Search results for: paper airplane problem

Authors: T. Mrna, R. Doubrava

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This paper describes airworthiness requirements with respect damage tolerance. Damage tolerance determines the amount and magnitude of damage on parts of the airplane. Airworthiness requirements determine the amount of damage that can still be in flight capable of the condition. Component damage can be defined as barely visible impact damage, visible impact damage or clear visible impact damage. Damage is also distributed it according to the velocity. It is divided into low or high velocity impact damage. The severity of damage to the part of airplane divides the airworthiness requirements into several categories according to severity. Airworthiness requirements are determined by type airplane. All types of airplane do not have the same conditions for airworthiness requirements. This knowledge is important for designing and operating an airplane.

Keywords: airworthiness requirements, composite, damage tolerance, low and high velocity impact

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Authors: Neslihan Şahin Çelik, Ali Eraslan

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In recent years, mathematics educators have frequently stressed the necessity of instructing students about models and modeling approaches that encompass cognitive and metacognitive thought processes, starting from the first years of school and continuing on through the years of higher education. The purpose of this study is to examine the thought processes of 4th-grade primary school students in their modeling activities and to explore the difficulties encountered in these processes, if any. The study, of qualitative design, was conducted in the 2015-2016 academic year at a public state-school located in a central city in the Black Sea Region of Turkey. A preliminary study was first implemented with designated 4th grade students, after which the criterion sampling method was used to select three students that would be recruited into the focus group. The focus group that was thus formed was asked to work on the model eliciting activity of the Paper Airplane Problem and the entire process was recorded on video. The Paper Airplane Problem required the students to determine the winner with respect to: (a) the plane that stays in the air for the longest time; (b) the plane that travels the greatest distance in a straight-line path; and (c) the overall winner for the contest. A written transcript was made of the video recording, after which the recording and the students' worksheets were analyzed using the Blum and Ferri modeling cycle. The results of the study revealed that the students tested the hypotheses related to daily life that they had set up, generated ideas of their own, verified their models by making connections with real life, and tried to make their models generalizable. On the other hand, the students had some difficulties in terms of their interpretation of the table of data and their ways of operating on the data during the modeling processes.

Keywords: primary school students, model eliciting activity, mathematical modeling, modeling process, paper airplane problem

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Authors: Huang Bingling, Jia Yuhong, Liu Yanhui

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As an important part of aircraft, landing gears are responsible for taking-off and landing function. In recent years, big airplane's structural quality increases a lot. As a result, landing gears have stricter technical requirements than ever before such as structure strength and etc. If the structural strength of the landing gear is enhanced through traditional methods like increasing structural quality, the negative impacts on the landing gear's function would be very serious and even counteract the positive effects. Thus, in order to solve this problem, the impact of pre-rotating of landing gears on performance of landing gears is studied from the theoretical and experimental verification in this paper. By increasing the pre-rotating speed of the wheel, it can improve the performance of the landing gear and reduce the structural quality, the force of joint parts and other properties. In addition, the pre-rotating of the wheels also has other advantages, such as reduce the friction between wheels and ground and extend the life of the wheel. In this paper, the impact of the pre-rotating speed on landing gears and the connecting between landing gears performance and pre-rotating speed would be researched in detail. This paper is divided into three parts. In the first part, large airplane landing gear model is built by CATIA and LMS. As most general landing gear type in big plane, four-wheel landing gear is picked as model. The second part is to simulate the process of landing in LMS motion, and study the impact of pre-rotating of wheels on the aircraft`s properties, including the buffer stroke, efficiency, power; friction, displacement and relative speed between piston and sleeve; force and load distribution of tires. The simulation results show that the characteristics of the different pre-rotation speed are understood. The third part is conclusion. Through the data of the previous simulation and the relationship between the pre-rotation speed of the aircraft wheels and the performance of the aircraft, recommended speed interval is proposed. This paper is of great theoretical value to improve the performance of large airplane. It is a very effective method to improve the performance of aircraft by setting wheel pre-rotating speed. Do not need to increase the structural quality too much, eliminating the negative effects of traditional methods.

