Geometry of Curves and Surfaces with MAPLE

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:
MAPLE V: A Quick Reference.- I Functions and Graphs with MAPLE.- 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.- 2 Graphs of Composed Functions.- 2.1 Graphs of Piecewise-Continuous Functions.- 2.2 Graphs of Piecewise-Differentiable Functions.- 3 Interpolation of Functions.- 3.1 Polynomial Interpolation of Functions.- 3.2 Spline Interpolation of Functions.- 3.3 Constructing Curves Using Spline Functions.- 4 Approximation of Functions.- 4.1 Method of Least Squares.- 4.2 Bezier Curves.- 4.3 Rational Bezier Curves.- II Curves with MAPLE.- 5 Plane Curves in Rectangular Coordinates.- 5.1 What Is a Curve?.- 5.2 Plotting Cycloidal Curves.- 5.3 Experiment with Polar Coordinates.- 5.4 Some Other Remarkable Curves.- 5.5 Level Curves, Vector Fields, and Trajectories.- 5.6 Level Curves of Functions and Extremal Problems.- 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.- 7 Asymptotes of Curves.- 8 Space Curves.- 8.1 Introduction.- 8.2 Knitting on Surfaces of Revolution.- 8.3 Plotting Curves (Tubes) with Shadow.- 8.4 Trajectories of Vector Fields in Space.- 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.- 10 Singular Points on Curves.- 10.1 Singular Points on Parametrized Curves.- 10.2 Singular Points on Implicitly Defined Plane Curves..- 10.3 Unusual Singular Points on Plane Curves.- 11 Length and Center of Mass of a Curve.- 11.1 Basic Facts.- 11.2 Calculation of Length and Center of Mass.- 12 Curvature and Torsion of Curves.- 12.1 Basic Facts.- 12.2 Curvature and Osculating Circle of a Curve in the Plane.- 12.3 Curvature and Torsion of a Curve in Space.- 12.4 Natural Equations of a Curve.- 13 Fractal Curves and Dimension.- 13.1 Sierpi?ski’s Curves.- 13.2 Peano Curves.- 13.3 Koch Curves.- 13.4 Dragon Curve (or Polygon).- 13.5 The Menger Curve.- 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 Curves.- 14.6 Composed Catmull-Rom Spline Curves.- 15 Non-Euclidean Geometry in the Half-Plane.- 15.1 Preliminary Facts.- 15.2 Examples of Visualization.- 16 Convex Hulls.- III Polyhedra with MAPLE.- 17 Regular Polyhedra.- 17.1 What Is a Polyhedron?.- 17.2 Platonic Solids.- 17.3 Star-Shaped Polyhedra.- 18 Semi-Regular Polyhedra.- 18.1 What Are Semi-Regular Polyhedra?.- 18.2 Programs for Plotting Semi-Regular Polyhedra.- IV Surfaces with MAPLE.- 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 The Osculating Paraboloid and a Type of Smooth Point.- 19.6 Singular Points on Surfaces.- 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.- 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.

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'.... The title of [this] book sounds appealing for such a purpose.... 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 author’s web page).... 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 who are familiar with the mathematical aspect of curves and surfaces but unfamiliar with the computation and plotting possibilities providing by Maple; second, people who are familiar with Maple but unfamiliar with curves and surfaces; third, people who are unfamiliar with both topics." — Computer Graphics Forum "The book can be recommended to students of mathematics, engineering or computer science, who have already a basic knowledge of MAPLE and are interested in the visualizations of geometry." ---Zentralblatt MATH