Log in

Relevant bibliographies by topics / Geometry, Projective. Curves, Plane / Journal articles

To see the other types of publications on this topic, follow the link: Geometry, Projective. Curves, Plane.

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

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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

Select a source type:

Consult the top 50 journal articles for your research on the topic 'Geometry, Projective. Curves, Plane.'

Next to every source in the list of references, there is an 'Add to bibliography' button. Press on it, and we will generate automatically the bibliographic reference to the chosen work in the citation style you need: APA, MLA, Harvard, Chicago, Vancouver, etc.

You can also download the full text of the academic publication as pdf and read online its abstract whenever available in the metadata.

Browse journal articles on a wide variety of disciplines and organise your bibliography correctly.

1

DE BOBADILLA, J. FERNÁNDEZ, I. LUENGO-VELASCO, A. MELLE-HERNÁNDEZ, and A. NÉMETHI. "ON RATIONAL CUSPIDAL PROJECTIVE PLANE CURVES." Proceedings of the London Mathematical Society 92, no. 1 (December 19, 2005): 99–138. http://dx.doi.org/10.1017/s0024611505015467.

Full text

Abstract:

In 2002, L. Nicolaescu and the fourth author formulated a very general conjecture which relates the geometric genus of a Gorenstein surface singularity with rational homology sphere link with the Seiberg--Witten invariant (or one of its candidates) of the link. Recently, the last three authors found some counterexamples using superisolated singularities. The theory of superisolated hypersurface singularities with rational homology sphere link is equivalent with the theory of rational cuspidal projective plane curves. In the case when the corresponding curve has only one singular point one knows no counterexample. In fact, in this case the above Seiberg--Witten conjecture led us to a very interesting and deep set of `compatibility properties' of these curves (generalising the Seiberg--Witten invariant conjecture, but sitting deeply in algebraic geometry) which seems to generalise some other famous conjectures and properties as well (for example, the Noether--Nagata or the log Bogomolov--Miyaoka--Yau inequalities). Namely, we provide a set of `compatibility conditions' which conjecturally is satisfied by a local embedded topological type of a germ of plane curve singularity and an integer $d$ if and only if the germ can be realized as the unique singular point of a rational unicuspidal projective plane curve of degree $d$. The conjectured compatibility properties have a weaker version too, valid for any rational cuspidal curve with more than one singular point. The goal of the present article is to formulate these conjectured properties, and to verify them in all the situations when the logarithmic Kodaira dimension of the complement of the corresponding plane curves is strictly less than 2.

APA, Harvard, Vancouver, ISO, and other styles

2

Malara, Grzegorz, Piotr Pokora, and Halszka Tutaj-Gasińska. "On 3-syzygy and unexpected plane curves." Geometriae Dedicata 214, no. 1 (February 4, 2021): 49–63. http://dx.doi.org/10.1007/s10711-021-00602-5.

Full text

Abstract:

AbstractIn this note we study curves (arrangements) in the complex projective plane which can be considered as generalizations of free curves. We construct families of arrangements which are nearly free and possess interesting geometric properties. More generally, we study 3-syzygy curve arrangements and we present examples that admit unexpected curves.

APA, Harvard, Vancouver, ISO, and other styles

3

Kamel, Alwaleed, and Waleed Khaled Elshareef. "Weierstrass points of order three on smooth quartic curves." Journal of Algebra and Its Applications 18, no. 01 (January 2019): 1950020. http://dx.doi.org/10.1142/s0219498819500208.

Full text

Abstract:

In this paper, we study the [Formula: see text]-Weierstrass points on smooth projective plane quartic curves and investigate their geometry. Moreover, we use a technique to determine in a very precise way the distribution of such points on any smooth projective plane quartic curve. We also give a variety of examples that illustrate and enrich the subject.

APA, Harvard, Vancouver, ISO, and other styles

4

Uchino, K. "Arnold's Projective Plane and -Matrices." Advances in Mathematical Physics 2010 (2010): 1–9. http://dx.doi.org/10.1155/2010/956128.

Full text

Abstract:

We will explain Arnold's 2-dimensional (shortly, 2D) projective geometry (Arnold, 2005) by means of lattice theory. It will be shown that the projection of the set of nontrivial triangular -matrices is the pencil of tangent lines of a quadratic curve on Arnold's projective plane.

APA, Harvard, Vancouver, ISO, and other styles

5

Taflin, Johan. "Invariant Elliptic Curves as Attractors in the Projective Plane." Journal of Geometric Analysis 20, no. 1 (August 18, 2009): 219–25. http://dx.doi.org/10.1007/s12220-009-9104-9.

Full text

APA, Harvard, Vancouver, ISO, and other styles

6

Pignoni, Roberto. "Integral relations for pointed curves in a real projective plane." Geometriae Dedicata 45, no. 3 (March 1993): 263–87. http://dx.doi.org/10.1007/bf01277967.

Full text

APA, Harvard, Vancouver, ISO, and other styles

7

ROUYER, JOËL. "A CHARACTERIZATION OF THE REAL PROJECTIVE PLANE." International Journal of Mathematics 21, no. 12 (December 2010): 1605–17. http://dx.doi.org/10.1142/s0129167x10006653.

Full text

Abstract:

It is proved in this article, that in the framework of Riemannian geometry, the existence of large sets of antipodes (i.e. farthest points) for diametral points of a smooth surface has very strong consequences on the topology and the metric of this surface. Roughly speaking, if the sets of antipodes of diametral points are closed curves, then the surface is nothing but the real projective plane.