Keywords: large airplane, landing gear, pre-rotating, simulation

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Authors: Birs Isabela, Muresan Cristina, Folea Silviu, Prodan Ovidiu

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The wing is one of the most important parts of an airplane because it ensures stability, sustenance and maneuverability of the airplane. Because of its shape, the airplane wing can be simplified to a smart beam. Active vibration suppression is realized using piezoelectric actuators that are mounted on the surface of the beam. This work presents a tuning procedure of fractional order controllers based on a graphical approach of the frequency domain representation. The efficacy of the method is proven by practically testing the controller on a laboratory scale experimental stand.

Keywords: fractional order control, piezoelectric actuators, smart beam, vibration suppression

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Authors: Djendi Mohmaed

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In this paper, we propose a dual adaptive algorithm, which is based on the combination between the forward blind source separation (FBSS) structure and the affine projection algorithm (APA). This proposed algorithm combines the advantages of the source separation properties of the FBSS structure and the fast convergence characteristics of the APA algorithm. The proposed algorithm needs two noisy observations to provide an enhanced speech signal. This process is done in a blind manner without the need for ant priori information about the source signals. The proposed dual forward blind source separation affine projection algorithm is denoted (DFAPA) and used for the first time in an airplane cockpit context to enhance the communication from- and to- the airplane. Intensive experiments were carried out in this sense to evaluate the performance of the proposed DFAPA algorithm.

Keywords: adaptive algorithm, speech enhancement, system mismatch, SNR

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Authors: M. Rutkowski, J. Cwiklak, M. Grzegorzewski, M. Adamski

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All class A Airplanes have to be equipped with Mod. S transponder for ATC surveillance purposes. This technology was designed to provide a robust and dependable solution to localize, identify and exchange data with the airplane. The purpose of this paper is to analyze potential hazards that are a result of lack of any security or encryption on a design level. Secondary Surveillance Radars rely on an active response from an airplane. SSR radar installation is broadcasting a directional interrogation signal to the planes in range on 1030MHz frequency with DPSK modulation. If the interrogation is correctly received by the transponder located on the plane, a proper answer is sent on 1090MHz with PPM modulation containing plane’s SQUAWK, barometric altitude, GPS coordinates and 24bit unique address code. This technology does not use any kind of encryption. All of the specifications from the previous chapter can be found easily on the internet. Since there is no encryption or security measure to ensure the credibility of the sender and message, it is highly hazardous to use such technology to ensure the safety of the air traffic. The only thing that identifies the airplane is the 24-bit unique address. Most of the planes have been sniffed by aviation enthusiasts and cataloged in web databases. In the moment of writing this article, The PoFung Technologies has announced that they are planning to release all band SDR transceiver – this device would be more than enough to build your own Mod. S Transponder. With fake transponder, a potential terrorist can identify as a different airplane. By replacing the transponder in a poorly controlled airspace, hijackers can enter another airspace identifying themselves as another plane and land in the desired area.

Keywords: flight safety, hijack, mod S transponder, security analysis

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Authors: Reza Hedayati, Meysam Jahanbakhshi

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The typical airplane stabilizer structures consist of two main similar segments (outer and inner parts), one of them a little larger than the other. In this study, bird impact on four different spots of the stabilizer structure: (a) between two ribs of smaller segment, (b) between two ribs of larger segment, (c) on the rib connecting the two segments, and (d) on a middle rib of the smaller segment, is investigated and their results are compared by means of energy absorption, displacement, and bird’s mass diagrams as well as visible damage induced on the stabilizer structure.

Keywords: airplane, bird strike, LS-DYNA, stabilizer

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Authors: Feri̇t Çakici, M. Kemal Leblebi̇ci̇oğlu

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In this study, an unmanned aerial vehicle (UAV) with level flight, vertical take-off and landing (VTOL) and mode-transitioning capability is designed and analyzed. The platform design combines both multirotor and fixed-wing (FW) conventional airplane structures and control surfaces; therefore named as VTOL-FW. The aircraft is modeled using aerodynamical principles and linear models are constructed utilizing small perturbation theory for trim conditions. The proposed method of control includes implementation of multirotor and airplane mode controllers and design of an algorithm to transition between modes in achieving smooth switching maneuvers between VTOL and FW flight. Thus, VTOL-FW UAV’s flight characteristics are expected to be improved by enlarging operational flight envelope through enabling mode-transitioning, agile maneuvers and increasing survivability. Experiments conducted in simulation and real world environments shows that VTOL-FW UAV has both multirotor and airplane characteristics with extra benefits in an enlarged flight envelope.

