Journal articles: 'Geometry, Analytic. Curves, Plane'

1

Гирш and A. Girsh. "Foci of Algebraic Curves." Geometry & Graphics 3, no. 3 (November 30, 2015): 4–17. http://dx.doi.org/10.12737/14415.

Full text

Abstract:

Curves have always been part geometry. Initially, there were lines and circle, then it was added to a conic section and later, with the advent of analytic geometry, they added more complex curves. Particularly in a number of lines are algebraic curves that are described by algebraic equations. Curves found application mostly in mechanics. Today algebraic curves used in engineering and in mathematics, in number theory, knot theory, computer science, criminology, etc. With the bringing to account of complex numbers became possible to consider curves in the complex plane. It has expanded the horizons of geometry and enriched their knowledge on curves, particularly on algebraic curves. Our goal is to give a geometric picture of the foci of algebraic curves clearly show the position of the foci in the plane, show how the number of foci associated with a class curve. The solution of this problem we see in the application we have developed ways to visualize imaginary images to the study of foci and focal centers of algebraic curves. This article explains the concept of the foci of algebraic curves shows the basic principle of the curve-theory and offers a method for the identification of the foci. The geometric picture of the foci is shown in a diagram, which is putted together from two tables. One table shows the real curve with her foci, the other table shows an imaginary cut of the curve, on which the isotropic line contacts the cut and under them intersects in a real point. The point is a focal point of the real curve. This project shows 16 diagrams for conic, cubes and quadrics.

APA, Harvard, Vancouver, ISO, and other styles

2

Chou, Kai-Seng, and Xi-Ping Zhu. "Shortening complete plane curves." Journal of Differential Geometry 50, no. 3 (1998): 471–504. http://dx.doi.org/10.4310/jdg/1214424967.

Full text

APA, Harvard, Vancouver, ISO, and other styles

3

Chen, Bingkui, Dong Liang, and Yane Gao. "Geometry design and mathematical model of a new kind of gear transmission with circular arc tooth profiles based on curve contact analysis." Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 228, no. 17 (March 11, 2014): 3200–3208. http://dx.doi.org/10.1177/0954406214526866.

Full text

Abstract:

A new meshing relationship for gear drive to characterize the conjugation geometry is studied in this paper based on conjugate curves. Conjugate curves are described as two smooth curves always keep continuous and tangent contact with each other in given contact direction under motion law. The general principle of curve meshing is developed for the given spatial or plane curve. The meshing equation along arbitrary direction of contact angle is derived. The properties of geometric and motion of the contact of conjugate curves are discussed. According to the equidistance-enveloping method, tubular meshing surfaces are proposed to build up circular arc tooth profiles, which inherit all properties of conjugate curves. The geometry design and mathematical model of the gears are established. Three types of contact pattern of tooth profiles are generated: convex-to-convex, convex-to-plane and convex-to-concave. A calculation example for convex-to-concave tooth profiles of gears is provided. Theoretical and numerical results demonstrate the feasibility and correctness of proposed conjugate curves theory and it lays the foundation for the design of high performance gear transmission.

APA, Harvard, Vancouver, ISO, and other styles

4

Гирш and A. Girsh. "Envelope of a Family of Curves." Geometry & Graphics 4, no. 4 (December 19, 2016): 14–18. http://dx.doi.org/10.12737/22839.

Full text

Abstract:

A one-parameter family of algebraic curves has an envelope line, which may be imaginary in certain cases. Jakob Steiner was right, considering the imaginary images as creation of analysis. In the analysis a real number is just a part of a complex number and in certain conditions the initial real values can give an imaginary result. But Steiner was wrong in denying the imaginary images in geometry. The geometry, in contrast to the single analytical space exists in several spaces: Euclidean geometry operates only on real figures valid and does not contain imaginary figures by definition; pseudo-Euclidean geometry operates on imaginary images and constructs their images, taking into account its own features. Geometric space is complex and each geometric object in it is the complex one, consisting of the real figure (core) having the "aura" of an imaginary extension. Thus, any analytical figure of the plane is present at every point of the plane or by its real part or by its imaginary extension. Would the figure’s imaginary extension be visible or not depends on the visualization method, whether the image has been assumed on superimposed epures – the Euclideanpseudo-Euclidean plane, or the image has been traditionally assumed only in the Euclidean plane. In this paper are discussed cases when a family of algebraic curves has an envelope, and is given an answer to a question what means cases of complete or partial absence of the envelope for the one-parameter family of curves. Casts some doubt on widely known categorical st

APA, Harvard, Vancouver, ISO, and other styles

5

Korotkiy, Viktor. "Cubic Curves in Engineering Geometry." Geometry & Graphics 8, no. 3 (November 24, 2020): 3–24. http://dx.doi.org/10.12737/2308-4898-2020-3-24.

Full text

Abstract:

In this paper are considered historically the first (the 60’s of the 20th century) computational methods for algebraic cubic curves constructing. The analysis of a general cubic curve equation r(t)=a3t3+a2t2+a1t+a0 has been carried out. As an example has been considered the simplest cubic curve r(t)=it3+jt2+kt. Based on the general cubic curve equation have been obtained equations of a cubic curve passing through two predetermined points and having predetermined tangents at these points. The equations have been presented both in Ferguson and Bézier forms. It has been shown that the cubic curve vector equation (for example, the standard equation of a Bezier curve) can be represented in a point form. Have been considered examples for constructing segments of cubic curves meeting the given boundary conditions. The generalized cubic curve equation, containing weight coefficients, has been obtained by the method of exit into four-dimensional space. Has been considered a vector parametric equation of a conical section, passing through two given points and touching predetermined straight lines at these points. The conical section is considered as a special case of a cubic curve. Curvature can be specified as an additional boundary condition. Has been considered the possibility for constructing a cubic curve with fixed positions of contacting planes at end points and given radii of curvature. Has been proposed an algorithm for constructing a plane cubic curve with a given curvature at the end points. Have been considered algorithms for constructing smooth compound Ferguson-Bezier curves. Smoothness conditions are imposed on a compound curve: 1) at any of its points, the curve must have a tangent (no fractures are allowed), 2) the curvature vector must be changed continuously from point to point (no discontinuous jump in the curvature vector is allowed neither in modulus no in direction). Have been proposed examples for constructing compound Ferguson-Bézier curves. Has been performed comparison of polynomial cubic spline with compound parametrically defined curves. Have been given examples for constructing cubic splines with fastened and free ends. The paper is for educational purposes, and intended for in-depth study of computer graphics basics.

