3rd Edition

Modern Differential Geometry of Curves and Surfaces with Mathematica

By Elsa Abbena, Simon Salamon, Alfred Gray Copyright 2006
    1016 Pages 531 B/W Illustrations
    by Chapman & Hall

    1016 Pages 531 B/W Illustrations
    by Chapman & Hall

    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.

    Curves in the Plane

    Euclidean Spaces
    Curves in Space
    The Length of a Curve
    Curvature of Plane Curves
    Angle Functions
    First Examples of Plane Curves
    The Semicubical Parabola and Regularity
    1.8 Exercises
    Notebook 1

    Famous Plane Curves

    Lemniscates of Bernoulli
    The Catenary
    The Cissoid of Diocles
    The Tractrix
    Pursuit Curves

    Alternative Ways of Plotting Curves

    Implicitly Defined Plane Curves
    The Folium of Descartes
    Cassinian Ovals
    Plane Curves in Polar Coordinates
    A Selection of Spirals
    Notebook 3

    New Curves from Old

    Iterated Evolutes
    Osculating Circles to Plane Curves
    Parallel Curves
    Pedal Curves
    Notebook 4
    Determining a Plane Curve from its Curvature
    Euclidean Motions
    Isometries of the Plane
    Intrinsic Equations for Plane Curves
    Examples of Curves with Assigned Curvature
    Notebook 5

    Global Properties of Plane Curves

    Total Signed Curvature
    Trochoid Curves
    The Rotation Index of a Closed Curve
    Convex Plane Curves
    The Four Vertex Theorem
    Curves of Constant Width
    Reuleaux Polygons and Involutes
    The Support Function of an Oval
    Notebook 6

    Curves in Space

    The Vector Cross Product
    Curvature and Torsion of Unit-Speed Curves
    The Helix and Twisted Cubic
    Arbitrary-Speed Curves in R3
    More Constructions of Space Curves
    Tubes and Tori
    Torus Knots
    Notebook 7

    Construction of Space Curves

    The Fundamental Theorem of Space Curves
    Assigned Curvature and Torsion
    Space Curves that Lie on a Sphere
    Curves of Constant Slope
    Loxodromes on Spheres
    8.7 Exercises
    Notebook 8

    Calculus on Euclidean Space

    Tangent Vectors to Rn
    Tangent Vectors as Directional Derivatives
    Tangent Maps or Differentials
    Vector Fields on R n
    Derivatives of Vector Fields
    Curves Revisited
    Notebook 9

    Surfaces in Euclidean Space

    Patches in Rn
    Patches in R3 and the Local Gauss Map
    The Definition of a Regular Surface
    Examples of Surfaces
    Tangent Vectors and Surface Mappings
    Level Surfaces in R3
    Notebook 10

    Nonorientable Surfaces

    Orientability of Surfaces
    Surfaces by Identification
    The Möbius Strip
    The Klein Bottle
    Realizations of the Real Projective Plane
    Twisted Surfaces
    Notebook 11

    Metrics on Surfaces

    The Intuitive Idea of Distance
    Isometries between Surfaces
    Distance and Conformal Maps
    The Intuitive Idea of Area
    Examples of Metrics
    Notebook 12

    Shape and Curvature

    The Shape Operator
    Normal Curvature
    Calculation of the Shape Operator
    Gaussian and Mean Curvature
    More Curvature Calculations
    A Global Curvature Theorem
    Nonparametrically Defined Surfaces
    Notebook 13

    Ruled Surfaces

    Definitions and Examples
    Curvature of a Ruled Surface
    Tangent Developables
    Noncylindrical Ruled Surfaces
    Notebook 14
    Surfaces of Revolution and Constant Curvature
    Surfaces of Revolution
    Principal Curves
    Curvature of a Surface of Revolution
    Generalized Helicoids
    Surfaces of Constant Positive Curvature
    Surfaces of Constant Negative Curvature
    More Examples of Constant Curvature
    Notebook 15

