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 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
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 DerivativesChapter 2. Graphs of Composed Functions
2.1 Graphs of Piecewise-Continuous Functions
2.2 Graphs of Piecewise-Differentiable FunctionsChapter 3. Interpolation of Functions
3.1 Polynomial Interpolation of Functions
3.2 Spline Interpolation of Functions
3.3 Construction of Curves Using Spline FunctionsChapter 4. Approximation of Functions
4.1 Method of Least Squares
4.2 Bezier Curves
4.3 Rational Bezier CurvesPart 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 TrajectoriesChapter 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 CrossesChapter 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 SpaceChapter 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 CurvesChapter 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 CurvesChapter 11. Length and Center of Mass of a Curve
11.1 Basic Facts
11.2 Calculation of Length and Center of MassChapter 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 CurveChapter 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 Curve14.1 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 CurvesChapter 15. Non-Euclidean Geometry on the Half-Plane
15.1 Preliminary Facts
15.2 Examples of VisualizationPart III. Polyhedra with MAPLE
17.1 What is a Polyhedron
17.2 Platonic Solids
17.3 Star-Shaped PolyhedraChapter 18. Semi-Regular Polyhedra
18.1 What Are Semi-Regular Polyhedra
18.2 Programs for Plotting Semi-Regular PolyhedraPart 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 SurfacesChapter 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 SurfacesChapter 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 SurfaceReferences
Index