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[资源] 【分享】CRC Standard Curves and Surfaces.CRC.1993【已搜无重复】

CRC Standard Curves and Surfaces
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Authors(Editors):
David von Seggern
Publisher: CRC
Pub Date: 2003
Pages: 416
ISBN: 0-8493-0196-3

Preface
Mathematical functions are a fundamental and prevalent ingredient in the
endeavors of scientists and engineers. The conclusions, predictions, and
analyses of such professionals are most often concisely contained in these
abstract relations, which are commonly referred to as "curves" when illustrated.
Many special curves can be found in mathematical tables, such as the
CRC Handbook of Mathematical Sciences,l and in mathematical dictionaries
(for example, James and James2 ). The National Bureau of Standard's handbook
(Abramowitz and Stegun3) is the acknowledged English-language source
for special functions in physics and engineering. The recent work entitled A
Catalog of Special Plane Curves4 is an excellent source for illustrations of
interesting functions in two dimensions. Yet, in spite of the frequent and
widespread use of mathematical functions, there has been, to date, no
volume in which a diversity of curves appear in graphic form. Unexpectedly,
there is no work which illustrates the spectrum of simple functions found in
most integral tables. Thus, there is not a single reference work which draws
together the entire gamut of forms which the modern scientist or engineer
uses within a career. Such a reference volume is long overdue, especially in
light of the fact that "curves" have become the ready tool of many other
disciplines due to the computational and storage powers of modern electronic
computers. Lastly, most of the curves appearing in older reference
works show the imprecision of hand drafting methods, and the reappearance
of familiar curves in precise, computer-plotted form should serve a useful
purpose in itself.
Curves are abstractions of the form and motion of the physical world.
Scientists have analyzed this world for millennia in order to render these
abstract expressions in the most minute detail, from gross astronomical
movements to infinitesimal atomic phenomena. It is now possible for a
remarkably detailed synthesis of natural phenomena to be created by the
proper use of these abstractions. Some such synthetic renditions have emerged
from the field of computer graphics already (mountainous terrain, cloud
formations, trees, to name a few) and are nearly indistinguishable from
reality. Modern scientists' skillful mathematical description of the motion of
nature, coupled with modern computing power, has also enabled them to
make increasingly accurate predictions of natural events, such as weather,
earthquakes, and oceanic currents. All such endeavors involve, as the quantitative
basis, functions whose curves are the visual representation of the
predicted motion. Scientists and engineers can use this reference work in two
ways to aid their work. In the fOlWard manner, they can look up the equation
of interest and see the corresponding visual form of the curve. In the inverse
manner, they can select a particular curve visually to serve in data fitting or in
computer modeling exercises.
This handbook, however, purports to serve a larger audience than those
engaged in mathematics, science, and engineering. Architects, designers,
draftsmen, and artists should benefit from this reference book of curves. New
expressions of form can be imagined through even a casual scanning of the
contents of this volume. And if one has general notion of the desired
appearance, the appropriate curve can be located in this volume and its
mathematical expression noted. The mathematical expressions given here can
be readily translated into high-level programming languages (for example,
FORTRAN) in order to generate a given curve in a particular environment of
application. Recent graphics languages enable cells, segments, or symbols to
be created once and stored for future use. These abstractions, which can be
composed of one or more curve segments, may be placed in a computer-based
design at any scale or rotation angle to achieve the desired effect. The
computer revolution indeed makes curve generation easy and rapid and
eliminates the former laborious hand calculations necessary to graph even
the simplest curves. Achieving the most intricate and subtlest abstract forms,
as well as the simple and plain, is possible for those who have only a
rudimentary programming knowledge. Properly designed computer programs
can open up this possibility even to those who have no grasp of the
underlying equations.
This work is intended to contain all curves in common use in applied
mathematics. In order to be comprehensive, the notion of "curve" has been
extended beyond its usual connection with algebraic or transcendental functions.
Here "curve" means any line or surface in two or three dimensions
which can be generated by a rule or set of rules expressible in mathematical
terms. Such rules may be entirely smooth and deterministic, and the first part
of this handbook is devoted to the curves represented in this way: algebraic
forms, transcendental forms, and special integrals. Here mathematicians,
scientists, and engineers will find those curves familiar to them. Selection of
functions for curve fitting can be eased by use of this handbook, and
questions concerning the form of a given function can be quickly settled.
Designers can find curves appropriate to their design goals. The latter part of
this handbook comprises curves and surfaces which are not smoothly generated
by a single relation, such as piecewise continuous functions, polygons,
and polyhedra.
When the generating rules include random components, a new series of
curves and surfaces emerges-the subject of the final chapter. The need for
cataloging such curves is due to the work of Mandelbrot,5 who has shown
that the study and description of the random component of natural phenomena
is as important, if not more important, than that of the deterministic
component. A future volume will collect together many interesting and
unusual curves which are not normally considered in pure mathematics.
These curves will be most useful to artists and designers who are able to
employ modern computer-assisted art and drafting systems.
This handbook begins with a chapter containing a qualitative summary of
deterministic curve properties and a classification of such curves. An explanation
of the means and conventions of presentation in later chapters is also
given here. This first chapter is meant to acquaint the reader with fundamental
mathematical properties of curves in order that application of the material
of the handbook can be more knowledgeable and meaningful. Those with
a solid background in calculus will find little new information here. A section
on matrix transformations has been included to indicate how a given curve
can be made to appear in many different forms.
The following chapters are organized so that similar curves are grouped
together for easy reference. Early chapters deal with curves in two dimensions,
progressing from the simple to the complex. Later chapters extend the
notion of curve to curves and surfaces in three dimensions. Final chapters
deal with piecewise continuous functions in two and three dimensions.

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