Space Curves

If a fixed point of X corresponds to a curve, the support of this curve is contained in the tetrahedron of reference. Hence the curve is either three non coplanar edges (there are 16 such), one edge doubled in a plane (face) union a ...

Space Curves

The main topics of the conference on "Curves in Projective Space" were good and bad families of projective curves, postulation of projective space curves and classical problems in enumerative geometry.

Introduction to Differential Geometry of Space Curves and Surfaces

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 State the technical definition of space curve outlining the difference between a curve and its trace. What is the most common way of defining space ...

Introduction to Differential Geometry of Space Curves and Surfaces


Geometric Theory of Algebraic Space Curves

The main part was the proof to the Theorem (36.9), namely that "All irreducible nonsingular space curves of degree at most five and genus at most one over an algebraically closed ground field are complete intersections.

Geometric Theory of Algebraic Space Curves


Differential Geometry of Curves and Surfaces

The local theory of space curves is similar to the theory for plane curves, but differences arise in the richer variety of configurations available for loci of curves in R3. In the study of plane curves, we introduced the curvature ...

Differential Geometry of Curves and Surfaces

Differential Geometry of Curves and Surfaces, Second Edition takes both an analytical/theoretical approach and a visual/intuitive approach to the local and global properties of curves and surfaces. Requiring only multivariable calculus and linear algebra, it develops students' geometric intuition through interactive computer graphics applets suppor

COMPUTING SPECIAL SMARANDACHE CURVES ACCORDING TO DARBOUX FRAME IN EUCLIDEAN 4 SPACE

... Number 4 (2019), 479-502 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-479 COMPUTING SPECIAL SMARANDACHE CURVES ACCORDING TO DARBOUX FRAME IN EUCLIDEAN 4-SPACE M. KHALIFA SAAD1,2,∗ AND M. A. ABD-RABO3 ...

COMPUTING SPECIAL SMARANDACHE CURVES ACCORDING TO DARBOUX FRAME IN EUCLIDEAN 4 SPACE

In this paper, we study some special Smarandache curves and their di erential geometric properties according to Darboux frame in Euclidean 4-space E4. Also, we compute some of these curves which lie fully on a hypersurface in E4. Moreover, we defray some computational examples in support our main results.

Modern Differential Geometry of Curves and Surfaces with Mathematica

Curves. in. Space. In previous chapters, we have seen that the curvature κ2[α] of a plane curve α measures the failure of α to be a straight line. In the present chapter, we define a similar curvature κ[α] for a curve in Rn; ...

Modern Differential Geometry of Curves and Surfaces with Mathematica

Presenting theory while using Mathematica in a complementary way, Modern Differential Geometry of Curves and Surfaces with Mathematica, the third edition of Alfred Gray’s famous textbook, covers how to define and compute standard geometric functions using Mathematica for constructing new curves and surfaces from existing ones. Since Gray’s death, authors Abbena and Salamon have stepped in to bring the book up to date. While maintaining Gray's intuitive approach, they reorganized the material to provide a clearer division between the text and the Mathematica code and added a Mathematica notebook as an appendix to each chapter. They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject. For example, Brioshi’s formula for the Gaussian curvature in terms of the first fundamental form can be too complicated for use in hand calculations, but Mathematica handles it easily, either through computations or through graphing curvature. Another part of Mathematica that can be used effectively in differential geometry is its special function library, where nonstandard spaces of constant curvature can be defined in terms of elliptic functions and then plotted. Using the techniques described in this book, readers will understand concepts geometrically, plotting curves and surfaces on a monitor and then printing them. Containing more than 300 illustrations, the book demonstrates how to use Mathematica to plot many interesting curves and surfaces. Including as many topics of the classical differential geometry and surfaces as possible, it highlights important theorems with many examples. It includes 300 miniprograms for computing and plotting various geometric objects, alleviating the drudgery of computing things such as the curvature and torsion of a curve in space.

Differential Geometry of Curves and Surfaces

In this section, we consider curves in 3-dimensional Euclidean space, or more briefly “space curves”. To do this, we will need the vector product (or cross product) of two vectors in space (cf. Appendix A.3). Space curves.