APA, Harvard, Vancouver, ISO, and other styles

6

Tsai, Dong-Ho. "Geometric expansion of starshaped plane curves." Communications in Analysis and Geometry 4, no. 3 (1996): 459–80. http://dx.doi.org/10.4310/cag.1996.v4.n3.a5.

Full text

APA, Harvard, Vancouver, ISO, and other styles

7

Le Maire, Pauline, and Marc Munschy. "2D potential theory using complex algebra: New equations and visualization for the interpretation of potential field data." GEOPHYSICS 83, no. 2 (March 1, 2018): J1—J13. http://dx.doi.org/10.1190/geo2016-0611.1.

Full text

Abstract:

The shape of an anomaly (magnetic or gravity) along a profile provides information on the geometry, horizontal location, depth, and magnetization of the source. For a 2D source, the horizontal location, depth, and geometry of a source are determined through the analysis of the curve of the analytic signal. However, the amplitude of the analytic signal is independent of the dips of the structure, the apparent inclination of magnetization, and the regional magnetic field. To better characterize the parameters of the source, we have developed a new approach for studying 2D potential field equations using complex algebra. Complex equations for different geometries of the sources are obtained for gravity and magnetic anomalies in the spatial and spectral domains. In the spatial domain, these new equations are compact and correspond to logarithmic or power functions with a negative integer exponent. We found that modifying the shape of the source changes the exponent of the power function, which is equivalent to differentiation or integration. We developed anomaly profiles using plots in the complex plane, which is called mapping. The obtained complex curves are loops passing through the origin of the plane. The shape of these loops depends only on the geometry and not on the horizontal location of the source. For source geometries defined by a single point, the loop shape is also independent of the source depth. The orientation of the curves in the complex plane is related to the order of differentiation or integration, the geometry and dips of the structures, and the apparent inclination of magnetization and of the regional magnetic field. The application of these equations and mapping on total field magnetic anomalies across a magmatic dike in Norway shows coherent results, allowing us to determine the geometry and the apparent inclination of magnetization.

APA, Harvard, Vancouver, ISO, and other styles

8

Petrovic, Maja, Bojan Banjac, and Branko Malesevic. "The geometry of trifocal curves with applications in architecture, urban and spatial planning." Spatium, no. 32 (2014): 28–33. http://dx.doi.org/10.2298/spat1432028p.

Full text

Abstract:

In this paper we consider historical genesis of trifocal curve as an optimal curve for solving the Fermat?s problem (minimizing the sum of distance of one point to three given points in the plane). Trifocal curves are basic plane geometric forms which appear in location problems. We also analyze algebraic equation of these curves and some of their applications in architecture, urbanism and spatial planning. The area and perimeter of trifocal curves are calculated using a Java application. The Java applet is developed for determining numerical value for the Fermat-Torricelli-Weber point and optimal curve with three foci, when starting points are given on an urban map. We also present an application of trifocal curves through the analysis of one specific solution in South Stream gas pipeline project.

APA, Harvard, Vancouver, ISO, and other styles

9

Steward, David R., Philippe Le Grand, Igor Janković, and Otto D. L. Strack. "Analytic formulation of Cauchy integrals for boundaries with curvilinear geometry." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 464, no. 2089 (October 30, 2007): 223–48. http://dx.doi.org/10.1098/rspa.2007.0138.

Full text

Abstract:

A general framework for analytic evaluation of singular integral equations with a Cauchy kernel is developed for higher order line elements of curvilinear geometry. This extends existing theory which relies on numerical integration of Cauchy integrals since analytic evaluation is currently published only for straight lines, and circular and hyperbolic arcs. Analytic evaluation of Cauchy integrals along straight elements is presented to establish a context coalescing new developments within the existing body of knowledge. Curvilinear boundaries are partitioned into sectionally holomorphic elements that are conformally mapped from a local curvilinear Z -plane to a straight line in the -plane. Cauchy integrals are evaluated in these planes to achieve a simple representation of the complex potential using Chebyshev polynomials and a Taylor series expansion of the conformal mapping. Bell polynomials and the Faà di Bruno formula provide this Taylor series for mappings expressed as inverse mappings and/or compositions. Examples illustrate application of the general framework to boundary-value problems with boundaries of natural coordinates, Bezier curves and B-splines. Strings formed by the union of adjacent curvilinear elements form a large class of geometries along which Dirichlet and/or Neumann conditions may be applied. This provides a framework applicable to a wide range of fields of study including groundwater flow, electricity and magnetism, acoustic radiation, elasticity, fluid flow, air flow and heat flow.

APA, Harvard, Vancouver, ISO, and other styles

10

Watanabe, S. "The genera of Galois closure curves for plane quartic curves." Hiroshima Mathematical Journal 38, no. 1 (March 2008): 125–34. http://dx.doi.org/10.32917/hmj/1207580347.

Full text

APA, Harvard, Vancouver, ISO, and other styles

11

Liang, Dong, Bingkui Chen, Yane Gao, Shuai Peng, and Siling Qin. "Geometric and Meshing Properties of Conjugate Curves for Gear Transmission." Mathematical Problems in Engineering 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/484802.

Full text

Abstract:

Conjugate curves have been put forward previously by authors for gear transmission. Compared with traditional conjugate surfaces, the conjugate curves have more flexibility and diversity in aspects of gear design and generation. To further extend its application in power transmission, the geometric and meshing properties of conjugate curves are discussed in this paper. Firstly, general principle descriptions of conjugate curves for arbitrary axial position are introduced. Secondly, geometric analysis of conjugate curves is carried out based on differential geometry including tangent and normal in arbitrary contact direction, characteristic point, and curvature relationships. Then, meshing properties of conjugate curves are further revealed. According to a given plane or spatial curve, the uniqueness of conjugated curve under different contact angle conditions is discussed. Meshing commonality of conjugate curves is also demonstrated in terms of a class of spiral curves contacting in the given direction for various gear axes. Finally, a conclusive summary of this study is given.

APA, Harvard, Vancouver, ISO, and other styles

12

FISH, JOEL W. "ESTIMATES FOR J-CURVES AS SUBMANIFOLDS." International Journal of Mathematics 22, no. 10 (October 2011): 1375–431. http://dx.doi.org/10.1142/s0129167x11007306.