APA, Harvard, Vancouver, ISO, and other styles

8

WALL, C. T. C. "Geometry of projection-generic space curves." Mathematical Proceedings of the Cambridge Philosophical Society 147, no. 1 (July 2009): 115–42. http://dx.doi.org/10.1017/s0305004108002168.

Full text

Abstract:

AbstractIn earlier work I defined a class of curves, forming a dense open set in the space of maps from S1 to P3, such that the family of projections of a curve in this class is stable under perturbations of C: we call the curves in the class projection-generic. The definition makes sense also in the complex case. The partition of projective space according to the singularities of the corresponding projection of C is a stratification. Its local structure outside C is the same as that of the versal unfoldings of the singularities presented.To study points on C we introduce the blow-up BC of P3 along C, and a family of plane curves, parametrised by z ∈ BC; we saw in the earlier work that this is a flat family.Here we show that near most z ∈ BC, the family gives a family of parametrised germs which versally unfolds the singularities occurring. Otherwise we find that the double point number δ of Γz drops by 1 for z ∉ EC. We establish a theory of versality for unfoldings of A or D singularities such that δ drops by at most 1, and show that in the remaining cases, we have an unfolding which is versal in this sense.This implies normal forms for the stratification of BC; further work allows us to derive local normal forms for strata of the stratification of P3.

APA, Harvard, Vancouver, ISO, and other styles

9

TSUKAMOTO, MASAKI. "Deformation of Brody curves and mean dimension." Ergodic Theory and Dynamical Systems 29, no. 5 (February 3, 2009): 1641–57. http://dx.doi.org/10.1017/s014338570800076x.

Full text

Abstract:

AbstractThe main purpose of this paper is to show that ideas of deformation theory can be applied to ‘infinite-dimensional geometry’. We develop the deformation theory of Brody curves. A Brody curve is a kind of holomorphic map from the complex plane to the projective space. Since the complex plane is not compact, the parameter space of the deformation can be infinite-dimensional. As an application we prove a lower bound on the mean dimension of the space of Brody curves.

APA, Harvard, Vancouver, ISO, and other styles

10

Wennink, Thomas. "Counting the number of trigonal curves of genus 5 over finite fields." Geometriae Dedicata 208, no. 1 (January 9, 2020): 31–48. http://dx.doi.org/10.1007/s10711-019-00508-3.

Full text

Abstract:

AbstractThe trigonal curves of genus 5 can be represented by projective plane quintics that have one singularity of delta invariant one. Combining this with a partial sieve method for plane curves we count the number of such curves over any finite field. The main application is that this gives the motivic Euler characteristic of the moduli space of trigonal curves of genus 5.

APA, Harvard, Vancouver, ISO, and other styles

11

Greuel, Gert-Martin, Christoph Lossen, and Eugenii Shustin. "Geometry of families of nodal curves on the blown-up projective plane." Transactions of the American Mathematical Society 350, no. 1 (1998): 251–74. http://dx.doi.org/10.1090/s0002-9947-98-02055-8.

Full text

APA, Harvard, Vancouver, ISO, and other styles

12

LANGE, H., and E. SERNESI. "SEVERI VARIETIES AND BRANCH CURVES OF ABELIAN SURFACES OF TYPE (1, 3)." International Journal of Mathematics 13, no. 03 (May 2002): 227–44. http://dx.doi.org/10.1142/s0129167x02001381.

Full text

Abstract:

A polarized abelian surface (A, L) of type (1, 3) induces a 6:1 covering of A onto the projective plane with branch curve, a plane curve B of degree 18. The main result of the paper is that for a general abelian surface of type (1, 3), the curve B is irreducible and reduced and admits 72 cusps, 36 nodes or tacnodes, each tacnode counting as 2 nodes, 72 flexes and 36 bitangents. The main idea of the proof is that for a general (A, L) the discriminant curve in the linear system |L| coincides with the closure of the Severi variety of curves in |L| admitting a node and is dual to the curve B in the sense of projective geometry.

APA, Harvard, Vancouver, ISO, and other styles

13

FRIEDMAN, MICHAEL, MAXIM LEYENSON, and EUGENII SHUSTIN. "ON RAMIFIED COVERS OF THE PROJECTIVE PLANE I: INTERPRETING SEGRE'S THEORY (WITH AN APPENDIX BY EUGENII SHUSTIN)." International Journal of Mathematics 22, no. 05 (May 2011): 619–53. http://dx.doi.org/10.1142/s0129167x11006945.

Full text

Abstract:

We study ramified covers of the projective plane ℙ2. Given a smooth surface S in ℙN and a generic enough projection ℙN → ℙ2, we get a cover π: S → ℙ2, which is ramified over a plane curve B. The curve B is usually singular, but is classically known to have only cusps and nodes as singularities for a generic projection. The main question that arises is with respect to the geometry of branch curves; i.e. how can one distinguish a branch curve from a non-branch curve with the same numerical invariants? For example, a plane sextic with six cusps is known to be a branch curve of a generic projection iff its six cusps lie on a conic curve, i.e. form a special 0-cycle on the plane. The classical work of Beniamino Segre gives a complete answer to the second question in the case when S is a smooth surface in ℙ3. We give an interpretation of the work of Segre in terms of relation between Picard and Chow groups of 0-cycles on a singular plane curve B. In addition, the appendix written by E. Shustin shows the existence of new Zariski pairs.

