Second-order Curve Research Articles

Breathing motion compensation in chest tomosynthesis: evaluation of the effect on image quality and presence of artifacts.

Chest tomosynthesis (CTS) has a relatively longer acquisition time compared with chest X-ray, which may increase the risk of motion artifacts in the reconstructed images. Motion artifacts induced by breathing motion adversely impact the image quality. This study aims to reduce these artifacts by excluding projection images identified with breathing motion prior to the reconstruction of section images and to assess if motion compensation improves overall image quality. In this study, 2969 CTS examinations were analyzed to identify examinations where breathing motion has occurred using a method based on localizing the diaphragm border in each of the projection images. A trajectory over diaphragm positions was estimated from a second-order polynomial curve fit, and projection images where the diaphragm border deviated from the trajectory were removed before reconstruction. The image quality between motion-compensated and uncompensated examinations was evaluated using the image quality criteria for anatomical structures and image artifacts in a visual grading characteristic (VGC) study. The resulting rating data were statistically analyzed using the software VGC analyzer. A total of 58 examinations were included in this study with breathing motion occurring either at the beginning or end ( ) or throughout the entire acquisition ( ). In general, no significant difference in image quality or presence of motion artifacts was shown between the motion-compensated and uncompensated examinations. However, motion compensation significantly improved the image quality and reduced the motion artifacts in cases where motion occurred at the beginning or end. In examinations where motion occurred throughout the acquisition, motion compensation led to a significant increase in ripple artifacts and noise. Compensation for respiratory motion in CTS by excluding projection images may improve the image quality if the motion occurs mainly at the beginning or end of the examination. However, the disadvantages of excluding projections may outweigh the benefits of motion compensation.

Modeling construct change over time amidst potential changes in construct measurement: A longitudinal moderated factor analysis approach.

In analyzing longitudinal data with growth curve models, a critical assumption is that changes in the observed measures reflect construct changes and not changes in the manifestation of the construct over time. However, growth curve models are often fit to a repeated measure constructed as a sum or mean of scale items, making an implicit assumption of constancy of measurement. This practice risks confounding actual construct change with changes in measurement (i.e., differential item functioning [DIF]), threatening the validity of conclusions. An improved method that avoids such confounding is the second-order growth curve (SGC) model. It specifies a measurement model at each occasion of measurement that can be evaluated for invariance over time. The applicability of the SGC model is hindered by key limitations: (a) the SGC model treats time as continuous when modeling construct growth but as discrete when modeling measurement, reducing interpretability and parsimony; (b) the evaluation of DIF becomes increasingly error-prone given multiple timepoints and groups; (c) DIF associated with continuous covariates is difficult to incorporate. Drawing on moderated nonlinear factor analysis, we propose an alternative approach that provides a parsimonious framework for including many time points and DIF from different types of covariates. We implement this model through Bayesian estimation, allowing for incorporation of regularizing priors to facilitate efficient evaluation of DIF. We demonstrate a two-step workflow of measurement evaluation and growth modeling, with an empirical example examining changes in adolescent delinquency over time. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Applied for Higher Order Bezier Curve using MATLAB

A Bézier curve is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. The piecewise nature of a Bezier curve means that its representative equation is a linear combination of Bezier of particular degrees. The aim of this study is to infer Bézier curve graphs from higher orders using MATLAB. Where the applied analytical method was used I have reached .Will be clarified, and the following results have been reached is used in the linear curve association to express the distance ( on the line between the points and , In a second-order Bezier curve, we can create intermediate points such as and . These points have a value located in the interval , To obtain a Bezier curve for higher degrees, note that the idea in arriving at drawing the curve is to divide each line by placing a point on it and then connect the new points to each other (note that we get rid of a point at each stage), and we repeat the process until we have only one line left. So, we put ( on it and then we draw the curve. By comparing the manual solution and the solution using MATLAB, we find that the solution using MATLAB is faster and more accurate, especially at higher levels. And the study recommends the following : It is noted that the curve lies completely within the convex closure of the points, so these points are used with ease to form the required curve, it is also possible to apply geometric transformations (such as rotation and sliding) to the curve by applying this transformation to the control points of this curve.