Full text

Abstract:

In this paper, we develop some basic analytic tools to study compactness properties of J-curves (i.e. pseudoholomorphic curves) when regarded as submanifolds. Incorporating techniques from the theory of minimal surfaces, we derive an inhomogeneous mean curvature equation for such curves by establishing an extrinsic monotonicity principle for nonnegative functions f satisfying Δf ≥ -c2f, we show that curves locally parametrized as a graph over a coordinate tangent plane have all derivatives a priori bounded in terms of curvature and ambient geometry, and we thus establish ϵ-regularity for the square length of their second fundamental forms. These results are all provided for J-curves either with or without Lagrangian boundary and hold in almost all Hermitian manifolds of arbitrary even dimension (i.e. Riemannian manifolds for which the almost complex structure is an isometry).

APA, Harvard, Vancouver, ISO, and other styles

13

Dresp-Langley, Birgitta. "2D Geometry Predicts Perceived Visual Curvature in Context-Free Viewing." Computational Intelligence and Neuroscience 2015 (2015): 1–9. http://dx.doi.org/10.1155/2015/708759.

Full text

Abstract:

Planar geometry was exploited for the computation of symmetric visual curves in the image plane, with consistent variations in local parameters such assagitta,chordlength, and the curves’ height-to-width ratio, an indicator of the visual area covered by the curve, also called aspect ratio. Image representations of single curves (no local image context) were presented to human observers to measure their visual sensation of curvature magnitude elicited by a given curve. Nonlinear regression analysis was performed on both the individual and the average data using two types of model: (1) a power function wherey(sensation) tends towards infinity as a function ofx(stimulus input), most frequently used to model sensory scaling data for sensory continua, and (2) an “exponential rise to maximum” function, which converges towards an asymptotically stable level ofyas a function ofx. Both models provide satisfactory fits to subjective curvature magnitude as a function of the height-to-width ratio of single curves. The findings are consistent with an in-built sensitivity of the human visual system to local curve geometry, a potentially essential ground condition for the perception of concave and convex objects in the real world.

APA, Harvard, Vancouver, ISO, and other styles

14

Yajima, Takahiro, Shunya Oiwa, and Kazuhito Yamasaki. "Geometry of curves with fractional-order tangent vector and Frenet-Serret formulas." Fractional Calculus and Applied Analysis 21, no. 6 (December 19, 2018): 1493–505. http://dx.doi.org/10.1515/fca-2018-0078.

Full text

Abstract:

Abstract This paper discusses a construction of fractional differential geometry of curves (curvature of curve and Frenet-Serret formulas). A tangent vector of plane curve is defined by the Caputo fractional derivative. Under a simplification for the fractional derivative of composite function, a fractional expression of Frenet frame of curve is obtained. Then, the Frenet-Serret formulas and the curvature are derived for the fractional ordered frame. The different property from the ordinary theory of curve is given by the explicit expression of arclength in the fractional-order curvature. The arclength part of the curvature takes a large value around an initial time and converges to zero for a long period of time. This trend of curvature may reflect the memory effect of fractional derivative which is progressively weaken for a long period of time. Indeed, for a circle and a parabola, the curvature decreases over time. These results suggest that the basic property of fractional derivative is included in the fractional-order curvature appropriately.

APA, Harvard, Vancouver, ISO, and other styles

15

Bauer, Martin, Martins Bruveris, Stephen Marsland, and Peter W. Michor. "Constructing reparameterization invariant metrics on spaces of plane curves." Differential Geometry and its Applications 34 (June 2014): 139–65. http://dx.doi.org/10.1016/j.difgeo.2014.04.008.

Full text

APA, Harvard, Vancouver, ISO, and other styles

16

Lentzos, Konstantinos, and Lillian F. Pasley. "Determinantal representations of invariant hyperbolic plane curves." Linear Algebra and its Applications 556 (November 2018): 108–30. http://dx.doi.org/10.1016/j.laa.2018.06.033.

Full text

APA, Harvard, Vancouver, ISO, and other styles

17

Mirman, Boris, and Pradeep Shukla. "A characterization of complex plane Poncelet curves." Linear Algebra and its Applications 408 (October 2005): 86–119. http://dx.doi.org/10.1016/j.laa.2005.05.016.

Full text

APA, Harvard, Vancouver, ISO, and other styles

18

TRÁNG, LÊ DŨNG, HÉLÈNE MAUGENDRE, and CLAUDE WEBER. "GEOMETRY OF CRITICAL LOCI." Journal of the London Mathematical Society 63, no. 3 (June 2001): 533–52. http://dx.doi.org/10.1017/s0024610701001995.

Full text

Abstract:

Let(formula here)be the germ of a finite (that is, proper with finite fibres) complex analytic morphism from a complex analytic normal surface onto an open neighbourhood U of the origin 0 in the complex plane C2. Let u and v be coordinates of C2 defined on U. We shall call the triple (π, u, v) the initial data.Let Δ stand for the discriminant locus of the germ π, that is, the image by π of the critical locus Γ of π.Let (Δα)α∈A be the branches of the discriminant locus Δ at O which are not the coordinate axes.For each α ∈ A, we define a rational number dα by(formula here)where I(–, –) denotes the intersection number at 0 of complex analytic curves in C2. The set of rational numbers dα, for α ∈ A, is a finite subset D of the set of rational numbers Q. We shall call D the set of discriminantal ratios of the initial data (π, u, v). The interesting situation is when one of the two coordinates (u, v) is tangent to some branch of Δ, otherwise D = {1}. The definition of D depends not only on the choice of the two coordinates, but also on their ordering.In this paper we prove that the set D is a topological invariant of the initial data (π, u, v) (in a sense that we shall define below) and we give several ways to compute it. These results are first steps in the understanding of the geometry of the discriminant locus. We shall also see the relation with the geometry of the critical locus.

APA, Harvard, Vancouver, ISO, and other styles

19

Li, Baokun, Yi Cao, Qiuju Zhang, and Zhen Huang. "Position-singularity analysis of a special class of the Stewart parallel mechanisms with two dissimilar semi-symmetrical hexagons." Robotica 31, no. 1 (April 20, 2012): 123–36. http://dx.doi.org/10.1017/s0263574712000148.

