CRC Standard Curves and Surfaces with Mathematica

David H. von Seggern

• Publisher: Chapman and Hall/CRC
• Year: 2017
• ISBN:
  9781315355290
• Edition: 3
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Since the publication of this book's bestselling predecessor, Mathematica® has matured considerably and the computing power of desktop computers has increased greatly. The Mathematica® typesetting functionality has also become sufficiently robust that the final copy for this edition could be transformed directly from Mathematica R notebooks to LaTex input.Incorporating these aspects, CRC Standard Curves and Surfaces with Mathematica®, Third Edition is a virtual encyclopedia of curves and functions that depicts nearly all of the standard mathematical functions and geometrical figures in use today. The overall format of the book is largely unchanged from the previous edition, with function definitions and their illustrations presented closely together.New to the Third Edition:

• A new chapter on Laplace transforms
• New curves and surfaces in almost every chapter
• Several chapters that have been reorganized
• Better graphical representations for curves and surfaces throughout
• Downloadable resources, including the entire book in a set of interactive CDF (Computable Document Format) files

The book presents a comprehensive collection of nearly 1, 000 illustrations of curves and surfaces often used or encountered in mathematics, graphics design, science, and engineering fields. One significant change with this edition is that, instead of presenting a range of realizations for most functions, this edition presents only one curve associated with each function. The graphic output of the Manipulate function is shown exactly as rendered in Mathematica, with the exact parameters of the curve's equation shown as part of the graphic display. This enables readers to gauge what a reasonable range of parameters might be while seeing the result of one particular choice of parameters.

1

Introduction

1.1 Concept of a Curve

Let Eⁿ be the Euclidean space of dimension n. (According to this definition, E¹ is a line, E² is a plane, and E³ is a volume.) A curve in n-space is defined as the set of points which result when a mapping from E¹ to Eⁿ is performed. In this reference work, only curves in E² and E³ will be considered. Let t represent the independent variable in E¹. An E² curve is then given by

$x=f(t),y=g(t)$

and an E³ curve by

$x=f(t),y=g(t),z=h(t)$

where f, g, and h mean “function of.” The domain of t is usually (0, 2π),(−∞, ∞), or (0, ∞). These are the parametric representations of a curve. However, in E² curves are commonly expressed as

$y=f(x)$

or as

$f(x,y)=0$

which are the explicit and implicit forms, respectively. The explicit form is readily reducible from the parametric form when x = f(t) = t in E² and when x = f(t) = t and y = g(t)= t in E³. The implicit form of a curve will often comprise more points than a corresponding explicit form. For example, y² – x = 0 has two ranges in y, one positive and one negative, while the explicit form derived from solving the above equation gives y = $\sqrt{x}$ for which the range of y is positive only.

Generally, the definition of a curve imposes a smoothness criterion,¹ meaning that the trace of the curve has no abrupt changes of direction (continuous first derivative). However, for purposes of this reference work, a broader definition of curve is proposed. Here, a curve may be composed of smooth branches, each satisfying the above definition, provided that the intervals over which the curve branches are distinctly defined and are contiguous. This definition will encompass forms such as polygons or sawtooth functions.

1.2 Concept of a Surface

This reference work defines surfaces as existing only in E³. Therefore a surface is defined as the mapping from E² to E³ according to

$\begin{array}{l}x=f(s,t),\\ y=g(s,t),\\ z=h(s,t).\end{array}$

FIGURE 1.1
The Cartesian coordinate system for two dimensions.

As for curves, the conversion from this parametric form to more common forms

$z=f(x,y)$

or

$f(x,y,z)=0$

may not be possible in some cases. Again, a smoothness criterion¹ is desirable; but the generalized definition of surface requires that this smoothness criterion only be satisfied piecewise for all distinct mappings of the (s,t) plane over which the surface is defined. These generalized surfaces are termed manifolds. Cubes are examples of surfaces which can be defined in this deterministic manner.

1.3 Coordinate Systems

The number of available coordinate systems for representing curves is large and even larger for surfaces. However, to maintain uniformity of presentation throughout this volume, only the following will be used:

2 – D	3 – D
Cartesian, polar	Cartesian, cylindrical, spherical

The term parametric is often used as though it were a coordinate system, but it is really a representation of coordinates in terms of an additional independent parameter which is not itself a coordinate of the E³ space in ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface to the Third Edition
• Author
• 1 Introduction
• 2 Algebraic Functions
• 3 Transcendental Functions
• 4 Polynomial Sets
• 5 Special Functions in Mathematical Physics
• 6 Green’s Functions and Harmonic Functions
• 7 Special Functions in Probability and Statistics
• 8 Laplace Transforms
• 9 Nondifferentiable and Discontinuous Functions
• 10 Random Processes
• 11 Polygons
• 12 Three-Dimensional Curves
• 13 Algebraic Surfaces
• 14 Transcendental Surfaces
• 15 Complex Variable Surfaces
• 16 Minimal Surfaces
• 17 Regular and Semi-Regular Solids with Edges
• 18 Irregular and Miscellaneous Solids
• Index