Fermat Point Research Articles

Model for Rapidly Computing the Highest Temperature Based on Fermat Point in Chips

This paper proposes an analytical model for computing the highest temperature based on the Fermat point in chips. The transient temperature is determined through the heat source method based on the Fourier transform. Thermal accumulation at the Fermat point in chips of heat sources with identical powers is proposed based on the thermal path, and the expression of the highest temperature solved here can be applied in multiple hot spots with different powers in chips. The results compared with the finite element simulations show that the proposed model can offer an accurate solution to rapidly compute the highest temperature in the representative chips. The temperature of the Fermat point can be estimated to predict the highest temperature of the chip.

A hybrid recommendation system for Q&A documents

Question and answer (Q&A) documents are a new type of knowledge document composed of a question part and an answer part. The questions represent knowledge needs, and the answers contain the knowledge that meets these knowledge needs. An overload of accumulated Q&A documents decreases the reuse of valuable knowledge. In this paper, we propose a novel hybrid system to recommend Q&A documents to alleviate overload. First, knowledge needs are partitioned, and current knowledge needs are identified by sequentially clustering the Q&A documents. Second, a content-based (CB) recommendation method, a collaborative filtering (CF) recommendation method and a complementarity-based recommendation method are used to find the Q&A documents that are potentially helpful for the user. Third, the three initial recommendation lists of Q&A documents derived from the three recommendation methods are combined to form a more comprehensive recommendation based on the Fermat point. Because reading all Q&A documents in the recommendation list consumes an enormous amount of time and users prefer to read Q&A documents one by one starting from the top, a novel ranking mechanism is proposed to ensure that users obtain comprehensive knowledge to the greatest extent possible from the limited number of Q&A documents at the top of the list. The proposed approach is evaluated and compared based on an experimental dataset. Our experimental results show that the approach is feasible, performs well, and provides a more effective way to recommend Q&A documents.

A New Design Method for PI‐PD Control of Unstable Fractional‐Order System with Time Delay

In this paper, a practical PI‐PD controller parameter tuning method is proposed, which uses the incenter of the triangle and the Fermat point of the convex polygon to optimize the PI‐PD controller. Combined with the stability boundary locus method, the PI‐PD controller parameters that can ensure stability for the unstable fractional‐order system with time delay are obtained. Firstly, the parameters of the inner‐loop PD controller are determined by the centre coordinates of the CSR in the kd − kf plane. Secondly, a new graphical method is used to calculate the parameters of the PI controller, in which Fermat points in the CSR of (kp − ki) plane are selected. Furthermore, the method is extended to uncertain systems, and the PI‐PD controller parameters are obtained by using the proposed method through common stable region of all stable regions. The proposed graphical method not only ensures the stability of the closed‐loop system but also avoids the complicated optimization calculations. The superior control performance of this method is illustrated by simulation.

I-Min: An Intelligent Fermat Point Based Energy Efficient Geographic Packet Forwarding Technique for Wireless Sensor and Ad Hoc Networks

I-Min: An Intelligent Fermat Point Based Energy Efficient Geographic Packet Forwarding Technique for Wireless Sensor and Ad Hoc Networks

I-Min: An Intelligent Fermat Point Based Energy Efficient Geographic Packet Forwarding Technique for Wireless Sensor and Ad Hoc Networks

I-Min: An Intelligent Fermat Point Based Energy Efficient Geographic Packet Forwarding Technique for Wireless Sensor and Ad Hoc Networks

New Wireless Sensor Network Localization Algorithm for Outdoor Adventure

For a long time, it has been frequently reported that outdoor adventurers get lost and die. During the course of an outdoor adventure, it is a matter of life and death to locate the lost adventurer and start the emergency rescue after determining that someone was lost. In this paper, a new wireless sensor network localization algorithm for outdoor adventures is proposed. This algorithm is notably easy to implement, with rather low implementation overhead. In the proposed localization algorithm, first, the Fermat point model is used to divide the original triangular pyramid constructed by the neighbor anchor nodes of the unknown node into four smaller sub-triangular pyramids. Next, the available sub-triangular pyramid in which the unknown node is located can be further divided into a set of irregularly shaped subspaces by the mid-perpendicular plane model. Finally, the centroid of the available subspace can be computed and regarded as the location of the unknown node. Simulation results show that the proposed localization algorithm can remarkably improve the localization accuracy.