Full text

Abstract:

SUMMARYIn this paper, for a special class of the Stewart parallel mechanism, whose moving platform and base one are two dissimilar semi-symmetrical hexagons, the position-singularity of the mechanism for a constant-orientation is analyzed systematically. The force Jacobian matrix [J]T is constructed based on the principle of static equilibrium and the screw theory. After expanding the determinant of the simplified matrix [D], whose rank is the same as the rank of the matrix [J]T, a cubic symbolic expression that represents the 3D position-singularity locus of the mechanism for a constant-orientation is derived and graphically represented. Further research shows that the 3D position-singularity surface is extremely complicated, and the geometric characteristics of the position-singularity locus lying in a general oblique plane are very difficult to be identified. However, the position-singularity locus lying in the series of characteristic planes, where the moving platform coincides, are all quadratic curves compromised of infinite many sets of hyperbolas, four pairs of intersecting lines and a parabola. For some special orientations, the quadratic curve can degenerate into two lines or even one line, all of which are parallel to the ridgeline. Two theorems are presented and proved for the first time when the geometric characteristics of the position-singularity curves in the characteristic plane are analyzed. Moreover, the kinematic property of the position-singularity curves is obtained using the Grassmann line geometry and the screw theory. The theoretical results are demonstrated with several numeric examples.

APA, Harvard, Vancouver, ISO, and other styles

20

Ballico, Edoardo, and Changho Keem. "On plane curves with several singular points with high multiplicity." Hiroshima Mathematical Journal 26, no. 1 (1996): 117–26. http://dx.doi.org/10.32917/hmj/1206127492.

Full text

APA, Harvard, Vancouver, ISO, and other styles

21

Castro, Ildefonso, and Ildefonso Castro-Infantes. "Plane curves with curvature depending on distance to a line." Differential Geometry and its Applications 44 (February 2016): 77–97. http://dx.doi.org/10.1016/j.difgeo.2015.11.002.

Full text

APA, Harvard, Vancouver, ISO, and other styles

22

Buckley, Anita. "Elementary transformations of pfaffian representations of plane curves." Linear Algebra and its Applications 433, no. 4 (October 2010): 758–80. http://dx.doi.org/10.1016/j.laa.2010.04.005.

Full text

APA, Harvard, Vancouver, ISO, and other styles

23

Tsai, Dong-Ho. "Blowup and convergence of expanding immersed convex plane curves." Communications in Analysis and Geometry 8, no. 4 (2000): 761–94. http://dx.doi.org/10.4310/cag.2000.v8.n4.a3.

Full text

APA, Harvard, Vancouver, ISO, and other styles

24

Oger, Francis. "The number of paperfolding curves in a covering of the plane." Hiroshima Mathematical Journal 47, no. 1 (March 2017): 1–14. http://dx.doi.org/10.32917/hmj/1492048844.

Full text

APA, Harvard, Vancouver, ISO, and other styles

25

Poon, Chi-Cheung, and Dong-Ho Tsai. "Contracting convex immersed closed plane curves with fast speed of curvature." Communications in Analysis and Geometry 18, no. 1 (2010): 23–75. http://dx.doi.org/10.4310/cag.2010.v18.n1.a2.

Full text

APA, Harvard, Vancouver, ISO, and other styles

26

Liu, Jing, Hai Wang Li, Yue Nan Sun, and Xue Feng Shu. "The Research on the Nonlinear Stability of the Steel Spatial Arch Truss with 120m." Applied Mechanics and Materials 226-228 (November 2012): 1236–39. http://dx.doi.org/10.4028/www.scientific.net/amm.226-228.1236.

Full text

Abstract:

In this paper, the elastic stability and elasto-plastic stability analysis on the steel spatial arch truss with 120m span and 0.2 rise-span ratio is carried out with the finite element program ANSYS. In the analyses, three conditions which are the geometric ,the geometric and material nonlinear, double nonlinear under different initial geometric defects are considered. The results show that the instability mode of arch trusses are all asymmetric instability of plane, and that the buckling load of considering double nonlinear is 13.84% that of considering geometry nonlinear, and that the coupling effect of geometry nonlinear and material nonlinear has accelerated reducing the stiffness of structure and bearing stable capacity ,and that the ratio for the buckling load of the four arch trusses with L/500, L/400, L/300 and L/200 initial geometric defects is 1.0: 0.927: 0.904: 0.87: 0.804, and that load-displacement process curves of feature node are the same change for the structures under different initial geometric defects.

APA, Harvard, Vancouver, ISO, and other styles

27

Pal, Janos, and Dana Schlomiuk. "Summing up the Dynamics of Quadratic Hamiltonian Systems With a Center." Canadian Journal of Mathematics 49, no. 3 (June 1, 1997): 582–99. http://dx.doi.org/10.4153/cjm-1997-027-0.

Full text

Abstract:

AbstractIn this work we study the global geometry of planar quadratic Hamiltonian systems with a center and we sum up the dynamics of these systems in geometrical terms. For this we use the algebro-geometric concept of multiplicity of intersection Ip(P,Q) of two complex projective curves P(x, y, z) = 0, Q(x,y,z) = 0 at a point p of the plane. This is a convenient concept when studying polynomial systems and it could be applied for the analysis of other classes of nonlinear systems.

APA, Harvard, Vancouver, ISO, and other styles

28

Valera Sifontes, Rimary, Hédison Kiuity Sato, and Zoukaneri Ibrahim Moumoni. "Relief geometric effects on frequency-domain electromagnetic data." GEOPHYSICS 81, no. 5 (September 2016): E287—E296. http://dx.doi.org/10.1190/geo2015-0344.1.

Full text

Abstract:

A perpendicular transmitter-receiver coils arrangement used in the frequency-domain electromagnetic survey can have deviations in relation to its standard geometric definition due to the relief geometry of the surveyed area when combined with large transmitter-receiver distance and large transmitter loop size. This happens because the local relief characteristics along the transmitter loop wire laid on the ground can deviate the equivalent magnetic moment axis from the vertical, and the global characteristics locate the transmitter and receiver positions at different elevations. A study about that is carried on here substituting the rugged relief by an inclined plane. We have developed a new formulation for the [Formula: see text]-layered model that allowed us to investigate the relief geometry effects on FDEM data but restricting the analysis to the two-layer earth model, considering three cases of transmitter-receiver situations controlled by the relief conditions. The curves representing the Argand diagram and the apparent polarization parameter as a function of the apparent induction number were obtained for each relief model. Such procedures resulted to be very useful to demonstrate their behavior departing from those curves obtained for an inclined and a horizontal ground. These results show that small deviations in the verticality of the transmitter loop axis or in the horizontality of the surficial plane causes significant deviations, even for angles as small as 1°.