Developmental trajectories of children's playfulness in two- to six-year-olds

Even though playfulness has been found to be highly relevant to the development and wellbeing of young children, hardly any longitudinal findings are available on stability and changes in children's playfulness. This study examined developmental trajectories of children's playfulness in two- to six-year-olds over a two-year period and analyzed whether individual and family characteristics and transition to primary school explain these trajectories. The sample included 839 children (47.3% girls, MAge = 4.87 years, SDAge = 1.39) from 38 childcare centers and 47 kindergartens in Switzerland. Children's playfulness was assessed at three measurement time points at one-year intervals using the multidimensional Children's Playfulness Scale to gather parent and teacher reports. Second-order linear growth curve models showed significant interindividual differences in children's playfulness at initial baseline. These differences correlated with effects for age, migrant background, and maternal education. The growth-related processes varied depending on children's age, and the individual dimension of children's playfulness whether an increase, stability, or decline was observed. In particular, social components of playfulness were found to change. On average, children's playfulness increased in early years but decreased from school age onwards. However, closeness in parent–child relationships proved to be supportive of children's playfulness at all time points. The results indicate the significance of positive parenting to children's playfulness, but they also draw attention to how playful qualities can be maintained and more highly valued in the school environment. In general, the study can contribute to a better understanding of the construct of playfulness in childhood.

3D surgical planning method for lower jaw osteotomies applied to facial feminization surgery

Our proposed method uses a three-dimensional (3D) measurement approach that focuses mainly on the lower jaw from basal, lateral, and frontal views applied to the volumetric skull model derived from a computed tomography (CT) of the head. Likewise, we discuss the geometrical features and clinical considerations involved in the 3D biomodeling of the surgical osteotomy. The workflow that allowed this virtual planning to be developed was composed of medical imaging processing software, data extraction software from images, and statistical software that allows the creation and generation of curve-fitting (nonlinear regression) graphs from data. Thirty-two (32) anatomical points were positioned, sixteen (16) measurements were taken, and two-dimensional (2D) sketches in three views (frontal, lateral, and inferior) were generated to overlap in a 3D environment, which informed the cutting of the desired bone segments. Implementing a nonlinear regression curve-fitting on the contours of the original jaws allowed optimal planning of the osteotomy. Desired cutting shapes were extrapolated for the front view by third-order equations, while for the side and bottom views, log-normal distribution curves and second-order polynomial curves were used, respectively. The reduction in the mandibular volume was between 6.55 and 10.27 %, with two of the most important measurements related to vertical reduction in the lateral views and the difference to determine gonion reduction.

Impact of stroke on cognition in old age: Comparison of two population-based cohorts, born up to 30 years apart and followed from age 70 to 85.

Little is known about birth cohort differences in the impact of stroke on cognitive aging. Given improved poststroke rehabilitation and better treatments for vascular health risk, we may expect a reduction in the stroke impact in later-born cohorts. We tested this prediction using data from two cohorts, born in 1901-1907 (n = 1,155) and 1930 (n = 919), identified from the same city population at the same age of 70 and subsequently measured on the same cognitive outcomes (i.e., spatial ability, perceptual-motor speed, and reasoning) at ages 70, 75, 79, and 85. We fitted multiple-group second-order latent growth curve models to the data, regressing the first-order cognitive factor on the time-varying stroke variable and controlling for relevant covariates. Findings revealed moderate to large average cognitive decline (d = -.45) following stroke, and the impact was relatively similar across cohorts (1901-1907: d = -.52; 1930: d = -.39). However, there was a stroke by age by cohort interaction, implying that the stroke impact increased with age in the 1901-1907 cohort (dage ≤ 75 = -.42; dage ≥ 79 = -.70) but decreased in the 1930 cohort (dage ≤ 75 = -.53; dage ≥ 79 = -.17). We found no evidence for lagged effect of stroke beyond the impact on measures following the incidence. Our hypothesis was only partially supported, as the impact of stroke was reduced in the later-born cohort but solely at higher ages. (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Novel Adaptive Path-Smoothening Optimization Method for Mobile Robots

