Differential Geometry Of Plane Curves
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Differential Geometry Of Plane Curves
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Author : Hilário Alencar
language : en
Publisher: American Mathematical Society
Release Date : 2022-04-27
Differential Geometry Of Plane Curves written by Hilário Alencar and has been published by American Mathematical Society this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022-04-27 with Mathematics categories.
This book features plane curves—the simplest objects in differential geometry—to illustrate many deep and inspiring results in the field in an elementary and accessible way. After an introduction to the basic properties of plane curves, the authors introduce a number of complex and beautiful topics, including the rotation number (with a proof of the fundamental theorem of algebra), rotation index, Jordan curve theorem, isoperimetric inequality, convex curves, curves of constant width, and the four-vertex theorem. The last chapter connects the classical with the modern by giving an introduction to the curve-shortening flow that is based on original articles but requires a minimum of previous knowledge. Over 200 figures and more than 100 exercises illustrate the beauty of plane curves and test the reader's skills. Prerequisites are courses in standard one variable calculus and analytic geometry on the plane.
The Elementary Differential Geometry Of Plane Curves
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Author : R. H. Fowler
language : en
Publisher:
Release Date : 2005
The Elementary Differential Geometry Of Plane Curves written by R. H. Fowler and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2005 with Mathematics categories.
This precise account of elementary differential properties of plane curves provides a link between analysis and more complicated geometrical theorems, offering background and practice to geometry and analysis students. 1920 edition.
The Elementary Differential Geometry Of Plane Curves
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Author : R. H. Fowler
language : en
Publisher: Forgotten Books
Release Date : 2015-06-12
The Elementary Differential Geometry Of Plane Curves written by R. H. Fowler and has been published by Forgotten Books this book supported file pdf, txt, epub, kindle and other format this book has been release on 2015-06-12 with Mathematics categories.
Excerpt from The Elementary Differential Geometry of Plane Curves This tract is intended to present a precise account of the elementary differential properties of plane curves. The matter contained is in no sense new, but a suitable connected treatment in the English language has not been available. As a result, a number of interesting misconceptions are current in English text books. It is sufficient to mention two somewhat striking examples, (a) According to the ordinary definition of an envelope, as the locus of the limits of points of intersection of neighbouring curves, a curve is not the envelope of its circles of curvature, for neighbouring circles of curvature do not intersect. (b) The definitions of an asymptote - (1) a straight line, the distance from which of a point on the curve tends to zero as the point tends to infinity; (2) the limit of a tangent to the curve, whose point of contact tends to infinity - are not equivalent. The curve may have an asymptote according to the former definition, and the tangent may exist at every point, but have no limit as its point of contact tends to infinity. The subjects dealt with, and the general method of treatment, are similar to those of the usual chapters on geometry in any Cours d' Analyse, except that in general plane curves alone are considered. At the same time extensions to three dimensions are made in a somewhat arbitrary selection of places, where the extension is immediate, and forms a natural commentary on the two dimensional work, or presents special points of interest (Frenet's formulae). To make such extensions systematically would make the tract too long. The subject matter being wholly classical, no attempt has been made to give full references to sources of information; the reader however is referred at most stages to the analogous treatment of the subject in the Cours or Traite d' Analyse of de la Vallée Poussin, Goursat, Jordan or Picard, works to which the author is much indebted. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Modern Differential Geometry Of Curves And Surfaces With Mathematica
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Author : Elsa Abbena
language : en
Publisher: CRC Press
Release Date : 2017-09-06
Modern Differential Geometry Of Curves And Surfaces With Mathematica written by Elsa Abbena and has been published by CRC Press this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-06 with Mathematics categories.
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.
The Elementary Differential Geometry Of Plane Curves
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Author : R. H. Fowler
language : en
Publisher: Createspace Independent Publishing Platform
Release Date : 2017-09-17
The Elementary Differential Geometry Of Plane Curves written by R. H. Fowler and has been published by Createspace Independent Publishing Platform this book supported file pdf, txt, epub, kindle and other format this book has been release on 2017-09-17 with categories.