APA, Harvard, Vancouver, ISO, and other styles

14

Cheng, Bin, Su Liu, Yu Sun, and Jie Zhang. "The Intersection and its Application for Non-Circular Quadric Curved Surface Based on the Theory of Projective Correspondence." Applied Mechanics and Materials 389 (August 2013): 860–65. http://dx.doi.org/10.4028/www.scientific.net/amm.389.860.

Full text

Abstract:

Based on the theory of projective correspondence, the principle of affine correspondence and projective correspondence is proposed to resolve the intersection of non-circular quadric curved surface (NCQCS). According to the theory of projective geometry, the intersection that NCQCS intersect with a set of parallel planes is a group of homothetic curves. Taking the homothetic curves as corresponding element, spatial projective correspondence between any two homothetic curves is constructed. Similarly, the intersection that any NCQCS intersected by a set of planes through it axis is a group of similar curves. Taking the similar curves as associated element, spatial affine correspondence between any two similar curves is constructed. The intersection and drawing theory, method, application and technology for the NCQCS provides the theoretical and practice support for the intersection of the NCQCS.

APA, Harvard, Vancouver, ISO, and other styles

15

Bertrand, Benoît, Erwan Brugallé, and Grigory Mikhalkin. "Genus 0 characteristic numbers of the tropical projective plane." Compositio Mathematica 150, no. 1 (November 18, 2013): 46–104. http://dx.doi.org/10.1112/s0010437x13007409.

Full text

Abstract:

AbstractFinding the so-called characteristic numbers of the complex projective plane$ \mathbb{C} {P}^{2} $is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given$d$and$g$one has to find the number of degree$d$genus$g$curves that pass through a certain generic configuration of points and at the same time are tangent to a certain generic configuration of lines. The total number of points and lines in these two configurations is$3d- 1+ g$so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when$g= 0$. Namely, we show that the tropical problem is well posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we express genus 0 characteristic numbers of$ \mathbb{C} {P}^{2} $in terms of open Hurwitz numbers.

APA, Harvard, Vancouver, ISO, and other styles

16

Ivashchenko, Andrey Viktorovich, and Tat’yana Mikhaylovna Kondrat’eva. "Projective configurations in projectivegeometrical drawings." Vestnik MGSU, no. 5 (May 2015): 141–47. http://dx.doi.org/10.22227/1997-0935.2015.5.141-147.

Full text

Abstract:

The article focuses on the optimization of the earlier discussed computer method of obtaining new forms of polyhedra based on projective geometry drawings (trace Dia- grams).While working on getting new multifaceted forms by projective geometry methods based on the well-known models of polyhedra on the first stage of the work it is required to calculate the parameters of projective geometry drawings, and then to build them. This is an often used apparatus of analytical geometry. According to it, at first the parameters of the polyhedron (core system of planes) are calculated, then we obtain the equation of the plane of the face of the polyhedron, and finally we obtain the equations of lines - the next plane faces on the selected curve plane. At each stage of application such a method requires the use of the algorithms of floating point arithmetic, on the one hand, leads to some loss of accuracy of the results and, on the other hand, the large amount of com- puter time to perform these operations in comparison with integer arithmetic operations.The proposed method is based on the laws existing between the lines that make up the drawing - the known configurations of projective geometry (complete quadrilaterals, configuration of Desargues, Pappus et al.).The authors discussed in detail the analysis procedure of projective geometry draw- ing and the presence of full quadrilaterals, Desargues and Pappus configurations in it.Since the composition of these configurations is invariant with respect to projec- tive change of the original nucleus, knowing them, you can avoid the calculations when solving the equations for finding direct projective geometry drawing analytically, getting them on the basis of belonging to a particular configuration. So you can get a definite advantage in accuracy of the results, and in the cost of computer time. Finding these basic configurations significantly enriches the set of methods and the use of projective geometry drawings.

APA, Harvard, Vancouver, ISO, and other styles

17

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

18

Sierra, José Carlos. "The smooth surfaces in ℙ4 with few apparent triple points." Communications in Contemporary Mathematics 18, no. 01 (January 29, 2016): 1550013. http://dx.doi.org/10.1142/s0219199715500133.

Full text

Abstract:

We classify smooth complex projective surfaces in [Formula: see text] with [Formula: see text] apparent triple points, thus recovering and extending the results of Ascione [Sulle superficie immerse in un [Formula: see text], le cui trisecanti costituiscono complessi di [Formula: see text] ordine, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur.[Formula: see text]5[Formula: see text] 6 (1897) 162–169] and Severi [Intorno ai punti doppi impropri di una superficie generale dello spazio a quattro dimensioni, e a’ suoi punti tripli apparenti, Rend. Circ. Mat. Palermo 15 (1901) 33–51] for [Formula: see text], Marletta [Le superficie generali dell’ [Formula: see text] dotate di due punti tripli apparenti, Rend. Circ. Mat. Palermo 34 (1912) 179–186] for [Formula: see text], and Aure [The smooth surfaces in [Formula: see text] without apparent triple points, Duke Math. J. 57 (1988) 423–430] for [Formula: see text]. This is done thanks to a new projective character that can be introduced as a consequence of the main result of [K. Ranestad, On smooth plane curve fibrations in [Formula: see text], in Geometry of Complex Projective Varieties, Sem. Conf., Vol. 9 (Mediterranean, 1993), pp. 243–255; J. C. Sierra and A. L. Tironi, Some remarks on surfaces in [Formula: see text] containing a family of plane curves, J. Pure Appl. Algebra 209 (2007) 361–369; V. Beorchia and G. Sacchiero, Surfaces in [Formula: see text] with a family of plane curves, J. Pure Appl. Algebra 213 (2009) 1750–1755]. Going a bit further, we obtain some bounds on the Euler characteristic [Formula: see text] in terms of the degree [Formula: see text] and the sectional genus [Formula: see text] of a smooth surface in [Formula: see text]. For [Formula: see text], these bounds were first obtained in [A. B. Aure and K. Ranestad, The smooth surfaces of degree [Formula: see text] in [Formula: see text], in Complex Projective Geometry, London Mathematical Society Lecture Note Series, Vol. 179 (Cambridge University Press, Cambridge, 1992), pp. 32–46; K. Ranestad, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Ph.D. thesis, Oslo (1988); S. Popescu, On smooth surfaces of degree [Formula: see text] in the projective fourspace, Dissertation, Saarbrücken (1993)]. Here we give a different argument based on liaison that works also for [Formula: see text] and that allows us to determine the triples [Formula: see text] of the smooth surfaces with [Formula: see text] apparent triple points.

APA, Harvard, Vancouver, ISO, and other styles

19

Yamanoi, Katsutoshi. "Holomorphic curves in algebraic varieties of maximal albanese dimension." International Journal of Mathematics 26, no. 06 (June 2015): 1541006. http://dx.doi.org/10.1142/s0129167x15410062.

Full text

Abstract:

We prove a second main theorem type estimate in Nevanlinna theory for holomorphic curves f : Y → X from finite covering spaces Y → ℂ of the complex plane ℂ into complex projective manifolds X of maximal albanese dimension. If X is moreover of general type, then this implies that the special set of X is a proper subset of X. For a projective curve C in such X, our estimate also yields an upper bound of the ratio of the degree of C to the geometric genus of C, provided that C is not contained in a proper exceptional subset in X.

APA, Harvard, Vancouver, ISO, and other styles

20

Короткий and Viktor Korotkiy. "Plane Fields’ Quadratic Cremona Correspondence Set By Imaginary F-Points." Geometry & Graphics 5, no. 1 (April 17, 2017): 21–31. http://dx.doi.org/10.12737/25120.

Full text

Abstract:

Birational (Cremona) correspondences of two planes П, П' or Cremona transformations on the combined plane П = П' represent an effective tool for design of smooth dynamic curves and two-dimensional lines. The simplest birational correspondence is a quadratic map Ω of plane fields on one another. In the projective definition of the quadratic correspondence can participate two pairs of imaginary complex conjugate F-points set as double points of elliptic involutions on the lines associated with the third pair of F-points. In this case, the imaginary projective F-bundles cannot be used for generation of points corresponding in Ω. A generic constructive algorithm for design of corresponding points in a quadratic mapping Ω(П ↔ П'), set both by real and imaginary F-points is proposed in this paper. The algorithm is based on the use of auxiliary projective correspondence Δ between the points of the planes П, П' and Hirst’s transformation Ψ with the center in the one of real F-points. A theorem on the existence of an invariant conic common to Ω and Δ mappings has been proved. Has been demonstrated a possibility for quadratic mapping’s presentation as a product of collinearity and Hirst’s transformation: Ω = ΔΨ. Has been considered a solution for auxiliary problems arising during the generic constructive algorithm’s implementation: buildup a conic section, that is incident to imaginary points, and plotting the corresponding points in collinearity set by imaginary points. It has been demonstrated that there are two or four possible versions of collinearity for plane fields П, П', set by with participation of the imaginary corresponding points, due to an uncertainty related to the order of their relative correspondence. Have been completely solved the problem of mapping a straight line in a conic section within the quadratic Cremona correspondance of fields П ≠ П', set by a pair of real fundamental points, and two pairs of imaginary ones. It has been demonstrated that in general case the problem has two solutions. The obtained results are useful for the development of the geometric theory related to imaginary elements and this theory’s application in linear and non-linear descriptive geometry, operating projective images of first and second orders.

APA, Harvard, Vancouver, ISO, and other styles

21

Kopacz, Piotr. "ON GEOMETRIC PROPERTIES OF SPHERICAL CONICS AND GENERALIZATION OF Π IN NAVIGATION AND MAPPING." Geodesy and Cartography 38, no. 4 (December 21, 2012): 141–51. http://dx.doi.org/10.3846/20296991.2012.756995.

Full text

Abstract:

First, we cover the conical curves on 2-dimensional modeling sphere S 2 showing their geometric properties affecting the hyperbolic navigation. We place emphasis on the geometric definition of spherical parabola and relate it to the notions of spherical ellipse and hyperbola and give simple geometric proofs for relations between conical curves on the sphere. In the second part of the paper function representing the ratio of the circle's circumference to its diameter has been defined and researched to analyze the potential discrepancies in the spherical and conical projective models on which the navigational computations are based on. We compare some non-Euclidean geometric properties of curved surfaces and its Euclidean plane model in reference to the local and global approximation. As a working tool we use function for geometric comparison analysis in the theory of long-range navigation and cartographic projection. We state the existence of the infinite number of the circles having the same radius but different circumference on the conical surface. Finally, we survey the exemplary proposals of generalization of function . In particular, we focus on the geometric structure of applied model treated as a metric space showing the differences in the outputting computations if the changes in a metric are made. We also relate the function to Tissot's indicatrix of distortion.