A safe and smooth operating path is a prerequisite for mobile robots to accomplish tasks. Although the existing path optimization methods improve the smoothness of the planned path by introducing Bezier curve to locally optimize the path with regard to turning points, most of these methods manually select the position of control points and subjectively analyze the feasibility of the optimized path. It is argued unfavorably that it exhibits strong subjectivity and cumbersome selection process. To fill this gap, an adaptive path-smoothening optimization method is proposed in this study, which combines neural network, genetic algorithm, and Bezier curve to transform the path smoothing problem into an optimization problem. It rapidly determines the position of the optimal control point based on comprehension of constraints, e.g., path safety, curvature and kinematic restrains of the robot. The currently proposed method resolves the long-standing problems of strong subjectivity, cumbersome steps, and thus low efficiency in the selection process of control points, and lays the theoretical groundwork for smoothening the locus and path. To start with, according to the actual working conditions, the dataset corresponding to the position of the control point and the path deviation is constructed, and the neural network algorithm is used to solve the prediction model of the path deviation, so as to obtain the mapping relationship between the length and included angle of the control edge in the second-order Bezier curve and the path deviation. Subsequently, with reference to the prediction model of path deviation, a reliability evaluation function is formulated by comprehending multiple influential factors of mobile robot motion safety and path smoothness. The genetic algorithm is then introduced to detect the satisfactory control points in different environments. The currently proposed method is verified by experiments in different operating environments. The study results show that the currently proposed adaptive path-smoothening optimization method exhibits remarkably superior applicability and effectiveness compared to the currently prevailing methods. It demonstrates advantages of fast path planning, reduced path turning points, and desirable path smoothness. In addition, it can also ensure the safety of mobile robot along the planned path as availed by a pre-set criterion.

СОБСТВЕННЫЕ ЧАСТОТЫ И ФОРМЫ КОЛЕБАНИЙ ОБОЛОЧЕК ВРАЩЕНИЯ.

A method is proposed for constructing a solution to the problem of natural oscillations of shells of revolution described by surfaces of rotation of the second order. The initial equations of shell theory are arranged in a sequence that allows one to calculate the unknowns in an iterative process. The system of the first two momentless equations and the first two equations of infinitesimal bending in the proposition that their inverse operators can be constructed, can be reduced to a single equation for the vector of tangential deformations. The possibility of such a representation is due to the research of V.Z. Vlasov shells of revolution, the meridians of which are described by second-order curves. It is shown that the results of calculating the natural frequencies of vibrations of loose shells given in the well-known reference book on strength, stability and vibrations are incorrect.

Performance evaluation of a novel ball screw raceway profile based on geometric close contact theory

The raceway profile greatly influences the performance of a ball screw. Although the traditional circular raceway profile is simple in structure, it can be a limitation to the load carrying capacity of ball screws. In this paper, a novel type of ball screw raceway profile is proposed based on geometrical close contact theory. Firstly, the second-order approximation curve of the ball is derived using the McLaughlin formula in the normal plane of the raceway helix, which serves as the basis for constructing the close raceway profile. The mathematical expression of the contact stress distribution of the close raceway is then obtained using Chebyshev polynomials. A comparison and analysis of the contact stress distribution between the close raceway and the conventional raceway is carried out. Furthermore, simulation models of ball screws with the close raceway profile are established and the contact behavior of the ball with both types of raceways is statically simulated using ANSYS software. The dynamic performance of both ball screws is compared and analyzed using ADAMS software. The analysis results show that compared with the conventional raceway ball screw, the close raceway ball screw has lower stress, strain, and friction torque under the same load conditions. These research results have significant implications for the structural optimization of ball screws.

Theoretical, numerical, and experimental investigation on second-order Bezier curve flexure hinges

This paper proposes the second-order Bezier curve flexure hinge. The flexure hinge can obtain different notch profiles by changing the position of the control points. Firstly, the compliance and precision of the flexure hinge are modeled based on the force deformation formula of the cantilever beam in material mechanics. Compared with the simulation data of ANSYS Workbench, the modeling error is less than 7%. Secondly, the influence of geometric parameters on compliance and precision is analyzed. The flexure hinge is compared with circular, elliptical, and parabolic hinges. The results show that the second-order Bezier curve flexure hinges have a higher compliance precision ratio. In addition, the use of second-order Bezier curve flexure hinges provides better performance than the use of traditional circular and elliptical flexure hinges in bridge amplification mechanisms. Finally, a flexure hinge was manufactured and measured. The experimental results show that the error between the experimental and modeling values is 8.76%. Both the simulation and experimental results verified the effectiveness of the model.

