Abstract A surprising connection has recently been made between the amplitudes for Tr(Φ3) theory and the non-linear sigma model (NLSM). A simple shift of kinematic variables naturally suggested by the associahedron/stringy representation of Tr(Φ3) theory yields pion amplitudes at all loops. In this note we provide an elementary motivation and proof for this link going in the opposite direction, starting from the non-linear sigma model and discovering its formulation as a sum over triangulations of surfaces with simple numerator factors. This uses an ancient connection between “circles” and “triangles”, interpreting the equation $$ y=\sqrt{1-{x}^2} $$ y = 1 − x 2 both as parametrizing points a circle as well as generating the number of triangulations of polygons. A further simplification of the numerator factors exposes them as arising from the kinematically shifted Tr(Φ3) theory, and gives rise to novel tropical representations of NLSM amplitudes. The connection to Tr(Φ3) theory defines a natural notion of “surface-soft limit” intrinsic to curves on surfaces. Remarkably, with this definition, the soft limit of pion amplitudes vanishes directly at the level of the integrand, via obvious pairwise cancellations. We also give simple, explicit expressions for the multi-soft factors for tree and loop-level integrands in the limit as any number of pions are taken “surface-soft”.

The rapid evolution of digital technologies has heightened the demand for cryptographic systems that are both secure and adaptable. Catalan number-based cryptography has recently attracted attention for its ability to leverage combinatorial structures in the design of block ciphers and encryption protocols. This review presents a critical analysis of the current state of the field, examining core algorithms such as tweakable ciphers, polygon triangulation schemes, and variable-length block mechanisms from both theoretical and practical perspectives. Particular emphasis is placed on their resistance to advanced cryptanalytic methods and their potential applicability to quantum-resilient security. At the same time, the review identifies key challenges, including efficiency optimization, interoperability with established standards, and the need for systematic benchmarking. By integrating mathematical foundations with implementation-oriented insights, this article highlights the promise of Catalan numbers as a basis for cryptographic innovation while outlining the gaps and future directions necessary for their broader adoption.

An important task of pattern recognition and map generalization is to partition a set of disjoint polygons into groups and to aggregate the polygons within each group to a representative output polygon. We introduce a new method for this task called bicriteria shapes. Following a classical approach, we define the output polygons by merging the input polygons with a set of triangles that we select from a conforming Delaunay triangulation of the input polygons’ exterior. The innovation is that we control the selection of triangles with a bicriteria optimization model that is efficiently solved via graph cuts. In particular, we minimize a weighted sum that combines the total area of the output polygons and their total perimeter. In a basic problem, we ask for a single solution that is optimal for a preset parameter value. In a second problem, we ask for a set containing an optimal solution for every possible value of the parameter. We discuss how this set can be approximated with few solutions and show that it is hierarchically nested. Hence, the output is a hierarchical clustering that corresponds to multiple levels of detail. An evaluation with building footprints as input and a comparison with α-shapes that are based on the same underlying triangulation conclude the article. An advantage of bicriteria shapes compared to α-shapes is that the sequence of solutions for decreasing values of the parameter is monotone with respect to the total perimeter of the output polygons, resulting in a monotonically decreasing visual complexity.

Flip graphs of coloured triangulations of convex polygons

A Linear-Time Greedy Approach With Directional Optimization for Minimum Weight Triangulation of Convex Polygons

Geopositioning is the process of determining or estimating the geographic position of an object through the global positioning system (GPS). The calculations in geopositioning require measurements of distances or angles relative to known reference positions. In Android devices, achieving accuracy, speed, and power efficiency in geopositioning with GPS, cellular networks, and Wi-Fi can be challenging. This research aimed to improve the accuracy of the geopositioning process for cellular networks on Android devices through polygon triangulation using the Graham scan algorithm and determining a moment centroid for the improved estimation of geolocation data. The geolocation data were collected using an Android smartphone with a cellular network and disabled Wi-Fi. A filtering phase on the coordinates was established to obtain the closest distance coordinates from the other. The distances between each pair of coordinates were calculated using the haversine formula, and then the average distance of all pairs was calculated. Then, a polygon was formed by arranging the coordinates in a sequence, which was achieved using the Graham scan algorithm. After obtaining a set of triangles from the polygon triangulation results, the moment centroid of each formed triangle was determined. The centroid, as a result, was compared with another centroid calculation, the Lagrange interpolation polynomial. Based on the results obtained from quantifying the accuracy and precision using average Euclidean error (AEE) and root mean square error (RMSE), the coordinates derived from the moment centroid were more accurate and precise than the Lagrange interpolation polynomial.