APA, Harvard, Vancouver, ISO, and other styles

22

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

23

Nabarro, Ana Claudia, and Farid Tari. "Families of curve congruences on Lorentzian surfaces and pencils of quadratic forms." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 141, no. 3 (June 2011): 655–72. http://dx.doi.org/10.1017/s0308210510000454.

Full text

Abstract:

We define and study families of conjugate and reflected curve congruences associated to a self-adjoint operator on a smooth and oriented surface M endowed with a Lorentzian metric g. These families trace parts of the pencil joining the equations of the -asymptotic and the -principal curves, and the pencil joining the -characteristic and the -principal curves, respectively. The binary differential equations (BDEs) of these curves can be viewed as points in the projective plane. Using the polar lines of various BDEs with respect to the conic of degenerate quadratic forms, we obtain geometric results on the above pencils and their relation with the metric g, on the type of solutions of a given BDE, of its -conjugate equation and on BDEs with orthogonal roots.

APA, Harvard, Vancouver, ISO, and other styles

24

Pacheco, Rui, and Susana D. Santos. "Envelopes of circles and spacelike curves in the Lorentz–Minkowski 3-space." Forum Mathematicum 32, no. 3 (May 1, 2020): 693–711. http://dx.doi.org/10.1515/forum-2019-0092.

Full text

Abstract:

AbstractThe isotropy projection establishes a correspondence between curves in the Lorentz–Minkowski space {\mathbf{E}_{1}^{3}} and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane {\mathcal{L}^{2}}). In this paper, we shall give necessary and sufficient conditions for two given families of cycles to be related by a (extended) Laguerre transformation in terms of the well known Lorentzian invariants for smooth curves in {\mathbf{E}_{1}^{3}}. We shall discuss the causal character of the second derivative of unit speed spacelike curves in {\mathbf{E}_{1}^{3}} in terms of the geometry of the corresponding families of oriented circles and their envelopes. Several families of circles whose envelopes are well-known plane curves are investigated and their Laguerre invariants computed.

APA, Harvard, Vancouver, ISO, and other styles

25

Сальков and Nikolay Sal'kov. "Dupin Cyclide and Second-Order Curves. Part 1." Geometry & Graphics 4, no. 2 (June 18, 2016): 19–28. http://dx.doi.org/10.12737/19829.

Full text

Abstract:

Making smooth shapes of various products is caused by the following requirements: aerodynamic, structural, aesthetic, etc. That’s why the review of the topic of second-order curves is included in many textbooks on descriptive geometry and engineering graphics. These curves can be used as a transition from the one line to another as the first and second order smoothness. Unfortunately, in modern textbooks on engineering graphics the building of Konik is not given. Despite the fact that all the second-order curves are banded by a single analytical equation, geometrically they unites by the affiliation of the quadric, projective unites by the commonality of their construction, in the academic literature for each of these curves is offered its own individual plot. Considering the patterns associated with Dupin cyclide, you can pay attention to the following peculiarity: the center of the sphere that is in contact circumferentially with Dupin cyclide, by changing the radius of the sphere moves along the second-order curve. The circle of contact of the sphere with Dupin cyclide is always located in a plane passing through one of the two axes, and each of these planes intersects cyclide by two circles. This property formed the basis of the graphical constructions that are common to all second-order curves. In addition, considered building has a connection with such transformation as the dilation or the central similarity. This article considers the methods of constructing of second-order curves, which are the lines of centers tangent of the spheres, applies a systematic approach.

APA, Harvard, Vancouver, ISO, and other styles

26

Bruce, J. W. "Lines, surfaces and duality." Mathematical Proceedings of the Cambridge Philosophical Society 112, no. 1 (July 1992): 53–61. http://dx.doi.org/10.1017/s0305004100070754.

Full text

Abstract:

In the paper [12] Shcherbak studied some duality properties of projective curves and applied them to obtain information concerning central projections of surfaces in projective three space. He also states some interesting results relating the contact of a generic surface with lines and the contact of its dual with lines in the dual space. In this paper we extend this duality to cover non-generic surfaces. Our proof is geometric, and uses deformation theory. The basic idea is the following. Given a surface X in projective 3-space we can consider the lines tangent to X, and measure their contact. The points on the surface with a line yielding at least 4-point contact are classically known as the flecnodal. (The reason is that the tangent plane meets the surface in a nodal curve, one branch of which has an inflexion at the point in question; see Proposition 7 below. The line in question is the inflexional tangent, which is clearly asymptotic.)

APA, Harvard, Vancouver, ISO, and other styles

27

Короткий and Viktor Korotkiy. "Reconstruction of Quadratic Cremona Transformation." Geometry & Graphics 5, no. 2 (July 4, 2017): 59–68. http://dx.doi.org/10.12737/article_5953f3002a72d8.28689872.