Impact of Climate Change on the Number of Frost-free Winter Days in Prairies and Eastern Provinces of Canada

Highlights In northern region of Prairies and eastern provinces of Canada, the change in Frost-Free Days (FFDs) for the period of 1938–2022 is nearly negligible. In the stations located at the south of Ontario (called southern stations here), a maximum increase of 4.4 to 5.8 in FFDs per month was noted during the study period at different locations. The rates of change in FFDs per unit °C of the monthly mean of the daily minimum temperature (MMDMT) vary from a minimum of 0.001 to a maximum value of 2.92 days at the selected stations. Abstract. Although a considerable number of studies related to the impact of climate change on temperature-related variables have been conducted, studies exploring the changes in indices of daily winter temperatures, such as variation in the number of frost-free days (FFDs), for the winter season or for each month of the season are very few. The main objective of this study was to explore temporal trends in FFDs for each month of the winter season as well as examine the relationship between FFDs and the monthly mean of daily minimum temperature (MMDMT) at the selected stations in the central and eastern provinces of Canada. Daily minimum temperature data between 1938 and 2022 from 11 meteorological stations situated along a wide range of latitudes across the Prairies and eastern provinces of Canada has been analyzed. An exceedance probability model is developed to predict the FFDs for each winter month. The analysis leads to the conclusion that the number of FFDs per month exhibited an exponential relationship with the MMDMT for November, April, and the entire winter season, while the relationship for December through March followed a second-order polynomial curve. The results further highlight that the rate of change in FFDs per month varies across stations for each month. Notably, while the change is minimal for stations such as Kuujjuaq, Kuujjuarapik, and The Pas, it has notably increased by 23.1, 17.9, and 14.2 days at the Toronto, Windsor, and London stations, respectively. Additionally, it was observed that the rates of change in FFDs per unit increase in MMDMT (in °C) are consistently positive for all months and selected stations. The rate of change varies, ranging from a minimum of 0.001 at The Pas in December to a maximum value of 2.92 days at Montreal in April.Moreover, coefficient of determination (R2) values of more than 0.96 were observed for both exponential and second-order polynomial relationships in the months of November and April, as well as for the entire winter. A rising trend in MMDMT values, along with a decreasing trend in the MMDMT standard deviation values, was observed across all stations from 1938 to 2022. This trend indicates a more pronounced and predictable rise in temperature over time. Overall, the number of FFDs is increasing for all winter months, albeit with a few exceptions, at central and eastern stations in Canada. These warming trends during the winter months contribute to the expansion in the number of FFDs, which is expected to significantly impact hydrological, agricultural, and environmental processes within the study region. Keywords: Climate change, Exceedance probability model, Frost-free days, Monthly mean of daily minimum temperature.

Study of Wheat Grain Structure by Digital Radiography

Introduction. Analysis of the quality of wheat grain using instrumental methods is an important task due to the permanent growth in annual wheat production and export. Digital radiography is a promising method for studying the internal structure of wheat grain. Along with other methods, digital radiography allows non-destructive testing and comprehensive description of grain parameters.Aim. To determine the capabilities of digital radiography for studying the structure of wheat grain.Materials and methods. Digital X-ray images were obtained using a multifunctional mobile x-ray diagnostic unit produced by JSC "ELTECH-Med" was used with the installation tube voltage of 9 kV. Durum wheat grain samples of different varieties were used. Operations for processing grain images were automated using the scripts developed in Python using the numpy and opencv libraries.Results. The attenuation of X-ray radiation by grain material was studied. Wheat bran was found to exhibit a greater attenuation coefficient than the endosperm, assumably due to the higher concentration of potassium and phosphorus in the wheat outer layer. The calculated ratio of the attenuation coefficients of the bran to the endosperm was 1.27, compared to the experimentally obtained value of 1.36. The influence of the geometric shape of wheat grain on the formation of its X-ray image was determined. A method for describing the geometric shape of wheat grain using second-order curves was presented. A mathematical model for the transmission of X-ray radiation in wheat grain was constructed, taking into account its complex shape and the heterogeneous distribution of macroelements therein. This model allows estimation of the attenuation coefficients of the grain endosperm and outer layer using a non-destructive method.Conclusion. Digital radiography is an effective method for studying wheat when carrying out research and breeding tasks. The proposed model for the transmission of X-ray radiation in wheat grain can be used to numerically determine some indicators of grain quality.