APA, Harvard, Vancouver, ISO, and other styles

29

Homma, Masaaki. "Fragments of plane filling curves of degree q + 2 over the finite field of q elements, and of affine-plane filling curves of degree q + 1." Linear Algebra and its Applications 589 (March 2020): 9–27. http://dx.doi.org/10.1016/j.laa.2019.12.012.

Full text

APA, Harvard, Vancouver, ISO, and other styles

30

Kobiera, Arkadiusz. "Ellipse, hyperbola and their conjunction." Recreational Mathematics Magazine 5, no. 10 (December 1, 2018): 29–38. http://dx.doi.org/10.2478/rmm-2018-0006.

Full text

Abstract:

Abstract This article presents a simple analysis of cones which are used to generate a given conic curve by section by a plane. It was found that if the given curve is an ellipse, then the locus of vertices of the cones is a hyperbola. The hyperbola has foci which coincidence with the ellipse vertices. Similarly, if the given curve is the hyperbola, the locus of vertex of the cones is the ellipse. In the second case, the foci of the ellipse are located in the hyperbola’s vertices. These two relationships create a kind of conjunction between the ellipse and the hyperbola which originate from the cones used for generation of these curves. The presented conjunction of the ellipse and hyperbola is a perfect example of mathematical beauty which may be shown by the use of very simple geometry. As in the past the conic curves appear to be very interesting and fruitful mathematical beings.

APA, Harvard, Vancouver, ISO, and other styles

31

Gan, Buntara Sthenly, and Ay Lie Han. "Two nodes cusp geometry beam element by using condensed IGA." MATEC Web of Conferences 258 (2019): 05031. http://dx.doi.org/10.1051/matecconf/201925805031.

Full text

Abstract:

A cusp is a curve which is made by projecting a smooth curve in the 3D Euclidean space on a plane. Such a projection results in a curve whose singularities are self-crossing points or ordinary cusps. Self-crossing points created when two different points of the curves have the same projection at a point. Ordinary cusps created when the tangent to the curve is parallel to the direction of projection on a single point. The study of a cusp geometry beam is more complex than that of a straight beam because the structural deformations of the cusp geometry beam depend also on the coupled tangential displacement caused by the singular geometry. The Isogeometric Approach (IGA) is a computational geometry based on a series of polynomial basis functions used to represent the exact geometry. In IGA, the cusp geometry of the beam element can be modeled exactly. A thick cusp geometry beam element can be developed based on the Timoshenko beam theory, which allows the vertical shear deformation and rotatory inertia effects. The shape of the beam geometry and the shape functions formulation of the element can be obtained from IGA. However, in IGA, the number of equations will increase according to the number of degree of freedom (DOF) at the control points. A new condensation method is adopted to reduce the number of equations at the control points so that it becomes a standard two-node 6-DOF beam element. This paper highlights the application of IGA of a cusp geometry Timoshenko beam element in the context of finite element analysis and proposes a new condensation method to eliminate the drawbacks elevated by the conventional IGA. Examples are given to verify the effectiveness of the condensation method in static and free vibration problems.

APA, Harvard, Vancouver, ISO, and other styles

32

López-López, Jorge L. "The area as a natural pseudo-Hermitian structure on the spaces of plane polygons and curves." Differential Geometry and its Applications 28, no. 5 (October 2010): 582–92. http://dx.doi.org/10.1016/j.difgeo.2010.05.004.

Full text

APA, Harvard, Vancouver, ISO, and other styles

33

Guan, Jianyun, and Haiming Liu. "The Sub-Riemannian Limit of Curvatures for Curves and Surfaces and a Gauss-Bonnet Theorem in the Group of Rigid Motions of Minkowski Plane with General Left-Invariant Metric." Journal of Function Spaces 2021 (August 8, 2021): 1–14. http://dx.doi.org/10.1155/2021/1431082.

Full text

Abstract:

The group of rigid motions of the Minkowski plane with a general left-invariant metric is denoted by E 1 , 1 , g λ 1 , λ 2 , where λ 1 ≥ λ 2 > 0 . It provides a natural 2 -parametric deformation family of the Riemannian homogeneous manifold Sol 3 = E 1 , 1 , g 1 , 1 which is the model space to solve geometry in the eight model geometries of Thurston. In this paper, we compute the sub-Riemannian limits of the Gaussian curvature for a Euclidean C 2 -smooth surface in E 1 , 1 , g L λ 1 , λ 2 away from characteristic points and signed geodesic curvature for the Euclidean C 2 -smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the group of rigid motions of the Minkowski plane with a general left-invariant metric.

APA, Harvard, Vancouver, ISO, and other styles

34

Kampczyk, Arkadiusz. "Magnetic-Measuring Square in the Measurement of the Circular Curve of Rail Transport Tracks." Sensors 20, no. 2 (January 20, 2020): 560. http://dx.doi.org/10.3390/s20020560.

Full text

Abstract:

In rail transport, measuring the actual condition of a circular curve of a railway track is a key element of track position monitoring not only during operation but also during final works. Predicting changes in its position in the horizontal plane is one of the most important related scientific issues. This paper presents the results of measurements performed with an innovative measuring device called the Magnetic-Measuring Square (MMS). The aim of the research was to demonstrate the acceptability of using the MMS. Horizontal versines of a rail track curve were measured as three neighboring points on a curve (using the method of lacing/stringlining, also called the three-point or the Hallade method), and the perpendicularity of rail joints and shortenings were measured. The MMS device presented in this article was used to measure versines and differences in rails lengths (rail shortenings in the curve) in the operating mode involving a laser distance meter with a laser beam (laser power P < 1 mW, laser wavelength λ = 635 nm) with a target cross, a camera, and a surveying measuring disk. The measurement results confirmed that it is possible to employ the MMS to monitor the geometry of railway track fragments such as track transition curves and railway track curves in rail transport.

