Geometry of Curves and Surfaces with MapleVladimir Rovenski, Technion & University of Haifa, Israel 0-8176-4074-6 * 2000 * \$49.95 tent. * Hardcover * 310 pages * 391 Illustrations

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.

Click chapter titles to see extracts from each chapter.

CONTENTS

Preface

Maple V: A Quick Reference

Part I. Functions and Graphs with MAPLE

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

2.1 Graphs of Piecewise-Continuous Functions
2.2 Graphs of Piecewise-Differentiable Functions

3.1 Polynomial Interpolation of Functions
3.2 Spline Interpolation of Functions
3.3 Construction of Curves Using Spline Functions

4.1 Method of Least Squares
4.2 Bezier Curves
4.3 Rational Bezier Curves

Part II. Curves with MAPLE

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

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

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

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

10.1 Singular Points of Parametrized Curves
10.2 Singular Points of Implicitly Defined Plane Curves
10.3 Unusual Singular Points of Plane Curves

11.1 Basic Facts
11.2 Calculation of Length and Center of Mass

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

13.1 Sierpinski's Curves
13.2 Peano Curves
13.3 Koch Curves
13.4 Dragon Curve (or Polygon)
13.5 Menger Curve

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

15.1 Preliminary Facts
15.2 Examples of Visualization

Chapter 16. Convex Hulls

Part III. Polyhedra with MAPLE

17.1 What is a Polyhedron
17.2 Platonic Solids
17.3 Star-Shaped Polyhedra

18.1 What Are Semi-Regular Polyhedra
18.2 Programs for Plotting Semi-Regular Polyhedra

Part IV. Surfaces with MAPLE

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

20.1 Algebraic Surfaces
20.2 Surfaces of Revolution
20.3 Ruled Surfaces
20.4 Envelope of a One-Parameter Family 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