A Multi-Regional Path-Planning Method for Rescue UAVs with Priority Constraints

This study focuses on the path-planning problem of rescue UAVs with regional detection priority. Initially, we propose a mixed-integer programming model that integrates coverage path planning (CPP) and the hierarchical traveling salesman problem (HTSP) to address multi-regional path planning under priority constraints. For intra-regional path planning, we present an enhanced method for acquiring reciprocating flight paths to ensure complete coverage of convex polygonal regions with shorter flight paths when a UAV is equipped with sensors featuring circular sampling ranges. An additional comparison was made for spiral flight paths, and second-order Bezier curves were employed to optimize both sets of paths. This optimization not only reduced the path length but also enhanced the ability to counteract inherent drone jitter. Additionally, we propose a variable neighborhood descent algorithm based on K-nearest neighbors to solve the inter-regional access order path-planning problem with priority. We establish parameters for measuring distance and evaluating the priority order of UAV flight paths. Simulation and experiment results demonstrate that the proposed algorithm can effectively assist UAVs in performing path-planning tasks with priority constraints, enabling faster information collection in important areas and facilitating quick exploration of three-dimensional characteristics in unknown disaster areas by rescue workers. This algorithm significantly enhances the safety of rescue workers and optimizes crucial rescue times in key areas.

Numerical and experimental investigation of a variable-curvature shell class I flextensional transducer.

Class I flextensional transducers cannot form broadband characteristics because of the deep valley between their first- and second-order response curves. This valley arises from the sound pressure cancellation produced by the vibration in different areas of the shell. This paper examines the cause of the deep valley by analyzing the three-dimensional equivalent radiated sound field of the transducer. By solving the three-dimensional vibration equation of the shell of revolution and reducing its dimensionality, it can be concluded that its vibration state is related to the positive and negative curvatures in both main directions. Therefore, a variable-curvature shell class I flextensional transducer is proposed, and the second-order vibration of the transducer is changed by the variable-curvature structure composed of positive and negative Gaussian curvature surfaces. The proposed transducer avoids sound pressure cancellation and achieves broadband characteristics. After optimization, a transducer prototype is fabricated, and a pool test is conducted. The test results show that the transducer couples the first three vibration modes, attaining broadband characteristics with an in-band fluctuation of less than 8 dB and a bandwidth of more than 5.4 kHz in the axial and circumferential directions.

Моделирование конхоидальных кривых с использованием квазисимметрии относительно эллиптической оси

. This article presents the results of a study of the geometric properties of the Nicomed conchoid and the oblique conchoid. In this paper, the oblique conchoid is modeled in a new way, namely by quasi-symmetry with respect to the elliptic axis. The method used is a fourth-order transformation of the plane relative to the second-order curve. That is, a straight line with quasi-symmetry is mapped into a fourth-order curve. The image of a straight line in this case consists of two branches that tend to two asymptotes. Quasi–symmetry makes it possible to obtain an oblique conchoid, as a special case under certain conditions, and in the general case, many other conchoidal curves. The use of this method made it possible to discover new geometric properties of conchoidal curves, in particular, to find a previously undescribed constructive correspondence between points belonging to different branches of the oblique conchoid. The paper formulates and proves three statements, namely: 1) The image of a straight line with its quasi-symmetry with respect to a circle is a Nicomedes conchoid, 2) the image of a circle with its quasi-symmetry with respect to a circle is a curve of the sixth order, 3) the image of a straight parallel major semiaxis of an ellipse with its quasi-symmetry with respect to a given ellipse is two symmetrical oblique conchoids with respect to the minor semiaxis of an ellipse. Also, the equations of the curves under consideration and their asymptotes in the general case are derived. 
 The results of the research carried out in this paper expand the possibilities of using conchoidal curves in solving problems of engineering geometry. For example, when modeling various physical phenomena and processes, as well as in engineering and architectural design.

A Screening Method for Retired Lithium-Ion Batteries Based on Support Vector Machine With a Multi-Class Kernel Function

Abstract With the retirement of a large number of lithium-ion batteries from electric vehicles, their reuse has received increasing attention. However, a retired battery pack is not suitable for direct reuse due to the poor consistency of in-pack batteries. This paper proposes a method of retired lithium-ion battery screening based on support vector machine (SVM) with a multi-class kernel function. First, ten new NCR18650B batteries were used to carry out the aging experiments for collecting the main parameters, such as capacity, voltage, and direct current resistance. Second, an SVM based on a multi-class kernel function was proposed to screen retired batteries. To improve the screening efficiency, a capacity/voltage second-order conductance curve was adopted to extract their capacity features quickly, and four new feature points were selected as the input of the SVM to classify retired batteries. Finally, the retired batteries are accurately divided into four classes by the trained model, and the classification accuracy can reach 97.0%. Compared with the traditional method, the feature extraction time can be reduced by four-fifths, and the screening efficiency is greatly improved.

