Search Results (19)

Freeform optical design is regarded as a key approach to overcoming the performance limits of traditional imaging systems. However, the existing Seed Curve Expansion (SCE) algorithm has two major limitations. First, the initial and ideal image points are selected randomly, causing unstable optical performance and low construction accuracy, especially under finite object distance and non-paraxial incidence. Multiple trials are often needed, reducing efficiency and repeatability. Second, the algorithm cannot constrain aperture, focal length, or geometry; thus, despite good imaging quality, the final system parameters often deviate from design requirements, limiting engineering applicability. To address these issues, this work proposes a Stable Initialization Seed Curve Expansion (SI-SCE) algorithm based on ray tracing and Fermat’s principle. The method accurately calibrates the initial point and the ideal image point, eliminating uncertainties caused by randomness. A virtual auxiliary surface strategy is introduced to achieve high-precision freeform construction under finite object distance. In addition, a parameter constraint mechanism is embedded in the algorithm, enabling the designed off-axis multi-reflective freeform system to directly meet specified requirements on pupil diameter, focal length, and geometric size. The feasibility of the SI-SCE algorithm was demonstrated by designing a freeform off-axis three-mirror imaging system with a rectangular $6°\times 6°$ field of view and a moderate F-number. The final system features an F-number of 3.4 and an entrance pupil diameter of 60 mm. It achieves diffraction-limited performance across the Visible–NIR $\left(0.38-2 \mathrm{\mu }\mathrm{m}\right)$ wavelength range. Full article

In 2014, Paul Yiu constructed two equilateral triangles inscribed in a Kiepert hyperbola associated with a reference triangle. It was asserted that each of the equilateral triangles is triply perspective with the reference triangle, and in each case, the corresponding three perspectors are collinear. In this note, we provide proof of his assertions. Furthermore, as an analogue of Lemoine’s problem, we formulated and answered the question of how to recover the reference triangle given a Kiepert hyperbola, one of the two Fermat points and one vertex of the reference triangle. Full article

In recent years, wireless sensor networks have been widely used, especially in three-dimensional environments such as underwater and mountain environments. However, in harsh environments, wireless sensor networks may be damaged and split into many isolated islands. Therefore, restoring network connectivity to transmit data effectively in a timely manner is particularly important. However, the problem of finding the minimum relay nodes is NP-hard, so heuristics methods are preferred. This paper presents a novel connectivity recovery strategy based on boundary nodes and spatial triangle Fermat points for three-dimensional wireless sensor networks. The isolated islands are represented as the boundary nodes, and the connectivity recovery problem is modeled as a graph connectivity problem. Three heuristics algorithms—the variant Kruskal algorithm, the variant Prim algorithm, and the spatial triangle Fermat point algorithm—are proposed to solve this problem. The variant Kruskal algorithm and the variant Prim algorithm connect the isolated islands by constructing the minimum spanning tree to link all the boundary nodes and placing relay nodes along the edges of this tree. We derive an accurate formula to determine the coordinates of spatial triangle Fermat points. Based on this formula, the spatial triangle Fermat point algorithm constructs a Steiner tree to restore network connectivity. Extensive simulation experiments demonstrate that our proposed algorithms perform better than the existing algorithm. Full article

Traditional multi-agent path planning algorithms often lead to path overlap and excessive energy consumption when dealing with cooperative tasks due to the single-agent-single-task configuration. For this reason, the “many-to-one” cooperative planning method has been proposed, which, although improved, still faces challenges in the vast search space for meeting points and unreasonable task handover locations. This paper proposes the Cooperative Dynamic Priority Safe Interval Path Planning with a multi-meeting-point and single-meeting-point solving mode switching (Co-${\mathrm{D}\mathrm{P}\mathrm{S}\mathrm{I}\mathrm{P}\mathrm{P}}_{\mathrm{m}\mathrm{s}}$) algorithm to achieve multi-agent path planning with task handovers at multiple or single meeting points. First, the initial priority is set based on the positional relationships among agents within the cooperative group, and the improved Fermat point method is used to locate multiple meeting points quickly. Second, considering that agents must pick up sub-tasks or conduct task handovers midway, a segmented path planning strategy is proposed to ensure that cooperative agents can efficiently and accurately complete task handovers. Finally, an automatic switching strategy between multi-meeting-point and single-meeting-point solving modes is designed to ensure the algorithm’s success rate. Tests show that Co-${\mathrm{D}\mathrm{P}\mathrm{S}\mathrm{I}\mathrm{P}\mathrm{P}}_{\mathrm{m}\mathrm{s}}$ outperforms existing algorithms in 1-to-1 and m-to-1 cooperative tasks, demonstrating its efficiency and practicality. Full article