APA, Harvard, Vancouver, ISO, and other styles

35

Krutitskii, P. A. "The Dirichlet problem for the dissipative Helmholtz equation in a plane domain bounded by closed and open curves." Hiroshima Mathematical Journal 28, no. 1 (1998): 149–68. http://dx.doi.org/10.32917/hmj/1206126877.

Full text

APA, Harvard, Vancouver, ISO, and other styles

36

Dub, Sergey N., Cetin Haftaoglu, and Vitaliy M. Kindrachuk. "Estimate of theoretical shear strength of C60 single crystal by nanoindentation." Journal of Materials Science 56, no. 18 (March 22, 2021): 10905–14. http://dx.doi.org/10.1007/s10853-021-05991-2.

Full text

Abstract:

AbstractThe onset of plasticity in a single crystal C60 fullerite was investigated by nanoindentation on the (111) crystallographic plane. The transition from elastic to plastic deformation in a contact was observed as pop-in events on loading curves. The respective resolved shear stresses were computed for the octahedral slip systems $$\langle{01}\overline{1}\rangle\left\{ {{111}} \right\}$$ ⟨ 01 1 ¯ ⟩ 111 , supposing that their activation resulted in the onset of plasticity. A finite element analysis was applied, which reproduced the elastic loading until the first pop-in, using a realistic geometry of the Berkovich indenter blunt tip. The obtained estimate of the C60 theoretical shear strength was about $${1}/{11}$$ 1 / 11 of the shear modulus on {111} planes. Graphical abstract

APA, Harvard, Vancouver, ISO, and other styles

37

YONEDA, KEISHI, AKIO YONEZU, HIROYUKI HIRAKATA, and KOHJI MINOSHIMA. "ESTIMATION OF ANISOTROPIC PLASTIC PROPERTIES OF ENGINEERING STEELS FROM SPHERICAL IMPRESSIONS." International Journal of Applied Mechanics 02, no. 02 (June 2010): 355–79. http://dx.doi.org/10.1142/s1758825110000536.

Full text

Abstract:

This study proposes a method of reverse analysis to estimate the anisotropic plastic properties of engineering steels by spherical indentation. The method takes into consideration materials that obey the work-hardening law and show in-plane anisotropic yield stress. Finite element analysis was first carried out to compute the indentation behavior of such materials, showing that a permanent impression exhibited an anisotropic shape which was strongly dependent on the orthotropic axis. Based on the anisotropy of the impression geometry, we developed a simple approach to determine the yield stress, work-hardening exponent and yield stress ratio. The approach consists of several functions related to the parameters of two impression geometries, produced by dual spherical indentations with different indentation forces. Since the present method uses only two impression geometries and does not necessitate indentation force — displacement curves (indentation curves), it is a particularly useful technique to evaluate "indistinguishable materials" which are special sets of materials with distinct plastic properties, yet yield almost identical indentation curves.

APA, Harvard, Vancouver, ISO, and other styles

38

Karbassi, Amin, and Pierino Lestuzzi. "Fragility Analysis of Existing Unreinforced Masonry Buildings through a Numerical-based Methodology." Open Civil Engineering Journal 6, no. 1 (November 16, 2012): 121–30. http://dx.doi.org/10.2174/1874149501206010121.

Full text

Abstract:

As an approach to the problem of seismic vulnerability evaluation of existing buildings using the predicted vul-nerability method, numerical models can be applied to define fragility curves of typical buildings which represent building classes. These curves can be then combined with the seismic hazard to calculate the seismic risk for a building class (or individual buildings). For some buildings types, mainly the unreinforced masonry structures, such fragility analysis is complicated and time consuming if a Finite Element-based method is used. The FEM model has to represent the structural geometry and relationships between different structural elements through element connectivity. Moreover, the FEM can face major challenges to represent large displacements and separations for progressive collapse simulations. Therefore, the Applied Element Method which combines the advantages of FEM with that of the Discrete Element Method in terms of accurately modelling a deformable continuum of discrete materials is used in this paper to perform the fragility analysis for unreinforced masonry buildings. To this end, a series of nonlinear dynamic analyses using the AEM has been per-formed for two unreinforced masonry buildings (a 6-storey stone masonry and a 4-storey brick masonry) using more than 50 ground motion records. Both in-plane and out-of-plane failure have been considered in the damage analysis. The dis-tribution of the structural responses and inter-storey drifts are used to develop spectral-based fragility curves for the five European Macroseismic Scale damage grades.

APA, Harvard, Vancouver, ISO, and other styles

39

Li, Guochao, Jiao Liu, Honggen Zhou, Guizhong Tian, and Lei Li. "Modeling and analysis of variable-core groove for integral cutting tools." Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 232, no. 4 (July 4, 2017): 471–79. http://dx.doi.org/10.1177/0954408917718084.

Full text

Abstract:

With good stiffness, the helical groove with variable-core radius has been widely used for end mills, drills and other integral cutting tools. However, ground by five-axis grinder, its machining process is high cost and time consuming. Thus, this paper reports a graphical analysis method to obtain the structure parameters and geometric shapes of variable-core grooves with the known wheel geometry and position before the practical machining. The wheel movement during the machining process is firstly decomposed into three simple motions and modeled as a translation matrix. Then, a family of wheel surfaces is calculated and the groove cross section line is deduced by splitting the surfaces with a cross section plane. Accordingly, the normal section line of the groove is expressed by a series of scattered points, which are the intersections of the normal section plane and the helical curves generated by the points on the groove cross section line which moves along the wheel trajectory. All the mathematic models are programed by Matlab and verified by experiment. Finally, the key parameters of the variable-core groove, including the rake angle, groove width on the cross and normal sections and the departure distance between the cross and normal groove section lines, are analyzed from the tool bit to the hilt.

APA, Harvard, Vancouver, ISO, and other styles

40

Mostafa, Yasser E., and M. Hesham El Naggar. "Dynamic analysis of laterally loaded pile groups in sand and clay." Canadian Geotechnical Journal 39, no. 6 (December 1, 2002): 1358–83. http://dx.doi.org/10.1139/t02-102.

