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

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.”
— In EMS Newsletter, June 2007