Keywords: aircraft design, linear analysis, mode transitioning control, UAV

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Authors: Reza Rezaeipour Honarmandzad

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Wavelet analysis is an exciting method for solving difficult problems in mathematics, physics, and engineering, with modern applications as diverse as wave propagation, data compression, signal processing, image processing, pattern recognition, etc. Wavelets allow complex information such as signals, images and patterns to be decomposed into elementary forms at different positions and scales and subsequently reconstructed with high precision. In this paper a wavelet-based signal processing algorithm for airplane cable fault location is proposed. An orthogonal discrete wavelet decomposition and reconstruction algorithm is used to eliminate the noise in the aircraft cable fault signal. The experiment result has shown that the character of emission pulse and reflect pulse used to test the aircraft cable fault point are reserved and the high-frequency noise are eliminated by means of the proposed algorithm in this paper.

Keywords: wavelet analysis, signal processing, orthogonal discrete wavelet, noise, aircraft cable fault signal

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Authors: Marine Segui, Simon Bezin, Ruxandra Mihaela Botez

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As part of a ‘Morphing-Wing’ idea, this study consists of measuring how a winglet, which is able to change its shape during the flight, is efficient. Conventionally, winglets are fixed-vertical platforms at the wingtips, optimized for a cruise condition that the airplane should use most of the time. However, during a cruise, an airplane flies through a lot of cruise conditions corresponding to altitudes variations from 30,000 to 45,000 ft. The fixed winglets are not optimized for these variations, and consequently, they are supposed to generate some drag, and thus to deteriorate aircraft fuel consumption. This research assumes that it exists a winglet position that reduces the fuel consumption for each cruise condition. In this way, the methodology aims to find these optimal winglet positions, and to further simulate, and thus estimate the fuel consumption of an aircraft wearing this type of adaptive winglet during several cruise conditions. The adaptive winglet is assumed to have degrees of freedom given by the various changes of following surfaces: the tip chord, the sweep and the dihedral angles. Finally, results obtained during cruise simulations are presented in this paper. These results show that an adaptive winglet can reduce, thus improve up to 2.12% the fuel consumption of an aircraft during a cruise.

Keywords: aerodynamic, Cessna, Citation X, optimization, winglet

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Authors: S. Aditya, K. T. Nideesh, N. Guruprasad

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Being a subclass of linear programing problem, the Transportation Problem is a classic Operations Research problem where the objective is to determine the schedule for transporting goods from source to destination in a way that minimizes the shipping cost while satisfying supply and demand constraints. In this paper, we are representing the transportation problem for various warehouses along with various dealers situated in Bangalore city to reduce the transportation cost incurred by them as of now. The problem is solved by obtaining the Initial Basic feasible Solution through various methods and further proceeding to obtain optimal cost.

Keywords: NW method, optimum utilization, transportation problem, Vogel’s approximation method

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Authors: Mohammad Farid

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In this paper, we introduce and study an explicit iterative method based on hybrid extragradient method to approximate a common solution of generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup in Hilbert space. Further, we prove that the sequence generated by the proposed iterative scheme converge strongly to the common solution of generalized mixed equilibrium problem and fixed point problem for a nonexpansive semigroup. This common solution is the unique solution of a variational inequality problem and is the optimality condition for a minimization problem. The results presented in this paper are the supplement, extension and generalization of the previously known results in this area.

Keywords: generalized mixed equilibrium problem, fixed-point problem, nonexpansive semigroup, variational inequality problem, iterative algorithms, hybrid extragradient method

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Authors: Terence Soule, Tami Al Ghamdi

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To study how the partial knowledge transfer may affect the Genetic Algorithm (GA) performance, we model the Transfer Learning (TL) process using GA as the model solver. The objective of the TL is to transfer the knowledge from one problem to another related problem. This process imitates how humans think in their daily life. In this paper, we proposed to study a case where the knowledge transferred from the S problem has less information than what the T problem needs. We sampled the transferred population using different strategies of TL. The results showed transfer part of the knowledge is helpful and speeds the GA process of finding a solution to the problem.