Full text

Abstract:

Pile foundations supporting bridge piers, offshore platforms, and marine structures are required to resist not only static loading but also lateral dynamic loading. The static py curves are widely used to relate pile deflections to nonlinear soil reactions. The p-multiplier concept is used to account for the group effect by relating the load transfer curves of a pile in a group to the load transfer curves of a single pile. Some studies have examined the validity of the p-multiplier concept for the static and cyclic loading cases. However, the concept of the p-multiplier has not yet been considered for the dynamic loading case, and hence it is undertaken in the current study. An analysis of the dynamic lateral response of pile groups is described. The proposed analysis incorporates the static py curve approach and the plane strain assumptions to represent the soil reactions within the framework of a Winkler model. The model accounts for the nonlinear behaviour of the soil, the energy dissipation through the soil, and the pile group effect. The model was validated by analyzing the response of pile groups subjected to lateral Statnamic loading and comparing the results with field measured values. An intensive parametric study was performed employing the proposed analysis, and the results were used to establish dynamic soil reactions for single piles and pile groups for different types of sand and clay under harmonic loading with varying frequencies applied at the pile head. "Dynamic" p-multipliers were established to relate the dynamic load transfer curves of a pile in a group to the dynamic load transfer curves for a single pile. The dynamic p-multipliers were found to vary with the spacing between piles, soil type, peak amplitude of loading, and the angle between the line connecting any two piles and the direction of loading. The study indicated the effect of pile material and geometry, pile installation method, and pile head conditions on the p-multipliers. The calculated p-multipliers compared well with p-multipliers back-calculated from full scale field tests.Key words: lateral, transient loading, nonlinear, pilesoilpile interaction, py curves, Statnamic.

APA, Harvard, Vancouver, ISO, and other styles

41

Zhao, Chun-jiang, Meng-ying Su, Zheng-yi Jiang, Jiang Lian-yun, Xiaorong Yang, and Hai-long Cui. "Three-directional contact force model for the ball spinning of a thin-walled tube." Proceedings of the Institution of Mechanical Engineers, Part E: Journal of Process Mechanical Engineering 233, no. 3 (April 18, 2018): 500–507. http://dx.doi.org/10.1177/0954408918769431.

Full text

Abstract:

This paper provides a computational model for calculating three-directional ball spinning force in accordance with the theory of space analytic geometry. The contact boundary equation of the ball and tube is obtained. By projection, the two-dimensional curve in each coordinate plane is acquired. The projected area of the contact zone in the coordinate plane is calculated through the curve integral. It is assumed that the average pressure of the forming region is nearly equal to that when the steel ball is pressed into the tube. Hence, the unit pressure of the deformation zone is obtained. Then, the spinning component force and total spinning force are calculated. Using a Tu1 thin-walled tube of oxygen-free copper as experimental object, a ball spinning experiment is conducted, the axial spinning components force are tested and the ball spinning force calculation model is verified. Based on deformation rate, backward sliding accumulation and extension and frictional heating, the factors influencing calculation error are analysed at the end of this paper.

APA, Harvard, Vancouver, ISO, and other styles

42

Ma, Bole, and Yongsheng Ren. "Nonlinear Dynamic Analysis of the Cutting Process of a Nonextensible Composite Boring Bar." Shock and Vibration 2020 (October 28, 2020): 1–13. http://dx.doi.org/10.1155/2020/5971540.

Full text

Abstract:

A nonlinear dynamic analysis of the cutting process of a nonextensible composite cutting bar is presented. The cutting bar is simplified as a cantilever with plane bending. The nonlinearity is mainly originated from the nonextensible assumption, and the material of cutting bar is assumed to be viscoelastic composite, which is described by the Kelvin–Voigt equation. The motion equation of nonlinear chatter of the cutting system is derived based on the Hamilton principle. The partial differential equation of motion is discretized using the Galerkin method to obtain a 1-dof nonlinear ordinary differential equation in a generalized coordinate system. The steady forced response of the cutting system under periodically varying cutting force is approximately solved by the multiscale method. Meanwhile, the effects of parameters such as the geometry of the cutting bar (including length and diameter), damping, the cutting coefficient, the cutting depth, the number of the cutting teeth, the amplitude of the cutting force, and the ply angle on nonlinear lobes and primary resonance curves during the cutting process are investigated using numerical calculations. The results demonstrate that the critical cutting depth is inversely proportional to the aspect ratio of the cutting bar and the cutting force coefficient. Meanwhile, the chatter stability in the milling process can be significantly enhanced by increasing the structural damping. The peak of the primary resonance curve is bent toward the right side. Due to the cubic nonlinearity in the cutting system, primary resonance curves show the characteristics of typical Duffing’s vibrator with hard spring, and jump and multivalue regions appear.

APA, Harvard, Vancouver, ISO, and other styles

43

Ramaraj, T. C., and E. Eleftheriou. "Analysis of the Mechanics of Machining with Tapered End Milling Cutters." Journal of Engineering for Industry 116, no. 3 (August 1, 1994): 398–404. http://dx.doi.org/10.1115/1.2901958.

Full text

Abstract:

The analytical prediction of forces in a tapered mill is of vital interest to the accurate machining of complex blade shapes for the aerospace industry. Failure of these cutters is more often by chipping and fracture as opposed to gradual wear. Geometry of the cutters along with duration and length during cutting is analyzed using known parameters. The influence of the geometrical parameters on the spatial distribution of cutting forces is derived from kinematics and the basic shearing processes. Shear strain is approximately obtained with the use of Stabler’s rule. Stress on the shear plane is found from appropriate shear stress-strain curves. A coefficient of friction on the cutting edge is used to evaluate the shear as well as normal forces. As a consequences, torque experienced by the cutter is evaluated. Finally, verification of the analysis is presented through sample calculations and correlations with experimental data measured with a dynamometer.

APA, Harvard, Vancouver, ISO, and other styles

44

LIZZI, FEDELE, and BERNARDINO SPISSO. "NONCOMMUTATIVE FIELD THEORY: NUMERICAL ANALYSIS WITH THE FUZZY DISK." International Journal of Modern Physics A 27, no. 24 (September 28, 2012): 1250137. http://dx.doi.org/10.1142/s0217751x12501370.

Full text

Abstract:

The fuzzy disk is a discretization of the algebra of functions on the two-dimensional disk using finite matrices which preserves the action of the rotation group. We define a φ4 scalar field theory on it and analyze numerically three different limits for the rank of the matrix going to infinity. The numerical simulations reveal three different phases: uniform and disordered phases already present in the commutative scalar field theory and a nonuniform ordered phase as noncommutative effects. We have computed the transition curves between phases and their scaling. This is in agreement with studies on the fuzzy sphere, although the speed of convergence for the disk seems to be better. We have also performed the limits for the theory in the cases of the theory going to the commutative plane or commutative disk. In this case the theory behaves differently, showing the intimate relationship between the nonuniform phase and noncommutative geometry.