    A Selection of Minimal Surfaces

    Normal Variation
    Deformation from the Helicoid to the Catenoid
    Minimal Surfaces of
    More Examples of Minimal Surfaces
    Monge Patches and Scherk’s Minimal Surface
    The Gauss Map of a Minimal Surface
    Isothermal Coordinates
    Notebook 16

    Intrinsic Surface Geometry

    Intrinsic Formulas for the Gaussian Curvature
    Gauss’s Theorema Egregium
    Christoffel Symbols
    Geodesic Curvature of Curves on Surfaces
    Geodesic Torsion and Frenet Formulas
    Notebook 17
    Asymptotic Curves and Geodesics on Surfaces
    Asymptotic Curves
    Examples of Asymptotic Curves and Patches
    The Geodesic Equations
    First Examples of Geodesics
    Clairaut Patches
    Use of Clairaut Patches
    Notebook 18

    Principal Curves and Umbilic Points

    The Differential Equation for Principal Curves
    Umbilic Points
    The Peterson-Mainardi-Codazzi Equations
    Hilbert’s Lemma and Liebmann’s Theorem
    Triply Orthogonal Systems of Surfaces
    Elliptic Coordinates
    Parabolic Coordinates and a General Construction
    Parallel Surfaces
    The Shape Operator of a Parallel Surface
    Notebook 19

    Canal Surfaces and Cyclides of Dupin

    Surfaces Whose Focal Sets are 2-Dimensional
    Canal Surfaces
    Cyclides of Dupin via Focal Sets
    The Definition of Inversion
    Inversion of Surfaces
    Notebook 20
    The Theory of Surfaces of Constant Negative Curvature
    Intrinsic Tchebyshef Patches
    Patches on Surfaces of Constant Negative Curvature
    The Sine–Gordon Equation
    Tchebyshef Patches on Surfaces of Revolution
    The Bianchi Transform
    Moving Frames on Surfaces in R3
    Kuen’s Surface as Bianchi Transform of the Pseudosphere
    The B¨ acklund Transform
    Notebook 21

    Minimal Surfaces via Complex Variables

    Isometric Deformations of Minimal Surfaces
    Complex Derivatives
    Minimal Curves
    Finding Conjugate Minimal Surfaces
    The Weierstrass Representation
    Minimal Surfaces via Björling’s Formula
    Costa’s Minimal Surface
    Notebook 22
    Rotation and Animation using Quaternions
    Orthogonal Matrices
    Quaternion Algebra
    Unit Quaternions and Rotations
    Imaginary Quaternions and Rotations
    Rotation Curves
    Euler Angles
    Further Topics
    Notebook 23

    Differentiable Manifolds

    The Definition of a Differentiable Manifold
    Differentiable Functions on Manifolds
    Tangent Vectors on Manifolds
    Induced Maps
    Vector Fields on Manifolds
    Tensor Fields
    Notebook 24

    Riemannian Manifolds

    Covariant Derivatives
    Pseudo-Riemannian Metrics
    The Classical Treatment of Metrics
    The Christoffel Symbols in Riemannian Geometry
    The Riemann Curvature Tensor
    Notebook 25

    Abstract Surfaces and their Geodesics

    Christoffel Symbols on Abstract Surfaces
    Examples of Abstract Metrics
    The Abstract Definition of Geodesic Curvature
    Geodesics on Abstract Surfaces
    The Exponential Map and the Gauss Lemma
    Length Minimizing Properties of Geodesics
    Notebook 26

    The Gauss–Bonnet Theorem

    Turning Angles and Liouville’s Theorem
    The Local Gauss–Bonnet Theorem
    An Area Bound
    A Generalization to More Complicated Regions
    The Topology of Surfaces
    The Global Gauss–Bonnet Theorem .
    Applications of the Gauss–Bonnet Theorem
    Name Index
    Subject Index
    Notebook Index


    Abbena, Elsa; Salamon, Simon; Gray, Alfred

    “This is a nicely readable textbook on differential geometry. It offers an outstanding, comprehensive presentation of both theoretical and computational aspects … There are hundreds of illustrations that help the reader visualize the concepts. … It is a nicely written book, strongly recommended to all with an interest in differential geometry, its computational aspects and related fields.”
    — In EMS Newsletter, June 2007