Monte Carlo Tree Search Algorithm for the Euclidean Steiner Tree Problem

This study is concerned with a novel Monte Carlo Tree Search algorithm for the problem of minimal Euclidean Steiner tree on a plane. Given p p p points (terminals) on a plane, the goal is to find a connection between all the points, so that the total sum of the lengths of edges is as low as possible, while an addition of extra points (Steiner points) is allowed. Finding the minimum Steiner tree is known to be np-hard. While exact algorithms exist for this problem in 2D, their efficiency decreases when the number of terminals grows. A novel algorithm based on Upper Confidence Bound for Trees is proposed. It is adapted to the specific characteristics of Steiner trees. A simple heuristic for fast generation of feasible solutions based on Fermat points is proposed together with a correction procedure. By combing Monte Carlo Tree Search and the proposed heuristics, the proposed algorithm is shown to work better than both the greedy heuristic and pure Monte Carlo simulations. Results of numerical experiments for randomly generated and benchmark library problems (from OR-Lib) are presented and discussed.

Soap films and GeoGebra in the study of Fermat and Steiner points

ABSTRACTWe discuss how mathematics and secondary mathematics education majors developed an understanding of Fermat points for the triangle as well as Steiner points for the square and regular pentagon, and also of soap film configurations between parallel plates where forces are in equilibrium. The activities included the use of soap films and the interactive geometry program GeoGebra. Students worked in small groups using these tools to investigate the properties of Fermat and Steiner points and then justified the results of their investigations using geometrical arguments. These activities are specific approaches of how to encourage prospective teachers to use physical experiments to support students’ development of mathematical curiosity and mathematical justifications.

Fermat point for a triangle in three dimensions using the taxicab metric

ABSTRACTThis article explores the process of finding the Fermat point for a triangle ABC in three dimensions. Three examples are presented in detail using geometrical methods. A delightfully simple general method is then presented that requires only the comparison of coordinates of the vertices A, B and C.

'페르마 점'을 활용한 중학교 수학 영재 교수·학습 자료 개발 및 적용

'페르마 점'을 활용한 중학교 수학 영재 교수·학습 자료 개발 및 적용

Ordinary and symbolic Rees algebras for ideals of Fermat point configurations

Fermat ideals define planar point configurations that are closely related to the intersection locus of the members of a specific pencil of curves. These ideals have gained recent popularity as counterexamples to some proposed containments between symbolic and ordinary powers [6]. We give a systematic treatment of the family of Fermat ideals, describing explicitly the minimal generators and the minimal free resolutions of all their ordinary powers as well as many symbolic powers. We use these to study the ordinary and the symbolic Rees algebra of Fermat ideals. Specifically, we show that the symbolic Rees algebras of Fermat ideals are Noetherian. Along the way, we give formulas for the Castelnuovo–Mumford regularity of powers of Fermat ideals and we determine their reduction ideals.

Restoration of lost connectivity of partitioned wireless sensor networks

Abstract The lost connectivity due to failure of large-scale nodes plays major role to degrade the system performance by generating unnecessary overhead or sometimes totally collapse the active network. There are many issues and challenges to restore the lost connectivity in an unattended scenario, i.e. how many recovery nodes will be sufficient and on which locations these recovery nodes have to be placed. A very few centralized and distributed approaches have been proposed till now. The centralized approaches are good for a scenario where information about the disjoint network, i.e. number of disjoint segments and their locations are well known in advance. However, for a scenario where such information is unknown due to the unattended harsh environment, a distributed approach is a better solution to restore the partitioned network. In this paper, we have proposed and implemented a semi-distributed approach called Relay node Placement using Fermat Point. The proposed approach is capable of restoring lost connectivity with small number of recovery relay nodes and it works for any number of disjoint segments. The simulation experiment results show effectiveness of our approach as compared to existing benchmark approaches.

Some properties of Kiepert lines of a triangle

This article describes an investigation into Kiepert lines, and leads to some surprising and little-known relationships between the Fermat, Napoleon and Vecten points of a triangle.If we draw similar isosceles triangles A'BC, B'CA and C'AB outwards on the sides of a given scalene triangle ABC as in Figure 1, Kiepert's theorem tells us that the lines A'A, B'B and C'C meet in a single point - a Kiepert point [1, Chapter 11]. Since its position depends on the common base angles θ of the isosceles triangles, I label it K(θ), taking θ as the parameter of this point.