APA, Harvard, Vancouver, ISO, and other styles

45

Maré, George R. De. "Abinitio study of rotational isomerism in acrolein." Canadian Journal of Chemistry 63, no. 7 (July 1, 1985): 1672–80. http://dx.doi.org/10.1139/v85-280.

Full text

Abstract:

Analytic gradient (force) methods at the STO-3G, 3-21G, and 6-31G* basis set levels have been used to optimize the geometry of acrolein completely at each critical point (minima, maximum) in the torsional potential energy curves for rotation about the single C—C bond (dihedral angle θ). The STO-3G and 6-31G* optimizations predict the planar trans conformation (θ = 180°) to be more stable than the cis conformation (θ = 0°) by 1.87 and 6.97 kJ/mol, respectively. The 3-21G optimizations, in disagreement with experiment, place the planar cis structure below the trans by 4.5 J/mol. The predicted relative energy (ΔE) and position for the transition state (TS) for rotation from the trans conformer are ΔE = 22.35, 37.14, and 34.41 kJ/mol and θ = 91.8, 91.6, and 91.0° for the STO-3G, 3-21G, and 6-31G* optimizations, respectively. The computed and experimental geometries, relative energies, dipole moments, and coefficients for the torsional potential expansion are compared.

APA, Harvard, Vancouver, ISO, and other styles

46

Dias, Fabio Scalco, and Luis Fernando Mello. "Geometry of plane curves." Bulletin des Sciences Mathématiques 135, no. 4 (June 2011): 333–44. http://dx.doi.org/10.1016/j.bulsci.2011.03.007.

Full text

APA, Harvard, Vancouver, ISO, and other styles

47

Sickafoose, A. A., A. S. Bosh, J. P. Emery, M. J. Person, C. A. Zuluaga, M. Womack, J. D. Ruprecht, F. B. Bianco, and A. M. Zangari. "Characterization of material around the centaur (2060) Chiron from a visible and near-infrared stellar occultation in 2011." Monthly Notices of the Royal Astronomical Society 491, no. 3 (November 12, 2019): 3643–54. http://dx.doi.org/10.1093/mnras/stz3079.

Full text

Abstract:

ABSTRACT The centaur (2060) Chiron exhibits outgassing behaviour and possibly hosts a ring system. On 2011 November 29, Chiron occulted a fairly bright star (R ∼ 15 mag) as observed from the 3-m NASA Infrared Telescope Facility (IRTF) on Mauna Kea and the 2-m Faulkes Telescope North (FTN) at Haleakala. Data were taken as visible wavelength images and simultaneous, low-resolution, near-infrared (NIR) spectra. Here, we present a detailed examination of the light-curve features in the optical data and an analysis of the NIR spectra. We place a lower limit on the spherical diameter of Chiron's nucleus of 160.2 ± 1.3 km. Sharp, narrow dips were observed between 280 and 360 km from the centre (depending on event geometry). For a central chord and assumed ring plane, the separated features are 298.5–302 and 308–310.5 km from the nucleus, with normal optical depth ∼0.5–0.9, and a gap of 9.1 ± 1.3 km. These features are similar in equivalent depth to Chariklo's inner ring. The absence of absorbing/scattering material near the nucleus suggests that these sharp dips are more likely to be planar rings than a shell of material. The region of relatively increased transmission is within the 1:2 spin-orbit resonance, consistent with the proposed clearing pattern for a non-axisymmetric nucleus. Characteristics of possible azimuthally incomplete features are presented, which could be transient, as well as a possible shell from ∼900–1500 km: future observations are needed for confirmation. There are no significant features in the NIR light curves, nor any correlation between optical features and NIR spectral slope.

APA, Harvard, Vancouver, ISO, and other styles

48

Georgiev, Georgi Hristov, Radostina Petrova Encheva, and Cvetelina Lachezarova Dinkova. "Geometry of cylindrical curves over plane curves." Applied Mathematical Sciences 9 (2015): 5637–49. http://dx.doi.org/10.12988/ams.2015.56456.

Full text

APA, Harvard, Vancouver, ISO, and other styles

49

Ali, Habboush, Sanbhal Noor, Shao Huiqi, Jiang Jinhua, and Chen Nanliang. "Characterization and analysis of wrinkling behavior of glass warp knitted non-crimp fabrics based on double-dome draping geometry." Journal of Engineered Fibers and Fabrics 15 (January 2020): 155892502095852. http://dx.doi.org/10.1177/1558925020958521.

Full text

Abstract:

The good formability of textile composite materials over complex mold geometries is one of the reasons to make their use expanding in various modern industries. However, different defects in these reinforcements could have occurred during the forming step in the manufacturing process. The defects are arising for many reasons; some are related to the fabric itself and others related to the draping parameters. Understanding the textile structure mechanics and draping behavior is essential to choose the proper reinforcement as well as to attain better simulation. Fabric wrinkles and local out-of-plane bucking of yarns were the fundamental defects in focus. The main objective of this part of the project was to experimentally investigate and compare the draping behavior of six commercially available glass fabrics from the same category of warp-knitted non-crimp fabrics (WKNCFs). The tested fabrics included two stitching patterns: tricot and chain. Also, they were relatively heavy with approximate mass per square meter. A double-dome punching test was performed to implement draping for each fabric; then, the defects were detected and characterized. Punching load-displacement curves were also recorded. In addition, a defect code was designated for the main defects to characterize forming defects at the meso-macroscopic scale. The structure and the number of fabric axes, stacking sequence, and stitching pattern all contribute to defect formation during draping. The studied configurations in this paper can help in studying the simulation of deformed technical fabric and provide a method to minimize and even eliminate the draping defects.

APA, Harvard, Vancouver, ISO, and other styles

50

IITAKA, Shigeru. "Birational Geometry of Plane Curves." Tokyo Journal of Mathematics 22, no. 2 (December 1999): 289–321. http://dx.doi.org/10.3836/tjm/1270041440.

Full text

APA, Harvard, Vancouver, ISO, and other styles

Journal articles on the topic 'Geometry, Analytic. Curves, Plane'

Create a spot-on reference in APA, MLA, Chicago, Harvard, and other styles