Full text

Abstract:

The geometric correspondence between the points of two planes can be considered well defined only when base data for its establishing is available, and a construction method by which its possible on the basis of these data for each point in one plane to find the corresponding points in the other one. Quadratic Cremona transformation can be specified by pointing out in the combined plane seven pairs of corresponding points. Naturally there is a need to establish a method for constructing any number of corresponding points. An outstanding Russian geometer K.A. Andreev indicated the linear construction based on the consideration of two correlations by which for each eighth point in the one plane is found the corresponding point of the other one. But in his work was not set up a problem to construct excluded (fundamental) points of quadratic Cremona transformation specified by seven pairs of points. There are many constructive ways to obtain the quadratic transformation in the plane. For example, it can be obtained by using two pairs of projective pencils of straight lines with vertices at the fundamental points (F-points). K.A. Andreev noted that this method for establishing of quadratic correspondence spread only to those cases when all F-points are the real ones. This statement is true for the 19th century’s level of geometric science, but today it’s too categorical. The theory of imaginary elements in geometry allows to develop a universal algorithm for construction of corresponding points in a quadratic transformation, given both by real and imaginary F-points. Summarizing the K.A. Andreev task, we come to the problem of finding the fundamental points (F-points) for a quadratic transformation specified by seven pairs of corresponding points. Almost one and half century the K.A. Andreev generalized task remained unsolved. The formation of this task’s constructive solution algorithm and its practical implementation has become possible by means of modern computer geometric modeling. According to proposed algorithm, the construction of F-points is reduced to the construction of second order auxiliary curves, on which intersection are marked the required F-points. The result received in this paper is used for development of the Cremona transformations’ theory, and for further application of this theory in the practice of geometric modeling.

APA, Harvard, Vancouver, ISO, and other styles

28

Боровиков, И., Ivan Borovikov, Геннадий Иванов, Gennadiy Ivanov, Н. Суркова, and N. Surkova. "On Application of Transformations at Descriptive Geometry’s Problems Solution." Geometry & Graphics 6, no. 2 (August 21, 2018): 78–84. http://dx.doi.org/10.12737/article_5b55a35d683a33.30813949.

Full text

Abstract:

This publication is devoted to the application of transformations at descriptive geometry’s problems solution. Using parametric calculus lets rationally select the number of transformations in the drawing. In Cartesian coordinates, on condition that an identical coordinate plane exists, the difference between parameters of linear forms, given and converted ones, is equal to the number of transformations in the composition. In affine space under these conditions, this difference is equal to two. Based on parameters calculation the conclusion is confirmed that the method of rotation around the level line, as providing the transformation of the plane of general position to the level plane, is a composition of two transformations: replacement of projections planes and rotation around the projection line. In various geometries (affine, projective, algebraic ones, and topology) the types of corresponding transformations are studied. As a result of these transformations are obtained affine, projective, bi-rational and topologically equivalent figures respectively. Such transformations are widely used in solving of applied problems, for example, in the design of technical surfaces of dependent sections. At the same time, along with transformation invariants, the simplicity of the algorithm for constructing of corresponding figures should be taken into account, with the result that so-called stratified transformations are preferred. A sign of transformation’s stratification is a value of dimension for a set of corresponding points’ carriers. This fact explains the relative simplicity of the algorithm for constructing the corresponding points in such transformations. In this paper the use of stratified transformations when finding the points of intersection of a curve with a surface, as well as in the construction of surfaces with variable cross-section shape are considered. The given examples show stratification idea possibilities for solving the problems of descriptive geometry.

APA, Harvard, Vancouver, ISO, and other styles

29

Popov, Yu I. "Fields of geometric objects associated with compiled hyperplane-distribution in affine space." Differential Geometry of Manifolds of Figures, no. 51 (2020): 103–15. http://dx.doi.org/10.5922/0321-4796-2020-51-12.

Full text

Abstract:

A compiled hyperplane distribution is considered in an n-dimensional projective space . We will briefly call it a -distribution. Note that the plane L(A) is the distribution characteristic obtained by displacement in the center belonging to the L-subbundle. The following results were obtained: a) The existence theorem is proved: -distribution exists with arbitrary (3n – 5) functions of n arguments. b) A focal manifold is constructed in the normal plane of the 1st kind of L-subbundle. It was obtained by shifting the center A along the curves belonging to the L-distribution. A focal manifold is also given, which is an analog of the Koenigs plane for the distribution pair (L, L). c) It is shown that a framed -distribution in the 1st kind normal field of H-distribution induces tangent and normal bundles. d) Six connection theorems induced by a framed -distribution in these bundles are proved. In each of the bundles , the framed -distribution induces an intrinsic torsion-free affine connection in the tangent bundle and a centro-affine connection in the corresponding normal bundle. e) In each of the bundles (d) in the differential neighborhood of the 2nd order, the covers of 2-forms of curvature and curvature tensors of the corresponding connections are constructed.

APA, Harvard, Vancouver, ISO, and other styles

30

Волошинов, Д., and Denis Voloshinov. "On the Peculiarities of the Constructive Solution For Dandelin Spheres Problem." Geometry & Graphics 6, no. 2 (August 21, 2018): 55–62. http://dx.doi.org/10.12737/article_5b559f018f85a7.77112269.