Differential Geometry of Curves and Surfaces

This engrossing volume on curve and surface theories is the result of many years of experience the authors have had with teaching the most essential aspects of this subject. The first half of the text is suitable for a university-level course, without the need for referencing other texts, as it is completely self-contained. More advanced material in the second half of the book, including appendices, also serves more experienced students well. Furthermore, this text is also suitable for a seminar for graduate students, and for self-study. It is written in a robust style that gives the student the opportunity to continue his study at a higher level beyond what a course would usually offer. Further material is included, for example, closed curves, enveloping curves, curves of constant width, the fundamental theorem of surface theory, constant mean curvature surfaces, and existence of curvature line coordinates. Surface theory from the viewpoint of manifolds theory is explained, and encompasses higher level material that is useful for the more advanced student. This includes, but is not limited to, indices of umbilics, properties of cycloids, existence of conformal coordinates, and characterizing conditions for singularities. In summary, this textbook succeeds in elucidating detailed explanations of fundamental material, where the most essential basic notions stand out clearly, but does not shy away from the more advanced topics needed for research in this field. It provides a large collection of mathematically rich supporting topics. Thus, it is an ideal first textbook in this field. Request Inspection Copy

The Classical Differential Geometry of Curves and Surfaces

Chapter XII THE THEORY OF SPACE CURVES Differential geometry developed concurrently with analysis . The problems relating to the curvature of plane curves were approached by Newton , Leibniz , and Huygens ; the curvature of surfaces was ...

The Classical Differential Geometry of Curves and Surfaces


CONTRIBUTIONS TO DIFFERENTIAL GEOMETRY OF SPACELIKE CURVES IN LORENTZIAN PLANE L2

If dm 1dθ = m2=0 = k2 = const., then from (37) we obtain f(θ)=0, and m1 = 0 Hence the equation (32) becomes as follows: φ∗ = φ + k2 N. References [1] Ahmad T. Ali, Special Smarandache Curves in the Euclidean Space, Math Combin.

CONTRIBUTIONS TO DIFFERENTIAL GEOMETRY OF SPACELIKE CURVES IN LORENTZIAN PLANE L2

In this work, first the differential equation characterizing position vector of spacelike curve is obtained in Lorentzian plane L2.

Mathematical Methods for Curves and Surfaces

Frames that incorporate the unit tangent and unit polar vector as one component are known as adapted and directed curve frames, respectively [7]. An infinitude of such frames exists on any given space curve, but among them the ...

Mathematical Methods for Curves and Surfaces

This volume constitutes the thoroughly refereed post-conference proceedings of the 7th International Conference on Mathematical Methods for Curves and Surfaces, MMCS 2008, held in Tønsberg, Norway, in June/July 2008. The 28 revised full papers presented were carefully reviewed and selected from 129 talks presented at the conference. The topics addressed by the papers range from mathematical analysis of various methods to practical implementation on modern graphics processing units.

Modeling of Curves and Surfaces with MATLAB

We recover a plane curve from its curvature function k(s) > 0. ... a plane curve (ellipse, logarithmic spiral, etc.) ... 6.4.3 Curvature and torsion of a space curve We compute the characteristics of a C3-regular space curve at an ...

Modeling of Curves and Surfaces with MATLAB

This text on geometry is devoted to various central geometrical topics including: graphs of functions, transformations, (non-)Euclidean geometries, curves and surfaces as well as their applications in a variety of disciplines. This book presents elementary methods for analytical modeling and demonstrates the potential for symbolic computational tools to support the development of analytical solutions. The author systematically examines several powerful tools of MATLAB® including 2D and 3D animation of geometric images with shadows and colors and transformations using matrices. With over 150 stimulating exercises and problems, this text integrates traditional differential and non-Euclidean geometries with more current computer systems in a practical and user-friendly format. This text is an excellent classroom resource or self-study reference for undergraduate students in a variety of disciplines.

Differential Geometry of Curves and Surfaces

To familiarize the reader with calculations on space curves, we give three problems and end this section. Problem 1.4.1 Prove that the Taylor expansion of a space curve p(s) at s = 0 to its third term is given by p(s) = p(0) + e1 (0)s + ...

