### Synopses & Reviews

" This concise text on geometry with computer modeling presents some elementary methods for analytical modeling and visualization of curves and surfaces. The author systematically examines such powerful tools as 2-D and 3-D animation of geometrical images, transformations, shadows, and colors, and then further studies more complex problems in differential geometry. Well-illustrated with more than 350 figures--reproduceable using Maple programs in the book--, the work is devoted to three main areas: curves, surfaces, and polyhedra. Pedagogical benefits can be found in the large number of Maple programs, some of which are analogous to C++ programs, including those for splines and fractals. To avoid tedious typing, readers will be able to download many of the programs from the Birkhäuser web site. Aimed at a broad audience of students, instructors of mathematics, computer scientists and engineers who have a knowledge of analytical geometry, i.e., method of coordinates, this text will be an excellent classroom resource or self-study reference. With over 100 stimulating exercises, problems and solutions, Geometry of Curves and Surfaces with Maple will integrate traditional differential and non-Euclidean geometries with more current computer algebra systems in a practical and user-friendly format. CONTENTS Preface Maple V: A Quick Reference Part I. Functions and Graphs with MAPLE Chapter 1. Graphs of Tabular and Continuous Functions 1.1 Basic Two-Dimensional Plots 1.2 Graphs of Functions Obtained from Elementary Functions 1.3 Graphs of Special Functions 1.4 Transformations of Graphs 1.5 Investigation of Functions Using Derivatives Chapter 2. Graphs of Composed Functions 2.1 Graphs of Piecewise-Continuous Functions 2.2 Graphs of Piecewise-Differentiable Functions Chapter 3. Interpolation of Functions 3.1 Polynomial Interpolation of Functions 3.2 Spline Interpolation of Functions 3.3 Construction of Curves Using Spline Functions Chapter 4. Smoothing of Functions 4.1 Method of Least Squares 4.2 Bezier Curves 4.3 Rational Bezier Curves 4.4 Smoothing Polynomial Splines Part II. Curves with MAPLE Chapter 5. Plane Curves in Rectangular Coordinates 5.1 What is a Curve? 5.2 Plotting of Cycloidal Curves 5.3 Experiment With Polar Coordinates 5.4 Some Other Remarkable Curves 5.5 Level Curves, Vector Fields, and Trajectories Chapter 6. Curves in Polar Coordinates 6.1 Basic Plots in Polar Coordinates 6.2 Remarkable Curves in Polar Coordinates 6.3 Inversion of Curves 6.4 Spirals 6.5 Roses and Crosses Chapter 7. Asymptotes of Curves Chapter 8. Space Curves 8.1 Introduction 8.2 Knitting on Surfaces of Revolution 8.3 Plotting of Curves (Tubes) with Shadow 8.4 Trajectories of Vector Fields in Space Chapter 9. Tangent Lines to a Curve 9.1 Tangent Lines 9.2 Envelope Curve of a Family of Curves 9.3 Mathematical Embroidery 9.4 Evolute and Evolvent (Involute). Caustic 9.5 Parallel Curves Chapter 10. Singular Points of Curves 10.1 Singular Points of Parametrized Curves 10.2 Singular Points of Implicitly Defined Plane Curves 10.3 Unusual Singular Points of Plane Curves Chapter 11. Length and Center of Mass of a Curve 11.1 Basic Facts 11.2 Calculation of Length and Center of Mass Chapter 12. Curvature and Torsion of Curves 12.1 Basic Facts 12.2 Curvature and Osculating Circle of a Plane Curve 12.3 Curvature and Torsion of a Space Curve 12.4 Natural Equations of a Curve Chapter 13. Fractal Curves and Dimension 13.1 Sierpinski's Curves 13.2 Peano Curves 13.3 Koch Curves 13.4 Dragon Curve (or Polygon) 13.5 Menger Curve Chapter 14. Spline Curves 14.1 Preliminary Facts and Examples 14.2 Composed Bezier Curves 14.3 Composed B -Spline Curves 14.4 Beta-Spline Curves 14.5 Interpolation Using Cubic Hermite's Curves 14.6 Composed Catmull-Rom Spline Curves Chapter 15. Non-Euclidean Geometry on the Half-Plane 15.1 Preliminary Facts 15.2 Examples of Visualization Chapter 16. Convex Hulls Chapter 17. Polyhedra with MAPLE 17.1 Regular Polyhedra 17.2 What is a Polyhedron 17.3 Platonic Solids 17.4 Star-Shaped Polyhedra Chapter 18. Semi-Regular Polyhedra 18.1 What Are Semi-Regular Polyhedra 18.2 Programs for Plotting Semi-Regular Polyhedra Part IV. Surfaces with MAPLE Chapter 19. Surfaces in Space 19.1 What Is a Surface 19.2 Regular Parametrized Surface 19.3 Methods of Generating Surfaces 19.4 Tangent Planes and Normal Vectors 19.5 Osculating Paraboloid and Type of a Smooth Point 19.6 Singular Points on Surfaces Chapter 20. Some Classes of Surfaces 20.1 Algebraic Surfaces 20.2 Surfaces of Revolution 20.3 Ruled Surfaces 20.4 Envelope of a One-Parameter Family of Surfaces Chapter 21. Some Other Classes of Surfaces 21.1 Canal Surfaces and Tubes 21.2 Translation Surfaces 21.3 Twisted Surfaces 21.4 Parallel Surfaces (Equidistants) 21.5 Pedal and Podoid Surfaces 21.6 Cissoidal and Conchoidal Maps 21.7 Inversion of a Surface References Index"

