3rd Edition

# CRC Standard Curves and Surfaces with Mathematica

**Also available as eBook on:**

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.

**Introduction**Concept of a Curve

Concept of a Surface

Coordinate Systems

Qualitative Properties of Curves and Surfaces

Classification of Curves and Surfaces

Basic Curve and Surface Operations

Method of Presentation

References

**Algebraic Functions**

Plotting Information for This Chapter

Functions with x

^{n/m}

Functions with x

^{n}and (a + bx)

^{m}

Functions with (a

^{2}+ x

^{2}) and x

^{m}

Functions with (a

^{2}− x

^{2}) and x

^{m}

Functions with (a

^{3}+ x

^{3}) and x

^{m}

Functions with (a

^{3}− x

^{3}) and x

^{m}

Functions with (a

^{4}+ x

^{4}) and x

^{m}

Functions with (a

^{4}− x

^{4}) and x

^{m}

Functions with √a + bx and x

^{m}

Functions with √a

^{2}− x

^{2}and x

^{m}

Functions with √x

^{2}− a

^{2}and x

^{m}

Functions with √a

^{2}+ x

^{2}and x

^{m }

Miscellaneous Functions

Functions Expressible in Polar Coordinates

Functions Expressed Parametrically

**Transcendental Functions**

Plotting Information for This Chapter

Functions with sin

^{n }(2π ax) and cos

^{m}(2πbx)(n,m integers)

Functions with 1 ± sin

^{n }(2π ax) and 1 ±} cos

^{m }(2πbx)

Functions with c sin

^{n}(ax) + d cos

^{m }(bx)

Functions of More Complicated Arguments

Inverse Trigonometric Functions

Logarithmic Functions

Exponential Functions

Hyperbolic Functions

Inverse Hyperbolic Functions

Trigonometric Combined with Exponential Functions

Trigonometric Functions Combined with Powers of x

Logarithmic Functions Combined with Powers of x

Exponential Functions Combined with Powers of x

Hyperbolic Functions Combined with Powers of x

Combined Trigonometric Functions, Exponential Functions, and Powers of x

Miscellaneous Functions

Functions Expressible in Polar Coordinates

Functions Expressible Parametrically

**Polynomial Sets**

Plotting Information for This Chapter

Orthogonal Polynomials

Nonorthogonal Polynomials

References

**Special Functions in Mathematical Physics**

Plotting Information for This Chapter

Exponential and Related Integrals

Sine and Cosine Integrals

Gamma and Related Functions

Error Functions

Fresnel Integrals

Legendre Functions

Bessel Functions

Modified Bessel Functions

Kelvin Functions

Spherical Bessel Functions

Modified Spherical Bessel Functions

Airy Functions

Riemann Functions

Parabolic Cylinder Functions

Elliptic Integrals

Jacobi Elliptic Functions

References

**Green’s Functions and Harmonic Functions**

Plotting Information for This Chapter

Green’s Function for the Poisson Equation

Green’s Function for the Wave Equation

Green’s Function for the Diffusion Equation

Green’s Function for the Helmholtz Equation

Miscellaneous Green’s Functions

Harmonic Functions: Solutions to Laplace’s Equation

References

**Special Functions in Probability and Statistics**

Plotting Information for This Chapter

Discrete Probability Densities

Continuous Probability Densities

Sampling Distributions

**Laplace Transforms**

Plotting Information for This Chapter

Elementary Functions

Algebraic Functions

Exponential Functions

Trigonometric Functions

References

**Nondifferentiable and Discontinuous Functions**

Plotting Information for This Chapter

Functions with a Finite Number of Discontinuities

Functions with an Infinite Number of Discontinuities

Functions with a Finite Number of Discontinuities in First Derivative

Functions with an Infinite Number of Discontinuities in First Derivative

**Random Processes**

Plotting Information for This Chapter

Elementary Random Processes

General Linear Processes

Integrated Processes

Fractal Processes

Poisson Processes

References

**Polygons**

Plotting Information for This Chapter

Polygons with Equal Sides

Irregular Triangles

Irregular Quadrilaterals

Polyiamonds

Polyominoes

Polyhexes

Miscellaneous Polygons

**Three-Dimensional Curves**

Plotting Information for This Chapter

Helical Curves

Sine Waves in Three Dimensions

Miscellaneous 3-D Curves

Knots

Links

References

**Algebraic Surfaces**

Plotting Information for This Chapter

Functions with ax + by

Functions with x

^{2}/a

^{2}± y

^{2}/b

^{2}

Functions with x

^{2}/a

^{2}+ y

^{2}/b

^{2}±c

^{2})

^{1/2}

Functions with x

^{3}/a

^{3}± y

^{3}/b

^{3}

Functions with x

^{4}/a

^{4}± y

^{4}/b

^{4}

Miscellaneous Functions

Miscellaneous Functions Expressed Parametrically

**Transcendental Surfaces**

Plotting Information for This Chapter

Trigonometric Functions

Logarithmic Functions

Exponential Functions

Trigonometric and Exponential Functions Combined

Surface Spherical Harmonics

**Complex Variable Surfaces**

Plotting Information for This Chapter

Algebraic Functions

Transcendental Functions

**Minimal Surfaces**

Plotting Information for This Chapter

Elementary Minimal Surfaces

Complex Minimal Surfaces

References

**Regular and Semi-Regular Solids with Edges**

Plotting Information for This Chapter

Platonic Solids

Archimedean Solids

Duals of Platonic Solids

Stellated (Star) Polyhedra

References

**Irregular and Miscellaneous Solids**

Plotting Information for This Chapter

Irregular Polyhedra

Miscellaneous Closed Surfaces with Edges

**Index**

### Biography

**David H. von Seggern, PhD**, worked for Teledyne Geotech from 1967 to 1982 in Alexandria, Virginia, almost exclusively on analysis of seismic data related to underground nuclear explosions. This effort was supported by the Air Force Office of Scientific Research (AFOSR) and by the Defense Advanced Research Projects Agency (DARPA). His research there addressed detection and discrimination of explosions, physics of the explosive source, explosive yield estimation, wave propagation, and application of statistical methods. Dr. von Seggern earned his PhD at Pennsylvania State University in 1982. He followed that with a 10-year position in geophysics research at Phillips Petroleum Company, where he became involved with leading-edge implementation of seismic imaging of oil and gas prospects and with seismic-wave modeling.