique de la triangulation, au plus près de l’interface intrapsychique/interpersonnel : nous développons le concept d’une transitionnalité de la triangulation comme espace de rencontre entre le tiers interne et le tiers externe. La fonction de triangulation classiquement représentée par le père prend alors une dimension systémique : père, mère et enfant contribuent, chacun et ensemble, à l’émergence de cette fonction. La prise en compte de toutes ces considérations nous conduit vers une modernisation du concept d’Oedipe.

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38

Vivek, Ramakrishnan, Yogarajah Nanthagopan, and Sarmatha Piriyatharshan. "Beyond Methods: Theoretical Underpinnings of Triangulation in Qualitative and Multi-Method Studies”." SEEU Review 18, no. 2 (December 1, 2023): 105–22. http://dx.doi.org/10.2478/seeur-2023-0088.

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Abstract This paper explores the “triangulation approach” in social science research methods. Triangulation is an emerging method specifically in qualitative and multi-method studies. Researchers in the social sciences have increased their use of the triangulation approach recently due to the merits of this approach, especially to conduct in-depth studies through multiple inquiries. Simultaneously, this method has criticism and misconceptions among social researchers in terms of what it is, how it can be used, and the credibility of the findings when used. Therefore, this paper analyses in depth what triangulation is, the types of triangulations and how they can be applied in the social sciences. The study basically adopted systematic literature review and used 30 articles published in scientific journals. The paper critically reviewed journal articles related to triangulation, revealed how different types of triangulation methods have been applied in the social sciences and highlighted the merits and cautions of applying this method. The paper theoretically contributes to understanding the phenomenon of the triangulation approach and how it connects with qualitative and multi-method studies. In addition, it revealed that the following data collection techniques from multiple sources can be exploited: in-depth interviews, focus group discussions, literature, documents, and surveys on the triangulation method to improve the credibility and reliability of the research. Furthermore, the study empirically contributes to social science researchers applying this approach with proper understanding.

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39

Cerulli Irelli, Giovanni, and Daniel Labardini-Fragoso. "Quivers with potentials associated to triangulated surfaces, Part III: tagged triangulations and cluster monomials." Compositio Mathematica 148, no. 6 (November 2012): 1833–66. http://dx.doi.org/10.1112/s0010437x12000528.

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AbstractTo each tagged triangulation of a surface with marked points and non-empty boundary we associate a quiver with potential in such a way that whenever we apply a flip to a tagged triangulation the Jacobian algebra of the quiver with potential (QP) associated to the resulting tagged triangulation is isomorphic to the Jacobian algebra of the QP obtained by mutating the QP of the original one. Furthermore, we show that any two tagged triangulations are related by a sequence of flips compatible with QP-mutation. We also prove that, for each of the QPs constructed, the ideal of the non-completed path algebra generated by the cyclic derivatives is admissible and the corresponding quotient is isomorphic to the Jacobian algebra. These results, which generalize some of the second author’s previous work for ideal triangulations, are then applied to prove properties of cluster monomials, like linear independence, in the cluster algebra associated to the given surface by Fomin, Shapiro and Thurston (with an arbitrary system of coefficients).

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Glen, Marc Elliot. "Colourability and word-representability of near-triangulations." Pure Mathematics and Applications 28, no. 1 (October 1, 2019): 70–76. http://dx.doi.org/10.1515/puma-2015-0032.

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Abstract A graph G = (V;E) is word-representable if there is a word w over the alphabet V such that x and y alternate in w if and only if the edge (x; y) is in G. It is known [6] that all 3-colourable graphs are word-representable, while among those with a higher chromatic number some are word-representable while others are not. There has been some recent research on the word-representability of polyomino triangulations. Akrobotu et al. [1] showed that a triangulation of a convex polyomino is word-representable if and only if it is 3-colourable; and Glen and Kitaev [5] extended this result to the case of a rectangular polyomino triangulation when a single domino tile is allowed. It was shown in [4] that a near-triangulation is 3-colourable if and only if it is internally even. This paper provides a much shorter and more elegant proof of this fact, and also shows that near-triangulations are in fact a generalization of the polyomino triangulations studied in [1] and [5], and so we generalize the results of these two papers, and solve all open problems stated in [5].

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Sun, Peng Jiao, Xiao Man Wang, and Bo Huang. "The Design of Weldment Model Based on Improved Delaunay Triangulation Algorithm." Applied Mechanics and Materials 411-414 (September 2013): 1414–18. http://dx.doi.org/10.4028/www.scientific.net/amm.411-414.1414.

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By using Delaunay triangulation algorithm, triangulating with split-merge algorithm and the incremental insertion algorithm, improving the judgment that if the new inserted point meets the Delaunay triangulation "empty circle" criterion in synthesis algorithm, meanwhile, with the corresponding improvement of the LOP optimization algorithm, the design of weldment model in the virtual welding system has achieved good result.

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Bueno, Lucas Moutinho, and Jorge Stolfi. "4-Colored Triangulation of 3-Maps." International Journal of Computational Geometry & Applications 27, no. 04 (December 2017): 297–325. http://dx.doi.org/10.1142/s0218195917500091.