Full text

Abstract:

This paper is devoted to analysis of Dandelin spheres problem based on the constructive geometric approach. In the paper it has been demonstrated that the traditional approach used to this problem solving leads to obtaining for only a limited set of heterogeneous solutions. Consideration of the problem in the context of plane and space’s projective properties by structural geometry’s methods allows interpret this problem’s results in a new way. In the paper it has been demonstrated that the solved problem has a purely projective nature and can be solved by a unified method, which is impossible to achieve if conduct reasoning and construct proofs only on affine geometry’s positions. The research’s scientific novelty is the discovery and theoretical justification of a new classification feature allowing classify as Dandelin spheres the set of spheres pairs with imaginary tangents to the quadric, as well as pairs of imaginary spheres with a unified principle for constructive interrelation of images, along with real solutions. The work’s practical significance lies in the extension of application areas for geometric modeling’s constructive methods to the solution of problems, in the impro vement of geometric theory, in the development of system for geometric modeling Simplex’s functional capabilities for tasks of objects and processes design automation. The algorithms presented in the paper demonstrate the deep projective nature and interrelation of such problems as Apollonius circles and spheres one, Dandelin spheres one and others, as well as lay the groundwork for researches in the direction of these problems’ multidimensional interpretations. The problem solution can be useful for second-order curves’ blending function realization by means of circles with a view to improve the tools of CAD-systems’ design automation without use of mathematical numerical methods for these purposes.

APA, Harvard, Vancouver, ISO, and other styles

31

Короткий, Viktor Korotkiy, Гирш, and A. Girsh. "Graphic Reconstruction Algorithms of the Second-Order Curve, Given by the Imaginary Elements." Geometry & Graphics 4, no. 4 (December 19, 2016): 19–30. http://dx.doi.org/10.12737/22840.

Full text

Abstract:

Second-order curves are used as shape-generating elements in the design of technical devices and architectural structures. In such a case, a need for reconstruction task solution may emerge. The reconstruction is called the definition of the main axes and asymptotes of the second-order curve by its incomplete image containing n points and m tangents (n + m = 5). In CAD graphical systems there is no possibility for construction of the second order curve, given by real and imaginary points and tangents. Therefore, the second-order curve reconstruction cannot be made with the standard set of computer graphics tools. In this paper are proposed geometrically accurate algorithms for reconstruction of the secondorder curve, given by a mixed set of real and imaginary elements. A specialized software package has been developed for constructive realization of these algorithms. Imaginary geometric images are pair-conjugated, so there are only seven possible combinations of given data with imaginary elements participation: five points, two of which are imaginary ones; five points, four of which are imaginary ones; three real points, two imaginary tangents; a real point, four imaginary tangents; a real point, two imaginary points, two imaginary tangents; a real point, two imaginary points, two real tangents; two real points, two imaginary points, a real tangent. For reconstruction problem solution is used the main property of polar matching: if P and p are the pole and polar relative to the conic g, the harmonic homology with center P and axis p transforms the curve g in itself. The method of solution based on projective transformation of required conic into a circle. It has been shown that in some cases for reconstruction problem solution its necessary to apply the quadratic involution conversion, resting on plane by a conic beam. The developed technique and software package expand the capabilities of the computer geometric simulation for processes occurring with the second-order curves participation.

APA, Harvard, Vancouver, ISO, and other styles

32

Ganter, M. A., and D. W. Storti. "Object Extent Determination for Algebraic Solid Models." Journal of Mechanical Design 117, no. 1 (March 1, 1995): 20–26. http://dx.doi.org/10.1115/1.2826111.

Full text

Abstract:

This paper presents methods for determination of spatial extent of algebraic solid models. Algebraic solid models are a variation of implicit solid models defined by implicit polynomial functions with rational coefficients. Spatial extent information, which can be used to enhance the performance of visualization and property evaluation, includes silhouettes, outlines and profiles. Silhouettes are curves on the surface of the solid which separate portions of the surface which face towards or away from a given viewpoint. The projection of the silhouette onto the viewing plane gives the outline of the solid, and the bivariate implicit function which defines the area enclosed by the outline is called the profile. A method for outline determination is demonstrated using concepts from algebraic geometry including polar surfaces and variable elimination via the Gro¨bner basis method and/or resultants. Examples of outline generation are presented and a sample profile function is constructed.

APA, Harvard, Vancouver, ISO, and other styles

33

Talhofer, Václav, and Šárka Hošková-Mayerová. "Theoretical Foundations of Study of Cartography." Proceedings of the ICA 1 (May 16, 2018): 1–7. http://dx.doi.org/10.5194/ica-proc-1-110-2018.

Full text

Abstract:

Cartography and geoinformatics are technical-based fields which deal with modelling and visualization of landscape in the form of a map. The theoretical foundation is necessary to obtain during study of cartography and geoinformatics based mainly on mathematics. For the given subjects, mathematics is necessary for understanding of many procedures that are connected to modelling of the Earth as a celestial body, to ways of its projection into a plane, to methods and procedures of modelling of landscape and phenomena in society and visualization of these models in the form of electronic as well as classic paper maps. Not only general mathematics, but also its extension of differential geometry of curves and surfaces, ways of approximation of lines and surfaces of functional surfaces, mathematical statistics and multi-criterial analyses seem to be suitable and necessary. Underestimation of the significance of mathematical education in cartography and geoinformatics is inappropriate and lowers competence of cartographers and professionals in geographic information science and technology to solve problems.

APA, Harvard, Vancouver, ISO, and other styles

34

Gupta, K. C., and Suryansu Ray. "Fuzzy plane projective geometry." Fuzzy Sets and Systems 54, no. 2 (March 1993): 191–206. http://dx.doi.org/10.1016/0165-0114(93)90276-n.

Full text

APA, Harvard, Vancouver, ISO, and other styles

35

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

36

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

37

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

38

Caporaso, Lucia. "Enumerative Geometry of Plane Curves." Notices of the American Mathematical Society 67, no. 06 (June 1, 2020): 1. http://dx.doi.org/10.1090/noti2094.