Differential Geometry of Curves and Surfaces

This book is a posthumous publication of a classic by Prof. Shoshichi Kobayashi, who taught at U.C. Berkeley for 50 years, recently translated by Eriko Shinozaki Nagumo and Makiko Sumi Tanaka. There are five chapters: 1. Plane Curves and Space Curves; 2. Local Theory of Surfaces in Space; 3. Geometry of Surfaces; 4. Gauss–Bonnet Theorem; and 5. Minimal Surfaces. Chapter 1 discusses local and global properties of planar curves and curves in space. Chapter 2 deals with local properties of surfaces in 3-dimensional Euclidean space. Two types of curvatures — the Gaussian curvature K and the mean curvature H —are introduced. The method of the moving frames, a standard technique in differential geometry, is introduced in the context of a surface in 3-dimensional Euclidean space. In Chapter 3, the Riemannian metric on a surface is introduced and properties determined only by the first fundamental form are discussed. The concept of a geodesic introduced in Chapter 2 is extensively discussed, and several examples of geodesics are presented with illustrations. Chapter 4 starts with a simple and elegant proof of Stokes’ theorem for a domain. Then the Gauss–Bonnet theorem, the major topic of this book, is discussed at great length. The theorem is a most beautiful and deep result in differential geometry. It yields a relation between the integral of the Gaussian curvature over a given oriented closed surface S and the topology of S in terms of its Euler number χ(S). Here again, many illustrations are provided to facilitate the reader’s understanding. Chapter 5, Minimal Surfaces, requires some elementary knowledge of complex analysis. However, the author retained the introductory nature of this book and focused on detailed explanations of the examples of minimal surfaces given in Chapter 2.

Curves and Surfaces

Approximating Algebraic Space Curves by Circular Arcs⋆ Szilvia Béla1 and Bert Jüttler2 1 Doctoral Program in Computational Mathematics 2 Institute of Applied Geometry, Johannes Kepler University, Altenberger Str. 69, 4040 Linz, ...

Curves and Surfaces

This volume constitutes the thoroughly refereed post-conference proceedings of the 7th International Conference on Curves and Surfaces, held in Avignon, in June 2010. The conference had the overall theme: "Representation and Approximation of Curves and Surfaces and Applications". The 39 revised full papers presented together with 9 invited talks were carefully reviewed and selected from 114 talks presented at the conference. The topics addressed by the papers range from mathematical foundations to practical implementation on modern graphics processing units and address a wide area of topics such as computer-aided geometric design, computer graphics and visualisation, computational geometry and topology, geometry processing, image and signal processing, interpolation and smoothing, scattered data processing and learning theory and subdivision, wavelets and multi-resolution methods.

Discovering Curves and Surfaces with Maple

Figure 4.1.4 either included as additional parameters of the list defining the curve or specified outside the list. It is often preferable to draw space curves as thin tubes because tubes convey more information on the position and ...

Discovering Curves and Surfaces with Maple

Despite the fact that Maple is one of the most popular computer algebra systems on the market, surprisingly few users realise its potential for scientific visualisation. This book equips readers with the graphics tools needed on the voyage into the complex and beautiful world of curves and surfaces. A comprehensive treatment of Maples graphics commands and structures is combined with an introduction to the main aspects of visual perception, with priority given to the use of light, colour, perspective, and geometric transformations. Numerous examples cover all aspects of Maple graphics, and these may be easily tailored to the individual needs of the reader. The approach is context-independent, and as such will appeal to students, educators, and researchers in a broad spectrum of scientific disciplines. For the general user at any level of experience, this book will serve as a comprehensive reference manual. For the beginner, it offers a user-friendly introduction to the subject, with mathematical requirements kept to a minimum, while, for those interested in advanced mathematical visualisation, it explains how to maximise Maples graphical capabilities.

On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space

Curves are usually studied as subsets of an ambient space with a notion of equivalence. For example, one may study curves in the plane, the usual three dimensional space, the Minkowski space, curves on a sphere, etc.