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We describe an algorithm to triangulate a general 3-dimensional-map on an arbitrary space in such way that the resulting 3-dimensional triangulation is vertex-colorable with four colors. (Four-colorable triangulations can be efficiently represented and manipulated by the GEM data structure of Montagner and Stolfi.) The standard solution to this problem is the barycentric subdivision (BCS) of the map. Our algorithm yields a 4-colored triangulation that is provably smaller than the BCS, and in practice is often a small fraction of its size. When the input map is a shellable triangulation of a 3-ball, in particular, we can prove that the output size is less than [Formula: see text] times the size of the BCS.

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Krtolica, Predrag V., Predrag S. Stanimirović, Sead H. Mašović, Islam A. Elshaarawy, and Alena Stupina. "Convex Polygon Triangulation Based on Symmetry." Symmetry 15, no. 8 (August 2, 2023): 1526. http://dx.doi.org/10.3390/sym15081526.

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A polygon with n nodes can be divided into two subpolygons by an internal diagonal through node n. Splitting the polygon along diagonal δi,n and diagonal δn−i,n, i∈{2,…,⌊n/2⌋} results in mirror images. Obviously, there are ⌊n/2⌋−1 pairs of these reflectively symmetrical images. The influence of the observed symmetry on polygon triangulation is studied. The central result of this research is the construction of an efficient algorithm used for generating convex polygon triangulations in minimal time and without generating repeat triangulations. The proposed algorithm uses the diagonal values of the Catalan triangle to avoid duplicate triangulations with negligible computational costs and provides significant speedups compared to known methods.

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Koehl, Patrice, and Henri Orland. "A Physicist’s View on Partial 3D Shape Matching." Algorithms 16, no. 7 (July 18, 2023): 346. http://dx.doi.org/10.3390/a16070346.

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A new algorithm is presented to compute nonrigid, possibly partial comparisons of shapes defined by unstructured triangulations of their surfaces. The algorithm takes as input a pair of surfaces with each surface given by a distinct and unrelated triangulation. Its goal is to define a possibly partial correspondence between the vertices of the two triangulations, with a cost associated with this correspondence that can serve as a measure of the similarity of the two shapes. To find this correspondence, the vertices in each triangulation are characterized by a signature vector of features. We tested both the LD-SIFT signatures, based on the concept of spin images, and the wave kernel signatures obtained by solving the Shrödinger equation on the triangulation. A cost matrix C is constructed such that C(k,l) is the norm of the difference of the signature vectors of vertices k and l. The correspondence between the triangulations is then computed as the transport plan that solves the optimal transport or optimal partial transport problem between their sets of vertices. We use a statistical physics approach to solve these problems. The presentation of the proposed algorithm is complemented with examples that illustrate its effectiveness and manageable computing cost.

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Zhou Jian-Ming, Shao Ke-Ran, Zhou Ke-Ding, and Zhan Qiong-Hua. "Computing constrained triangulation and Delaunay triangulation: a new algorithm." IEEE Transactions on Magnetics 26, no. 2 (March 1990): 694–97. http://dx.doi.org/10.1109/20.106412.

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BERN, MARSHALL, and DAVID EPPSTEIN. "POLYNOMIAL-SIZE NONOBTUSE TRIANGULATION OF POLYGONS." International Journal of Computational Geometry & Applications 02, no. 03 (September 1992): 241–55. http://dx.doi.org/10.1142/s0218195992000159.

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We describe methods for triangulating polygonal regions of the plane so that no triangle has a large angle. Our main result is that a polygon with n sides can be triangulated with O(n2) nonobtuse triangles. We also show that any triangulation (without Steiner points) of a simple polygon has a refinement with O(n4) nonobtuse triangles. Finally we show that a triangulation whose dual is a path has a refinement with only O(n2) nonobtuse triangles.

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ABELLANAS, MANUEL, MERCÈ CLAVEROL, GREGORIO HERNÁNDEZ, FERRAN HURTADO, VERA SACRISTÁN, MARIA SAUMELL, and RODRIGO I. SILVEIRA. "IMPROVING SHORTEST PATHS IN THE DELAUNAY TRIANGULATION." International Journal of Computational Geometry & Applications 22, no. 06 (December 2012): 559–76. http://dx.doi.org/10.1142/s0218195912500161.

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We study a problem about shortest paths in Delaunay triangulations. Given two nodes s, t in the Delaunay triangulation of a point set S, we look for a new point p ∉ S that can be added, such that the shortest path from s to t, in the Delaunay triangulation of S∪{p}, improves as much as possible. We study several properties of the problem, and give efficient algorithms to find such a point when the graph-distance used is Euclidean and for the link-distance. Several other variations of the problem are also discussed.

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Silva, Luis F., Luiz F. Scheidegger, Tiago Etiene, João L. D. Comba, Luis G. Nonato, and Cláudio T. Silva. "A Weighted Delaunay Triangulation Framework for Merging Triangulations in a Connectivity Oblivious Fashion." Computer Graphics Forum 33, no. 6 (January 29, 2014): 18–30. http://dx.doi.org/10.1111/cgf.12274.

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Wilson, Virginia. "Research Methods: Triangulation." Evidence Based Library and Information Practice 9, no. 1 (March 5, 2014): 74. http://dx.doi.org/10.18438/b8ww3x.

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Wilson, Virginia. "Research Methods: Triangulation." Evidence Based Library and Information Practice 11, no. 1(S) (March 4, 2016): 66. http://dx.doi.org/10.18438/b86s5f.

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