3rd Edition

# Modern Differential Geometry of Curves and Surfaces with Mathematica

**Also available as eBook on:**

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

Cycloids

Lemniscates of Bernoulli

Cardioids

The Catenary

The Cissoid of Diocles

The Tractrix

Clothoids

Pursuit Curves

Exercises

Notebook

## Alternative Ways of Plotting Curves

Implicitly Defined Plane Curves

The Folium of Descartes

Cassinian Ovals

Plane Curves in Polar Coordinates

A Selection of Spirals

Exercises

Notebook 3

## New Curves from Old

Evolutes

Iterated Evolutes

Involutes

Osculating Circles to Plane Curves

Parallel Curves

Pedal Curves

Exercises

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

Exercises

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

Exercises

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

Exercises

Notebook 7

## Construction of Space Curves

The Fundamental Theorem of Space Curves

Assigned Curvature and Torsion

Contact

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

Exercises

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

Exercises

Notebook 10

## Nonorientable Surfaces

Orientability of Surfaces

Surfaces by Identification

The Möbius Strip

The Klein Bottle

Realizations of the Real Projective Plane

Twisted Surfaces

Exercises

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

Exercises

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

Exercises

Notebook 13

## Ruled Surfaces

Definitions and Examples

Curvature of a Ruled Surface

Tangent Developables

Noncylindrical Ruled Surfaces

Exercises

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

Exercises

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

Exercises

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

Exercises

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

Exercises

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

Exercises

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

Exercises

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

Exercises

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

Exercises

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

Exercises

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

Exercises

Notebook 24

## Riemannian Manifolds

Covariant Derivatives

Pseudo-Riemannian Metrics

The Classical Treatment of Metrics

The Christoffel Symbols in Riemannian Geometry

The Riemann Curvature Tensor

Exercises

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

Exercises

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

Exercises

Notebook

*Bibliography*

*Name Index*

*Subject Index*

*Notebook Index*

### Biography

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.”

— InEMS Newsletter, June 2007