Analytic Approach to Optimal Routing for Commercial Formation Flight

This paper explores an analytic, geometric approach to finding optimal routes for commercial formation flight. A weighted extension of the classical Fermat point problem is used to develop a scalable methodology for the formation routing problem, enabling quick calculation of formation costs. This rapid evaluation allows the large-scale fleet assignment problem to be solved via a mixed integer linear program in reasonable time. Weighting schemes for aircraft performance characteristics are first introduced and then extended to allow for differential rates of fuel burn. Finally, a case study for 210 transatlantic flight routes is presented, with results showing possible average fuel-burn savings against solo flight of around 8.7% for formations of two and 13.1% for formations of up to three.

The Fermat point for a taxicab triangle

The Fermat point P for a triangle ABC is the point P the sum of whose distances from the vertices A, B and C is a minimum. This note will show how to find the Fermat point for any triangle using the taxicab metric.

Fermat's point from five perspectives

The Fermat point of a triangle is the point such that minimizes the sum of the distances from that point to the three vertices. Five approaches to study the Fermat point of a triangle are presented in this article. First, students use a mechanical device using masses, strings and pulleys to study the Fermat point as the one that minimizes the potential energy of the system. Second, students use soap films between parallel planes connecting three pegs. The tension on the film will be minimal when the sum of distances is minimal. Third, students use an empirical approach, measuring distances in an interactive GeoGebra page. Fourth, students use Euclidean geometry arguments for two proofs based on the Torricelli configuration, and one using Viviani's Theorem. And fifth, the kinematic method is used to gain additional insight on the size of the angles between the segments joining the Fermat point with the vertices.

Look the Partial Equilibrium with the Global Optimum from the Problem of the Shortest Circuit for Three Villages

We explore the relationship between the partial equilibrium and the global optimum from the problem of the shortest circuit for three villages (PSCTV). Using the related properties of Fermat point and the circumcenter of the triangle, we can come to the conclusion that the partial equilibrium sometimes brings into correspondence with the global optimum but there is a great deal of difference between the two by analysing the difference between the sum of distances from Fermat point to all vertices and the ones from circumcenter to all vertices.

An $\bm{O(n\log n)}$ Shortest Path Algorithm Based on Delaunay Triangulation

In Euclidean and/or λ-geometry planes with obstacles, the shortest path problem involves determining the shortest path between a source and a destination. There are three different approaches to solve this problem in the Euclidean plane: roadmaps, cell decomposition, and potential field. In the roadmaps approach, a visibility graph is considered to be one of the most widely used methods. In this paper, we present a novel method based on the concepts of Delaunay triangulation, an improved Dijkstra algorithm and the Fermat points to construct a reduced visibility graph that can obtain the near-shortest path in a very short amount of computational time. The length of path obtained using our algorithm is the shortest in comparison to the other fastest algorithms with O(n log n) time complexity. The proposed fast algorithm is especially suitable for those applications which require determining the shortest connectivity between points in the Euclidean plane, such as the robot arm path planning and motion planning for a vehicle.

An Improved Refinement Algorithm of Triangular Mesh Subdivision Based on Minimum Weight Theory

The triangular mesh subdivision to any planar field has been widely adopted in such applicable fields as configurable engineer, computer graphics, and scientific computation visualization and so on because of its well approach to the borderline. Thus, developing and researching on one certain effective and reliable triangular mesh subdivision algorithm has important theoretical and practical meanings. This paper firstly describes a refined algorithm about triangular mesh based on geometrical multi-grid method, and discusses its advantages and disadvantages. Secondly a new refined algorithm about triangular mesh subdivision is put forward by applying Fermat point and its properties as well as the minimum weight theory of triangular meshed subdivision. Finally, this paper proves that this refined algorithm can actually improve the efficiency of triangular mesh subdivision and generate grids amount and quality.

Trilateration-Based Localization Algorithm Using the Lemoine Point Formulation

ABSTRACTThe centroid of three most closely spaced intersections of constant-range loci is conventionally used as trilateration estimate without rigorous justification. In this paper, we address the quality of trilateration intersections through range scaling factors. Several triangle centres, including centroid, incentre, Lemoine point, and Fermat point, are discussed in detail. Lemoine point (LP) is proposed as the best trilateration estimator thanks to the desired property that the total distance to three triangle edges is minimized. It is demonstrated through simulation that LP outperforms centroid localization without additional computational load. In addition, severe trilateration scenarios such as two-intersection cases are considered in this paper, and enhanced trilateration algorithms are proposed.