On this paper we present a solution to detect and know if a point M is inside a polygon ( A(k), k∈{1,...,n} ) or outside. We are going to give a very simple, practical and explicit method of the triangulation of a convex polygon (convex polyhedron) after a definition and the concretization of the order relation of the points of a polygon in a plane following a well-chosen orientation in before and an arbitrary point of the vertices of the polygon. In the case where the point M is outside the polygon, a simple optimization method will be applied to determine the distance between the point M and the polygon A(1), ..., A(n) and the point P of the border of the polygon closest to M ”The neighboring Point”.

Abstract Conway and Coxeter have shown that frieze patterns over positive rational integers are in bijection with triangulations of polygons. An investigation of frieze patterns over other subsets of the complex numbers has recently been initiated by Jørgensen and the first two authors. In this paper, we first show that a ring of algebraic numbers has finitely many units if and only if it is an order in a quadratic number field where . We conclude that these are exactly the rings of algebraic numbers over which there are finitely many non‐zero frieze patterns for any given height. We then show that apart from the cases all non‐zero frieze patterns over the rings of integers for have only integral entries and hence are known as (twisted) Conway–Coxeter frieze patterns.

We give enumerative interpretations of the polynomials arising as numerators and denominators of the $q$-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.

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.

Uniformly Acute Triangulations of Polygons

Shortcut hulls: Vertex-restricted outer simplifications of polygons

Canopy height estimation using drone-based RGB images

Toric Richardson varieties of Catalan type and Wedderburn–Etherington numbers

A triangulation of a polygon has an associated Stanley–Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals and describe their separated models. More generally, we do this for stacked simplicial complexes, in particular for stacked polytopes.

Abstract With the advancement of the three‐dimensional (3D) modeling techniques in the recent decade, civil infrastructure design (CID) has been used to produce large‐scale 3D infrastructure models that contain a large set of 3D models with a significantly large number of vertices. In the meantime, such large‐scale data necessitates a higher requirement for efficient data processing techniques. Two‐dimensional (2D)‐plane cutting is one of the basic techniques used to observe and analyze 3D models in CID. This paper presents an efficient cutting scheme for a section view of large‐scale 3D infrastructure models in CID. The proposed cutting scheme contains a pipeline of locating the intersected faces, computing the intersection points, and the polygon triangulation for visualization. Furthermore, an optimized data structure is introduced to realize the parallel implementation of the efficiency cutting scheme. After integrating to our existing 3D design and visualization platform, the experimental results with different 3D infrastructure model datasets, such as the large‐scale electrical substation models, have demonstrated both the effectiveness and the efficiency of the proposed cutting scheme.

The famous theorem of Conway and Coxeter on frieze patterns gave a geometric interpretation to integral friezes via triangulations of polygons. In this article, we review this result and show some of the development it has led to. The last decade has seen a lot of activities on friezes. One reason behind this is the connection to cluster combinatorics.

It is well known that Heron’s equality provides an explicit formula for the area of a triangle, as a symmetric function of the lengths of its edges. It has been extended by Brahmagupta to quadrilaterals inscribed in a circle (cyclic quadrilaterals). A natural problem is trying to further generalize the result to cyclic polygons with a larger number of edges. Surprisingly, this has proved to be far from simple, and no explicit solutions exist for cyclic polygons having n>4 edges. In this paper we investigate such a problem by following a new and elementary approach, based on the idea that the simple geometry underlying Heron’s and Brahmagupta’s equalities hides the real players of the game. In details, we propose to focus on the dissection of the edges determined by the incircles of a suitable triangulation of the cyclic polygon, showing that this approach leads to an explicit formula for the area as a symmetric function of the lengths of these segments. We also show that such a symmetry can be rediscovered in Heron’s and Brahmagupta’s results, which consequently represent special cases of the provided general equality.

Flipping in spirals

We prove that it is $\#\mathsf{P}$-complete to count the triangulations of a (non-simple) polygon.