From the PREFACE. THIS tract is intended to present a precise account of the elementary differential properties of plane curves. The matter contained is in no sense new, but a suitable connected treatment in the English language has not been available. As a result, a number of interesting misconceptions are current in English text books. It is sufficient to mention two somewhat striking examples, (a) According to the ordinary definition of an envelope, as the locus of the limits of points of intersection of neighbouring curves, a curve is not the envelope of its circles of curvature, for neighbouring circles of curvature do not intersect. (b) The definitions of an asymptote-(1) a straight line, the distance from which of a point on the curve tends to zero as the point tends to infinity; (2) the limit of a tangent to the curve, whose point of contact tends to infinity-are not equivalent. The curve may have an asymptote according to the former definition, and the tangent may exist at every point, but have no limit as its point of contact tends to infinity. The subjects dealt with, and the general method of treatment, are similar to those of the usual chapters on geometry in any Cours d'Analyse, except that in general plane curves alone are considered. At the same time extensions to three dimensions are made in a somewhat arbitrary selection of places, where the extension is immediate, and forms a natural commentary on the two dimensional work, or presents special points of interest (Frenet's formulae). To make such extensions systematically would make the tract too long. The subject matter being wholly classical, no attempt has been made to give full references to sources of information; the reader however is referred at most stages to the analogous treatment of the subject in the Cours or Traite d'Analyse of de la Vallee Poussin, Goursat, Jordan or Picard, works to which the author is much indebted. In general the functions, which define the curves under consideration, are (as usual) assumed to have as many continuous differential coefficients as may be mentioned. In places, however, more particularly at the beginning, this rule is deliberately departed from, and the greatest generality is sought for in the enunciation of any theorem. The determination of the necessary and sufficient conditions for the truth of any theorem is then the primary consideration. In the proofs of the elementary theorems, where this procedure is adopted, it is believed that this treatment will be found little more laborious than any rigorous treatment, and that it provides a connecting link between Analysis and more complicated geometrical theorems, in which insistence on the precise necessary conditions becomes tedious and out of place, and suitable sufficient conditions can always be tacitly assumed. At an earlier stage the more precise formulation of conditions may be regarded as (1) an important grounding for the student of Geometry, and (2) useful practice for the student of Analysis. The introductory chapter is a collection of somewhat disconnected theorems which are required for reference. The reader can omit it, and to refer to it as it becomes necessary for the understanding of later chapters....
Differential Geometry Of Curves And Surfaces
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Author : Shoshichi Kobayashi
language : en
Publisher: Springer Nature
Release Date : 2019-11-13
Differential Geometry Of Curves And Surfaces written by Shoshichi Kobayashi and has been published by Springer Nature this book supported file pdf, txt, epub, kindle and other format this book has been release on 2019-11-13 with Mathematics categories.
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.
Differential Geometry Of Curves And Surfaces With Singularities
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Author : Masaaki Umehara
language : en
Publisher: Algebraic and Differential Geo
Release Date : 2021-09
Differential Geometry Of Curves And Surfaces With Singularities written by Masaaki Umehara and has been published by Algebraic and Differential Geo this book supported file pdf, txt, epub, kindle and other format this book has been release on 2021-09 with Mathematics categories.
This book provides a unique and highly accessible approach to singularity theory from the perspective of differential geometry of curves and surfaces. It is written by three leading experts on the interplay between two important fields - singularity theory and differential geometry. The book introduces singularities and their recognition theorems, and describes their applications to geometry and topology, restricting the objects of attention to singularities of plane curves and surfaces in the Euclidean 3-space. In particular, by presenting the singular curvature, which originated through research by the authors, the Gauss-Bonnet theorem for surfaces is generalized to those with singularities. The Gauss-Bonnet theorem is intrinsic in nature, that is, it is a theorem not only for surfaces but also for 2-dimensional Riemannian manifolds. The book also elucidates the notion of Riemannian manifolds with singularities. These topics, as well as elementary descriptions of proofs of the recognition theorems, cannot be found in other books. Explicit examples and models are provided in abundance, along with insightful explanations of the underlying theory as well. Numerous figures and exercise problems are given, becoming strong aids in developing an understanding of the material. Readers will gain from this text a unique introduction to the singularities of curves and surfaces from the viewpoint of differential geometry, and it will be a useful guide for students and researchers interested in this subject.
The Elementary Differential Geometry Of Plane Curves
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Author : Ralph Howard Fowler
language : en
Publisher:
Release Date : 1929
The Elementary Differential Geometry Of Plane Curves written by Ralph Howard Fowler and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 1929 with Curves, Plane categories.
The Elementary Differential Geometry Of Plane Curves Classic Reprint
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Author : R. H. Fowler
language : en
Publisher: Forgotten Books
Release Date : 2018-01-05
The Elementary Differential Geometry Of Plane Curves Classic Reprint written by R. H. Fowler and has been published by Forgotten Books this book supported file pdf, txt, epub, kindle and other format this book has been release on 2018-01-05 with Mathematics categories.
Excerpt from The Elementary Differential Geometry of Plane Curves A limited selection of examples is given at the ends of the chapters. Besides their more Obvious function, these are intended to provide a summary of some of the more important extensions of the theorems proved in the text. References or sketches of a proof are therefore given in such cases, which should enable the reader to complete the proofs. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.
Differential Geometry Of Plane Curves
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Author : Hilário Alencar
language : en
Publisher:
Release Date : 2022
Differential Geometry Of Plane Curves written by Hilário Alencar and has been published by this book supported file pdf, txt, epub, kindle and other format this book has been release on 2022 with Curves, Algebraic categories.
This book features plane curves--the simplest objects in differential geometry--to illustrate many deep and inspiring results in the field in an elementary and accessible way. After an introduction to the basic properties of plane curves, the authors introduce a number of complex and beautiful topics, including the rotation number (with a proof of the fundamental theorem of algebra), rotation index, Jordan curve theorem, isoperimetric inequality, convex curves, curves of constant width, and the four-vertex theorem.