Keywords: transfer learning, partial transfer, evolutionary computation, genetic algorithm

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Authors: Keng Keh Lim, Zaleha Ismail, Yudariah Mohammad Yusof

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This paper aims to find out how students using convergent and divergent thinking in creative problem solving to solve mathematical problems creatively. Eight engineering undergraduates in a local university took part in this study. They were divided into two groups. They solved the mathematical problems with the use of creative problem solving skills. Their solutions were collected and analyzed to reveal all the processes of problem solving, namely: problem definition, ideas generation, ideas evaluation, ideas judgment, and solution implementation. The result showed that the students were able to solve the mathematical problem with the use of creative problem solving skills.

Keywords: convergent thinking, divergent thinking, creative problem solving, creativity

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Authors: Ismail El Gayar, Ossama Ismail, Yousri El Gamal

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This paper presented a technique to solve one of the transportation problems that faces us in real life which is the Bus Scheduling Problem. Most of the countries using buses in schools, companies and traveling offices as an example to transfer multiple passengers from many places to specific place and vice versa. This transferring process can cost time and money, so we build a decision support system that can solve this problem. In this paper, a genetic algorithm with the shortest path technique is used to generate a competitive solution to other well-known techniques. It also presents a comparison between our solution and other solutions for this problem.

Keywords: bus scheduling problem, decision support system, genetic algorithm, shortest path

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Authors: Mrinal Jana, Geetanjali Panda

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In this paper a methodology is developed to solve a nonlinear fractional programming problem in which the coefficients of the objective function and constraints are interval parameters. This model is transformed into a general optimization problem and relation between the original problem and the transformed problem is established. Finally the proposed methodology is illustrated through a numerical example.

Keywords: fractional programming, interval valued function, interval inequalities, partial order relation

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Authors: Deyadeen Ali Alshibani

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In the literature, the multi-armed bandit problem as a statistical decision model of an agent trying to optimize his decisions while improving his information at the same time. There are several different algorithms models and their applications on this problem. In this paper, we evaluate the Regret-regression through comparing with Q-learning method. A simulation on determination of optimal treatment regime is presented in detail.

Keywords: optimal, bandit problem, optimization, dynamic programming

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Authors: Sukhveer Singh, Sandeep Singh

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A bi-objective fuzzy transportation problem with the objectives to minimize the total fuzzy cost and fuzzy time of transportation without according priorities to them is considered. To the best of our knowledge, there is no method in the literature to find efficient solutions of the bi-objective transportation problem under uncertainty. In this paper, a bi-objective transportation problem in an uncertain environment has been formulated. An algorithm has been proposed to find efficient solutions of the bi-objective transportation problem under uncertainty. The proposed algorithm avoids the degeneracy and gives the optimal solution faster than other existing algorithms for the given uncertain transportation problem.

Keywords: uncertain transportation problem, efficient solution, ranking function, fuzzy transportation problem

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Authors: Ravneil Nand

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Cooperative coevolution uses problem decomposition methods to solve a larger problem. The problem decomposition deals with breaking down the larger problem into a number of smaller sub-problems depending on their method. Different problem decomposition methods have their own strengths and limitations depending on the neural network used and application problem. In this paper we are introducing a new problem decomposition method known as Hidden-Neuron Level Decomposition (HNL). The HNL method is competing with established problem decomposition method in time series prediction. The results show that the proposed approach has improved the results in some benchmark data sets when compared to the standalone method and has competitive results when compared to methods from literature.

Keywords: cooperative coevaluation, feed forward network, problem decomposition, neuron, synapse

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Authors: Ebrahim H. Kapeel, Ehab Safwat, Hossam Hendy, Ahmed M. Kamel, Yehia Z. Elhalwagy

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Motivated by the problem of the availability of high-fidelity flight simulation models for small unmanned aerial vehicles (UAVs). This paper focuses on the geometric-mass inertia modeling and the actuation system modeling for the small fixed-wing UAVs. The UAV geometric parameters for the body, wing, horizontal and vertical tail are physically measured. Pendulum experiment with high-grade sensors and data analysis using MATLAB is used to estimate the airplane moment of inertia (MOI) model. Finally, UAV’s actuation system is modeled by estimating each servo transfer function by using the system identification, which uses experimental measurement for input and output angles through using field-programmable gate array (FPGA). Experimental results for the designed models are given to illustrate the effectiveness of the methodology. It also gives a very promising result to finalize the open-loop flight simulation model through modeling the propulsion system and the aerodynamic system.