#### Review

"I was hunting for a book that would provide a set of practical exercises for the students of a graduate course entitled ""Geometric Modeling for Computer Graphics"" that I teach at the University of Bordeaux. The title of the book sounds appealing for such a purpose, isn't it? Anyway, it sounded appealing enough to me to buy the book (well, actually I did not buy it because the editor of Computer Graphics Forum sent it free... but that's another story). Once I open the book, my hope came true. Almost every topic you could imagine about curves and surfaces is somewhere inside: this includes common, and less common, definitions and properties (parametric and implicit form, rectangular and polar form, tangent, asymptote, envelope, normal, curvature, torsion, twist, length, center of mass, evolute and involute, pedal and podoid, etc) as well as the whole menagerie of usual, and less usual, curves and surfaces (polynomials and rational polynomials, B-splines, Bezier, Hermite, Catmul-Rom, Beta-splines, scalar and vector fields, polygons and polyhedra, fractals, etc). Of course 310 pages is a bit short to present all these topics deeply, but for each of them, there is at least a definition, an example, a piece of Maple source code and the resulting figure generated by the code (note that all the code pieces can be downloaded from the authors web page). My only reproach is that the classification used by the author is not always clear. But fortunately, the index is rich enough to easily find a topic you are interested in. To conclude, the book is clearly valuable for at least three kinds of people: First, people that are familiar with the mathematical aspect of curves and surfaces but unfamiliar with the computation and plotting possibilities providing by Maple, Second, people that are familiar with Maple but unfamiliar with curves and surfaces, Third, people that are unfamiliar with both topics. Great work, Mr. Rovenski. Pr. Christophe Schlick Computer Graphics Forum"

#### Synopsis

This book utilizes the power of Maple V to visually chart a path to curves and surfaces. Highly illustrated, the book clearly explains the geometry of surfaces, covering many aspects of curves, surfaces, and polyhedrons.

#### Synopsis

This concise text on geometry with computer modeling presents some
elementary methods for analytical modeling and visualization of curves
and surfaces. The author systematically examines such powerful tools
as 2-D and 3-D animation of geometric images, transformations,
shadows, and colors, and then further studies more complex problems in
differential geometry.
Well-illustrated with more than 350 figures---reproducible using Maple
programs in the book---the work is devoted to three main areas:
curves, surfaces, and polyhedra. Pedagogical benefits can be found in
the large number of Maple programs, some of which are analogous to C++
programs, including those for splines and fractals. To avoid tedious
typing, readers will be able to download many of the programs from the
Birkhauser web site.
Aimed at a broad audience of students, instructors of mathematics,
computer scientists, and engineers who have knowledge of analytical
geometry, i.e., method of coordinates, this text will be an excellent
classroom resource or self-study reference. With over 100 stimulating
exercises, problems and solutions, {\it Geometry of Curves and
Surfaces with Maple} will integrate traditional differential and non-
Euclidean geometries with more current computer algebra systems in a
practical and user-friendly format.

### Table of Contents

[see attached for detailed TOC]

Preface * MAPLE V: A Quick Reference * Part I: Functions and Graphs

with MAPLE * 1. Graphs of Tabular and Continuous Functions * 2. Graphs

of Composed Functions * 3. Interpolation of Functions * 4. Smoothing

of Functions * Part II: Curves with MAPLE * 5. Plane Curves in

Rectangular Coordinates * 6. Curves in Polar Coordinates * 7.

Asymptotes of Curves * 8. Space Curves * 9. Tangent Lines to a Curve *

10. Singular Points of Curves * 11. Length and Center of Mass of a

Curve * 12. Curvature and Torsion of Curves * 13. Fractal Curves and

Dimension * 14. Spline Curves * 15. Non-Euclidean Geometry on the Half

-Plane * 16. Convex Hulls * Part III: Polyhedra with MAPLE * 17.

Regular Polyhedra * 18. Semi-Regular Polyhedra * Part IV: Surfaces

with MAPLE * 19. Surfaces in Space * 20. Some Classes of Surfaces *

21. Some Other Classes of Surfaces * References * Index