Reinterpretation of the Fermat principle governing the propagation of light in media within the Ramsey theory is suggested. Complete bi-colored graphs corresponding to light propagation in media are considered. The vertices of the graphs correspond to the points in real physical space in which the light sources or sensors are placed. Red links in the graphs correspond to the actual optical paths, emerging from the Fermat principle. A variety of optical events, such as refraction and reflection, may be involved in light propagation. Green links, in turn, denote the trial/virtual optical paths, which actually do not occur. The Ramsey theorem states that within the graph containing six points, inevitably, the actual or virtual optical cycle will be present. The implementation of the Ramsey theorem with regard to light propagation in metamaterials is discussed. The Fermat principle states that in metamaterials, a light ray, in going from point S to point P, must traverse an optical path length L that is stationary with respect to variations of this path. Thus, bi-colored graphs consisting of links corresponding to maxima or minima of the optical paths become possible. The graphs, comprising six vertices, will inevitably demonstrate optical cycles consisting of the mono-colored links corresponding to the maxima or minima of the optical path. The notion of the “inverse graph” is introduced and discussed. The total number of triangles in the “direct” (source) and “inverse” Ramsey optical graphs is the same. The applications of “Ramsey optics” are discussed, and an optical interpretation of the infinite Ramsey theorem is suggested. Full article

In a wireless sensor network (WSN), geocasting is a location-based routing protocol used for data collection or information delivery. In geocasting, a target region usually contains many sensor nodes with limited battery capacity, and sensor nodes in multiple target regions need to transmit data to the sink. Therefore, how to use location information to construct an energy efficient geocasting path is a very important issue. FERMA is a geocasting scheme for WSNs based on Fermat points. In this paper, an efficient grid-based geocasting scheme for WSNs, which is called GB-FERMA, is proposed. The scheme uses the Fermat point theorem to search for the specific nodes as Fermat points in a grid-based WSN, and it selects the optimal relay nodes (gateways) in the grid structure to realize energy-aware forwarding. In the simulations, when the initial power 0.25 J, the average energy consumption of GB-FERMA is about 53% of FERMA-QL, 37% of FERMA, and 23% of GEAR; however, when with the initial power 0.5 J, the average energy consumption of GB-FERMA is about 77% of FERMA-QL, 65% of FERMA, and 43% of GEAR. The proposed GB-FERMA can effectively reduce the energy consumption and thus prolong the lifetime of the WSN. Full article

A data point calculation method that does not require the use of Fermat′s principle and a simple and general design method of starting points of freeform off-axis multi-mirror optical systems are proposed in this paper, which aim to promote the realization of high-performance reflective systems containing freeform surfaces. Taking a planar system and the required parameters as the input, a good starting point for a freeform off-axis multi-mirror system can be automatically obtained using the proposed method. The design of a freeform off-axis five-mirror system with a low F-number is taken as an example to show the effectiveness of the proposed method. The method can also be used for the design of freeform reflective systems with other numbers of mirrors. Full article

Solving polynomial equations inevitably faces many severe challenges, such as easily occupying storage space and demanding prohibitively expensive computation resources. There has been considerable interest in exploiting the sparsity to improve computation efficiency, since asymmetry phenomena are prevalent in scientific and engineering fields, especially as most of the systems in real applications have sparse representations. In this paper, we propose an efficient parallel hybrid algorithm for constructing a Dixon matrix. This approach takes advantage of the asymmetry (i.e., sparsity) in variables of the system and introduces a heuristics strategy. Our method supports parallel computation and has been implemented on a multi-core system. Through time-complexity analysis and extensive benchmarks, we show our new algorithm has significantly reduced computation and memory overhead. In addition, performance evaluation via the Fermat–Torricelli point problem demonstrates its effectiveness in combinatorial geometry optimizations. Full article

This paper is motivated by the difference between the classical principal component analysis (PCA) in a Euclidean space and the tropical PCA in a tropical projective torus as follows. In Euclidean space, the projection of the mean point of a given data set on the principle component is the mean point of the projection of the data set. However, in tropical projective torus, it is not guaranteed that the projection of a Fermat-Weber point of a given data set on a tropical polytope is a Fermat-Weber point of the projection of the data set. This is caused by the difference between the Euclidean metric and the tropical metric. In this paper, we focus on the projection on the tropical triangle (the three-point tropical convex hull), and we develop one algorithm and its improved version, such that for a given data set in the tropical projective torus, these algorithms output a tropical triangle, on which the projection of a Fermat-Weber point of the data set is a Fermat-Weber point of the projection of the data set. We implement these algorithms in $\mathtt{R}$ language and test how they work with random data sets. We also use $\mathtt{R}$ language for numerical computation. The experimental results show that these algorithms are stable and efficient, with a high success rate. Full article

In this paper we present a survey of Fermat metrics and their applications to stationary spacetimes. A Fermat principle for light rays is stated in this class of spacetimes and we present a variational theory for the light rays and a description of the multiple image effect. Some results on variational methods, as Ljusternik-Schnirelmann and Morse Theory are recalled, to give a description of the variational methods used. Other applications of the Fermat metrics concern the global hyperbolicity and the geodesic connectedeness and a characterization of the Sagnac effect in a stationary spacetime. Finally some possible applications to other class of spacetimes are considered. Full article