Keywords: unmanned aerial vehicle, geometric-mass inertia model, system identification, Simulink

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Authors: Chaghoub Soraya, Zhang Xiaoyan

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This research presents the first constant approximation algorithm to the p-median network design problem with multiple cable types. This problem was addressed with a single cable type and there is a bifactor approximation algorithm for the problem. To the best of our knowledge, the algorithm proposed in this paper is the first constant approximation algorithm for the p-median network design with multiple cable types. The addressed problem is a combination of two well studied problems which are p-median problem and network design problem. The introduced algorithm is a random sampling approximation algorithm of constant factor which is conceived by using some random sampling techniques form the literature. It is based on a redistribution Lemma from the literature and a steiner tree problem as a subproblem. This algorithm is simple, and it relies on the notions of random sampling and probability. The proposed approach gives an approximation solution with one constant ratio without violating any of the constraints, in contrast to the one proposed in the literature. This paper provides a (21 + 2)-approximation algorithm for the p-median network design problem with multiple cable types using random sampling techniques.

Keywords: approximation algorithms, buy-at-bulk, combinatorial optimization, network design, p-median

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Authors: Sanam Haseen, Abdul Bari

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In this paper a three stage multivariate programming problem with random survey cost and variances as random variables has been formulated as a non-linear stochastic programming problem. The problem has been converted into an equivalent deterministic form using chance constraint programming and modified E-modeling. An empirical study of the problem has been done at the end of the paper using R-simulation.

Keywords: chance constraint programming, modified E-model, stochastic programming, stratified sample surveys, three stage sample surveys

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Authors: Mehmet Hakan Karaata

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In this paper, we first coin a new graph theocratic problem with numerous applications. Second, we provide two algorithms for the problem. The first solution is using a brute-force techniques, whereas the second solution is based on an initial identification of the cycles in the given graph. We then provide a correctness proof of the algorithm. The applications of the problem include graph analysis, graph drawing and network structuring.

Keywords: algorithm, cycle, graph algorithm, graph theory, network structuring

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Authors: Sujeet Kumar Singh, Shiv Prasad Yadav

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This paper develops an approach for solving intuitionistic fuzzy linear fractional programming (IFLFP) problem where the cost of the objective function, the resources, and the technological coefficients are triangular intuitionistic fuzzy numbers. Here, the IFLFP problem is transformed into an equivalent crisp multi-objective linear fractional programming (MOLFP) problem. By using fuzzy mathematical programming approach the transformed MOLFP problem is reduced into a single objective linear programming (LP) problem. The proposed procedure is illustrated through a numerical example.

Keywords: triangular intuitionistic fuzzy number, linear programming problem, multi objective linear programming problem, fuzzy mathematical programming, membership function

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Authors: Sourabh Joshi, Tarun Sharma, Anurag Sharma

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Ant Colony Optimization is heuristic Algorithm which has been proven a successful technique applied on number of combinatorial optimization problems. Two variants of Ant Colony Optimization algorithm named Ant System and Max-Min Ant System are implemented in MATLAB to solve travelling Salesman Problem and the results are compared. In, this paper both systems are analyzed by solving the some Travelling Salesman Problem and depict which system solve the problem better in term of cost and time.

Keywords: Ant Colony Optimization, Travelling Salesman Problem, Ant System, Max-Min Ant System

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Authors: Boris Melnikov, Dmitrii Chaikovskii, Elena Melnikova

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The traveling salesman problem (TSP) is a well-known optimization problem that seeks to find the shortest possible route that visits a set of points and returns to the starting point. In this paper, we apply some heuristics of the TSP for the task of restoring the DNA matrix. This restoration problem is often considered in biocybernetics. For it, we must recover the matrix of distances between DNA sequences if not all the elements of the matrix under consideration are known at the input. We consider the possibility of using this method in the testing of distance calculation algorithms between a pair of DNAs to restore the partially filled matrix.

Keywords: optimization problems, DNA matrix, partially filled matrix, traveling salesman problem, heuristic algorithms

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Authors: Ivan Baravy

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Interpolation problem, as it was initially posed in terms of polynomials, is well researched. However, further mathematical developments extended it significantly. Trigonometric interpolation is widely used in Fourier analysis, while its generalized representation as exponential interpolation is applicable to such problem of mathematical physics as modelling of Ziegler-Biersack-Littmark repulsive interatomic potentials. Formulated for finite fields, this problem arises in decoding Reed--Solomon codes. This paper shows the relation between different interpretations of the problem through the class of matrices of special structure - Hankel matrices.