Study on second-order buoyancy segment linearity in magnetic liquids

The study of second-order buoyancy is important for magnetic fluid applications, particularly for magnetic liquid sensors. Previous research focused solely on the second-order buoyancy change of the permanent magnet, without examining the state of the magnetic liquid or delving deeper into the reasons for the buoyancy change. This paper not only summarizes the laws governing the second-order buoyancy curve, but also employs a two-dimensional simulation model that includes the magnetic liquid in COMSOL Multiphysics software. Through simulation and experimental results, we demonstrate that the second-order buoyancy acting on the permanent magnet in the magnetic fluid produces a segmented linear curve in relation to the distance between the magnet and the container wall. By combining simulation results with theoretical calculations, we explain the segmented linear relationship between the second-order buoyancy and the axial movement of the permanent magnet as resulting from changes in the pressure gradient of the magnetic fluid near the permanent magnet.

Designing the working surfaces of rotary planetary mechanisms

Purpose. To design a method for smoothing the working surfaces of stator and rotor with the use of computer simulation to eliminate the impact of the rotor on the stator when they interact. Methodology. Special and general methods of research have been used: interpolation of the point series – to determine the contour nodes of the rotor and the stator of the rotary-planetary machine; formation of B-splines – to construct a point series whose coordinates are structurally determined; technology of automated formation of curves in CAD-system SolidWorks – for modelling of functional surfaces of a planetary-rotary compressor. Findings. Algorithms for formation of the contours representing curves defined analytically or constructively with a given accuracy have been developed. The obtained contours are used in the CAD system as linear elements of the surface model. The developed method has been tested in the simulation of functional surfaces of a planetary-rotary compressor. Optimization of the body shape and rotor profiles in order to increase the productivity of the rotary-planetary machine has been carried out. Originality. The developed algorithms make it possible to determine the original point series belonging to any curve and provide a given interpolation accuracy when forming a B-spline contour or second-order curve arcs. Computer models of the body surfaces of the rotor are formed on the basis of the gear ratio of the planetary-rotary mechanism and the rotor dimensions. In order to increase the performance of the compressor, the working surfaces of the rotor have been optimized. The maximum volume of the working chamber was increased by increasing the radius of the moving gear of the planetary-rotary mechanism. In order to prevent the rotor from jamming during compressor operation, the rotor contour was changed. Practical value. The method for modelling the surfaces of complex shape in CAD-system has been developed on the basis of creating contours which with given accuracy represent lines from the surface determinant. This method makes it possible to form computer models of complex surfaces on the basis of a framework consisting of curves absent in CAD libraries.

ANALYSIS OF THE PROBLEM OF CONSTRUCTING AN ELIPSE FROM TWO TANGENTS

This article discusses the problem of constructing an ellipse by two given tangents and the coordinates of the center of the ellipse, which arises in the development of CAD design systems. When analyzing the sufficiency of the completeness of the initial data, a symmetrical display of the given lines relative to the center of the ellipse is performed, which leads to a parallelogram with an ellipse inscribed in it. The subsequent affine transformation maps the parallelogram into a square with an ellipse inscribed in it. The analysis of the obtained results allows us to conclude that the initial data are incomplete and it is necessary to supplement the initial data with the coordinate of the point of contact of the ellipse with one of the straight lines. When constructing an ellipse from the supplemented initial data, using direct and inverse affine transformations, the points through which the ellipse must pass and the equations of tangents at these points are determined. This makes it possible to compose a system of linear equations in which the unknowns are the coefficients of the general equation of the second-order curve. The resulting solution is analyzed according to the classification of second-order curves for belonging to a real ellipse. The proposed method is compared with the well-known projective-graphical method for solving the same problem.

Tricritical Behavior in Dynamical Phase Transitions.

We identify a new scenario for dynamical phase transitions associated with time-integrated observables occurring in diffusive systems described by the macroscopic fluctuation theory. It is characterized by the pairwise meeting of first- and second-order bias-induced phase transition curves at two tricritical points. We formulate a simple, general criterion for its appearance and derive an exact Landau theory for the tricritical behavior. The scenario is demonstrated in three examples: the simple symmetric exclusion process biased by an activity-related structural observable; the Katz-Lebowitz-Spohn lattice gas model biased by its current; and in an active lattice gas biased by its entropy production.