Cybersecurity is a critical issue for Real-Time IoT applications since high performance and low latencies are required, along with security requirements to protect the large number of attack surfaces to which IoT devices are exposed. Elliptic Curve Cryptography (ECC) is largely adopted in an IoT context to provide security services such as key-exchange and digital signature. For Real-Time IoT applications, hardware acceleration for ECC-based algorithms can be mandatory to meet low-latency and low-power/energy requirements. In this paper, we propose a fast and configurable hardware accelerator for NIST P-256/-521 elliptic curves, developed in the context of the European Processor Initiative. The proposed architecture supports the most used cryptography schemes based on ECC such as Elliptic Curve Digital Signature Algorithm (ECDSA), Elliptic Curve Integrated Encryption Scheme (ECIES), Elliptic Curve Diffie-Hellman (ECDH) and Elliptic Curve Menezes-Qu-Vanstone (ECMQV). A modified version of Double-And-Add-Always algorithm for Point Multiplication has been proposed, which allows the execution of Point Addition and Doubling operations concurrently and implements countermeasures against power and timing attacks. A simulated approach to extract power traces has been used to assess the effectiveness of the proposed algorithm compared to classical algorithms for Point Multiplication. A constant-time version of the Shamir’s Trick has been adopted to speed-up the Double-Point Multiplication and modular inversion is executed using Fermat’s Little Theorem, reusing the internal modular multipliers. The accelerator has been verified on a Xilinx ZCU106 development board and synthesized on both 45 nm and 7 nm Standard-Cell technologies. Full article

For the law enforcement agencies, lawful interception is still one of the main means to intercept a suspect or address most illegal actions. Due to its centralized management, however, it is easy to implement in traditional networks, but the cost is high. In view of this restriction, this paper aims to exploit software-defined network (SDN) technology to contribute to the next generation of intelligent lawful interception technology, i.e., to optimize the deployment of intercept access points (IAPs) in hybrid software-defined networks where both SDN nodes and non-SDN nodes exist simultaneously. In order to deploy IAPs, this paper puts forward an improved equal-cost multi-path shortest path algorithm and accordingly proposes three SDN interception models: T interception model, ECMP-T interception model and Fermat-point interception model. Considering the location relevance of all intercepted targets and the operation and maintenance cost of operators from the global perspective, by the way, we further propose a restrictive minimum vertex cover algorithm (RMVCA) in hybrid SDN. Implementing different SDN interception algorithms based RMVCA in real-world topologies, we can reasonably deploy the best intercept access point and intercept the whole hybrid SDN with the least SDN nodes, as well as significantly optimize the deployment efficiency of IAPs and improve the intercept link coverage in hybrid SDN, contributing to the implementation of lawful interception. Full article

A 5G metasurface (MS) transmitarray (TA) feed by compact-antenna array with the performance of high gain and side-lobe level (SLL) reduction is presented. The proposed MS has two identical metallic layers etched on both sides of the dielectric substrate and four fixed vias connecting two metallic layers that works at 28 GHz to increase the transmission phase shift range. The proposed planar TA consisting of unit cells with different dimensional information can simulate the function as an optical lens according to the Fermat’s principle, so the quasi-spherical wave emitted by the compact Potter horn antenna at the virtual focal point will transform to the quasi-plane wave by the phase-adjustments. Then, the particle swarm optimization (PSO) is introduced to optimize the phase distribution on the TA to decrease the SLL further. It is found that the optimized TA could achieve 27 dB gain at 28 GHz, 11.8% 3 dB gain bandwidth, −30 dB SLL, and aperture efficiency of 23% at the operating bandwidth of 27.5–29.5 GHz, which performs better than the nonoptimized one. The advanced particularities of this optimized TA including low cost, low profile, and easy to configure make it great potential in paving the way to 5G communication and radar system. Full article

High-precision ultrasound imaging of void defects is critical for the performance and safety assessment of ballastless track structures. The sound propagation velocity of each layer in the ballastless track structure is quite different. However, the traditional concrete Synthetic Aperture Focusing Technique (SAFT) ultrasound imaging method is based on the assumption that the concrete has a single constant shear wave velocity. Thus, it is not a suitable method for the ultrasonic imaging of multilayer structures. In this paper, a Multilayer SAFT high-precision ultrasound imaging method is proposed. It is based on the ray-tracing technique and uses the Fermat principle to find the refraction point that minimizes the delay of the acoustic wave propagation path at the interface of the discrete layers. Then, the acoustic wave propagation path is segmented by the position of the refraction point, and the propagation delay of the ultrasonic wave is obtained segment by segment. Thus, the propagation delay of the ultrasonic wave is obtained one by one, so that the propagation delay of the ultrasonic wave in the multilayer structure can be accurately obtained. Finally, the focused image is obtained according to the SAFT imaging algorithm. The finite element simulation and experimental results show that the Multilayer SAFT imaging method can accurately track the propagation path of the ultrasonic wave in ballastless track structures, as well as accurately calculate the propagation delay of the ultrasonic wave and the lengths of void defects. The high accuracy of the Multilayer SAFT imaging represents a significant improvement compared to traditional SAFT imaging. Full article