Keywords: Berlekamp-Massey algorithm, exponential interpolation, finite fields, Hankel matrices, Hankel polynomials

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Authors: Divyesh Patel, Tanuja Srivastava

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This paper considers the NP-hard problem of reconstructing binary matrices satisfying exactly-1-4-adjacency constraint from its row and column projections. This problem is formulated into a maximization problem. The objective function gives a measure of adjacency constraint for the binary matrices. The maximization problem is solved by the simulated annealing algorithm and experimental results are presented.

Keywords: discrete tomography, exactly-1-4-adjacency, simulated annealing, binary matrices

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Authors: Bruce Wrightsman

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Construction and design are inexorably linked. Traditional building methodologies, including those using wood, comprise a series of material layers differentiated and separated from each other. This results in the separation of two agencies of building envelope (skin) separate from the structure. However, from a material performance position reliant on additional materials, this is not an efficient strategy for the building. The merits of traditional platform framing are well known. However, its enormous effectiveness within wood-framed construction has seldom led to serious questioning and challenges in defining what it means to build. There are several downsides of using this method, which is less widely discussed. The first and perhaps biggest downside is waste. Second, its reliance on wood assemblies forming walls, floors and roofs conventionally nailed together through simple plate surfaces is structurally inefficient. It requires additional material through plates, blocking, nailers, etc., for stability that only adds to the material waste. In contrast, when we look back at the history of wood construction in airplane and boat manufacturing industries, we will see a significant transformation in the relationship of structure with skin. The history of boat construction transformed from indigenous wood practices of birch bark canoes to copper sheathing over wood to improve performance in the late 18th century and the evolution of merged assemblies that drives the industry today. In 1911, Swiss engineer Emile Ruchonnet designed the first wood monocoque structure for an airplane called the Cigare. The wing and tail assemblies consisted of thin, lightweight, and often fabric skin stretched tightly over a wood frame. This stressed skin has evolved into semi-monocoque construction, in which the skin merges with structural fins that take additional forces. It provides even greater strength with less material. The monocoque, which translates to ‘mono or single shell,’ is a structural system that supports loads and transfers them through an external enclosure system. They have largely existed outside the domain of architecture. However, this uniting of divergent systems has been demonstrated to be lighter, utilizing less material than traditional wood building practices. This paper will examine the role monocoque systems have played in the history of wood construction through lineage of boat and airplane building industries and its design potential for wood building systems in architecture through a case-study examination of a unique wood construction approach. The innovative approach uses a wood monocoque system comprised of interlocking small wood members to create thin shell assemblies for the walls, roof and floor, increasing structural efficiency and wasting less than 2% of the wood. The goal of the analysis is to expand the work of practice and the academy in order to foster deeper, more honest discourse regarding the limitations and impact of traditional wood framing.

Keywords: wood building systems, material histories, monocoque systems, construction waste

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Authors: Ching-Shoei Chiang

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The Malfatti’s Problem solves the problem of fitting 3 circles into a right triangle such that these 3 circles are tangent to each other, and each circle is also tangent to a pair of the triangle’s sides. This problem has been extended to any triangle (called general Malfatti’s Problem). Furthermore, the problem has been extended to have 1+2+…+n circles inside the triangle with special tangency properties among circles and triangle sides; we call it extended general Malfatti’s problem. In the extended general Malfatti’s problem, call it Tri(Tn), where Tn is the triangle number, there are closed-form solutions for Tri(T₁) (inscribed circle) problem and Tri(T₂) (3 Malfatti’s circles) problem. These problems become more complex when n is greater than 2. In solving Tri(Tn) problem, n>2, algorithms have been proposed to solve these problems numerically. With a similar idea, this paper proposed an algorithm to find the radii of circles with the same tangency properties. Instead of the boundary of the triangle being a straight line, we use a convex circular arc as the boundary and try to find Tn circles inside this convex circular triangle with the same tangency properties among circles and boundary Carc. We call these problems the Carc(Tn) problems. The CPU time it takes for Carc(T16) problem, which finds 136 circles inside a convex circular triangle with specified tangency properties, is less than one second.

Keywords: circle packing, computer-aided geometric design, geometric constraint solver, Malfatti’s problem

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