On Special Curves According to Darboux Frame in the Three Dimensional Lorentz Space

In the light of great importance of curves and their frames in many different branches of science, especially differential geometry as well as geometric properties and the uses in various fields, we are interested here to study a special kind of curves called Smarandache curves in Lorentz 3-space.

Algebraic Curves over a Finite Field

7.9 DUAL CURVES OF SPACE CURVES In Section 5.11, the dual curve of a plane curve is investigated. Now duality for space curves is considered. Let Γ be a non-degenerate irreducible curve of PG(r, K). DEFINITION 7.82 The dual curve ofΓ is ...

Algebraic Curves over a Finite Field

This book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.

Lectures on Classical Differential Geometry

FUNDAMENTAL THEOREM for space curves : If two single - valued continuous functions « ( 8 ) and t ( s ) , > 0 , are given , then there exists one and only one space curve , determined but for its position in space , for which s is the ...

Lectures on Classical Differential Geometry

Elementary, yet authoritative and scholarly, this book offers an excellent brief introduction to the classical theory of differential geometry. It is aimed at advanced undergraduate and graduate students who will find it not only highly readable but replete with illustrations carefully selected to help stimulate the student's visual understanding of geometry. The text features an abundance of problems, most of which are simple enough for class use, and often convey an interesting geometrical fact. A selection of more difficult problems has been included to challenge the ambitious student. Written by a noted mathematician and historian of mathematics, this volume presents the fundamental conceptions of the theory of curves and surfaces and applies them to a number of examples. Dr. Struik has enhanced the treatment with copious historical, biographical, and bibliographical references that place the theory in context and encourage the student to consult original sources and discover additional important ideas there. For this second edition, Professor Struik made some corrections and added an appendix with a sketch of the application of Cartan's method of Pfaffians to curve and surface theory. The result was to further increase the merit of this stimulating, thought-provoking text — ideal for classroom use, but also perfectly suited for self-study. In this attractive, inexpensive paperback edition, it belongs in the library of any mathematician or student of mathematics interested in differential geometry.

Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type 2

Vol.4(2016), 29-43 Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type-2 Yasin Unlütürk ̈ Department of Mathematics ... Also, Smarandache breadth curves are defined according to this frame in Minkowski 3-space.

Smarandache Curves of a Spacelike Curve According to the Bishop Frame of Type 2

A third order vectorial differential equation of position vector of Smarandache breadth curves has been obtained in Minkowski 3-space.

A Treatise on Algebraic Plane Curves

curve . What singularities has this new plane curve ? It has singular points of the same description as those of the ... If V lie on a tangent to the space curve , the point of contact will go into a cusp ; we can avoid that by taking V ...

A Treatise on Algebraic Plane Curves

A thorough introduction to the theory of algebraic plane curves and their relations to various fields of geometry and analysis. Almost entirely confined to the properties of the general curve, and chiefly employs algebraic procedure. Geometric methods are much employed, however, especially those involving the projective geometry of hyperspace. 1931 edition. 17 illustrations.

Points and Curves in the Monster Tower

(B.11) Fix a δ appropriate to the case, (B.9), (B.10), or (B.11). Applying the classification results of [GH] we obtain: 1. A space curve germ of the form (B.9) is RL-equivalent to (t4,t5 + δt7,t11); 2. A space curve germ of the form ...

Points and Curves in the Monster Tower

Cartan introduced the method of prolongation which can be applied either to manifolds with distributions (Pfaffian systems) or integral curves to these distributions. Repeated application of prolongation to the plane endowed with its tangent bundle yields the Monster tower, a sequence of manifolds, each a circle bundle over the previous one, each endowed with a rank $2$ distribution. In an earlier paper (2001), the authors proved that the problem of classifying points in the Monster tower up to symmetry is the same as the problem of classifying Goursat distribution flags up to local diffeomorphism. The first level of the Monster tower is a three-dimensional contact manifold and its integral curves are Legendrian curves. The philosophy driving the current work is that all questions regarding the Monster tower (and hence regarding Goursat distribution germs) can be reduced to problems regarding Legendrian curve singularities.