Full text

APA, Harvard, Vancouver, ISO, and other styles

39

Nunemacher, Jeffrey. "Asymptotes, Cubic Curves, and the Projective Plane." Mathematics Magazine 72, no. 3 (June 1, 1999): 183. http://dx.doi.org/10.2307/2690881.

Full text

APA, Harvard, Vancouver, ISO, and other styles

40

LÊ, QUY THUONG. "ALEXANDER POLYNOMIALS OF COMPLEX PROJECTIVE PLANE CURVES." Bulletin of the Australian Mathematical Society 97, no. 3 (March 7, 2018): 386–95. http://dx.doi.org/10.1017/s0004972717001198.

Full text

Abstract:

We compute the Alexander polynomial of a nonreduced nonirreducible complex projective plane curve with mutually coprime orders of vanishing along its irreducible components in terms of certain multiplier ideals.

APA, Harvard, Vancouver, ISO, and other styles

41

Nunemacher, Jeffrey. "Asymptotes, Cubic Curves, and the Projective Plane." Mathematics Magazine 72, no. 3 (June 1999): 183–92. http://dx.doi.org/10.1080/0025570x.1999.11996729.

Full text

APA, Harvard, Vancouver, ISO, and other styles

42

Tomihisa, Toshio. "Geometry of projective plane and Poisson structure." Journal of Geometry and Physics 59, no. 5 (May 2009): 673–84. http://dx.doi.org/10.1016/j.geomphys.2009.02.005.

Full text

APA, Harvard, Vancouver, ISO, and other styles

43

BAYAR, A., S. EKMEKCI, and Z. AKCA. "A note on fibered projective plane geometry." Information Sciences 178, no. 4 (February 15, 2008): 1257–62. http://dx.doi.org/10.1016/j.ins.2007.10.003.

Full text

APA, Harvard, Vancouver, ISO, and other styles

44

Erdnüß, B. "MEASURING IN IMAGES WITH PROJECTIVE GEOMETRY." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XLII-1 (September 26, 2018): 141–48. http://dx.doi.org/10.5194/isprs-archives-xlii-1-141-2018.

Full text

Abstract:

<p><strong>Abstract.</strong> There is a fundamental relationship between projective geometry and the perspective imaging geometry of a pinhole camera. Projective scales have been used to measure within images from the beginnings of photogrammetry, mostly the cross-ratio on a straight line. However, there are also projective frames in the plane with interesting connections to affine and projective geometry in three dimensional space that can be utilized for photogrammetry. This article introduces an invariant on the projective plane, describes its relation to affine geometry, and how to use it to reduce the complexity of projective transformations. It describes how the invariant can be use to measure on projectively distorted planes in images and shows applications to this in 3D reconstruction. The article follows two central ideas. One is to measure coordinates in an image relatively to each other to gain as much invariance of the viewport as possible. The other is to use the remaining variance to determine the 3D structure of the scene and to locate the camera centers. For this, the images are projected onto a common plane in the scene. 3D structure not on the plane occludes different parts of the plane in the images. From this, the position of the cameras and the 3D structure are obtained.</p>

APA, Harvard, Vancouver, ISO, and other styles

45

Hsu, Yu-Wen. "Curve shortening flow and smooth projective planes." Communications in Analysis and Geometry 27, no. 6 (2019): 1281–324. http://dx.doi.org/10.4310/cag.2019.v27.n6.a4.

Full text

APA, Harvard, Vancouver, ISO, and other styles

46

Dydak, Jerzy, and Michael Levin. "Maps to the projective plane." Algebraic & Geometric Topology 9, no. 1 (March 30, 2009): 549–68. http://dx.doi.org/10.2140/agt.2009.9.549.

Full text

APA, Harvard, Vancouver, ISO, and other styles

47

Mandelkern, Mark. "A constructive real projective plane." Journal of Geometry 107, no. 1 (May 27, 2015): 19–60. http://dx.doi.org/10.1007/s00022-015-0272-4.

Full text

APA, Harvard, Vancouver, ISO, and other styles

48

Janic, Milan, and Dejan Tanikic. "Geometry of straight lines pencils." Facta universitatis - series: Architecture and Civil Engineering 2, no. 4 (2002): 291–94. http://dx.doi.org/10.2298/fuace0204291j.

Full text

Abstract:

This paper considers a pencil of straight Unes in the Euclidean plane as well as the same pencil of straight lines in the projective plane where the projective geometry model M" is defined with its points forming the sets of (n-l) collinear points, whose supporting straight lines belong to the considered pencil of straight lines.

APA, Harvard, Vancouver, ISO, and other styles

49

Pirola, Gian Pietro, and Enrico Schlesinger. "Monodromy of projective curves." Journal of Algebraic Geometry 14, no. 4 (2005): 623–42. http://dx.doi.org/10.1090/s1056-3911-05-00408-x.

Full text

APA, Harvard, Vancouver, ISO, and other styles

50

Yamazaki, Takayoshi, and Atsuko Yamada Yoshikawa. "Plane curves in Lie sphere geometry." Kodai Mathematical Journal 19, no. 3 (1996): 322–40. http://dx.doi.org/10.2996/kmj/1138043650.

Full text

APA, Harvard, Vancouver, ISO, and other styles

We offer discounts on all premium plans for authors whose works are included in thematic literature selections. Contact us to get a unique promo code!