Geometry of Curves and Surfaces with MAPLE

Przednia okładka
Springer Science & Business Media, 26 kwi 2000 - 310
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.
 

Spis treści

Graphs of Tabular and Continuous Functions
4
14 Transformations of Graphs
4
15 Investigation of Functions Using Derivatives
4
Graphs of Composed Functions
4
22 Graphs of PiecewiseDifferentiable Functions
ii
Interpolation of Functions
ix
32 Spline Interpolation of Functions
xi
33 Constructing Curves Using Spline Functions
xi
Fractal Curves and Dimension
71
132 Peano Curves
75
133 Koch Curves
78
134 Dragon Curve or Polygon
82
135 The Menger Curve
85
Spline Curves
89
141 Preliminary Facts and Examples
90
142 Composed Bezier Curves
92

Approximation of Functions
7
42 Bezier Curves
8
43 Rational Bezier Curves
9
Curves with MAPLE
11
Plane Curves in Rectangular Coordinates
13
51 What Is a Curve?
14
52 Plotting Cycloidal Curves
15
53 Experiment with Polar Coordinates
18
54 Some Other Remarkable Curves
19
55 Level Curves Vector Fields and Trajectories
20
56 Level Curves of Functions and Extremal Problems
23
Curves in Polar Coordinates
27
62 Remarkable Curves in Polar Coordinates
30
63 Inversion of Curves
33
64 Spirals
35
65 Roses and Crosses
36
Asymptotes of Curves
39
Space Curves
31
82 Knitting on Surfaces of Revolution
35
83 Plotting Curves Tubes with Shadow
41
84 Trajectories of Vector Fields in Space
47
Tangent Lines to a Curve
49
93 Mathematical Embroidery
49
Caustic
51
Singular Points on Curves
53
101 Singular Points on Parametrized Curves
53
102 Singular Points on Implicitly Defined Plane Curves
53
103 Unusual Singular Points on Plane Curves
53
Length and Center of Mass of a Curve
55
112 Calculation of Length and Center of Mass
57
Curvature and Torsion of Curves
63
122 Curvature and Osculating Circle of a Curve in the Plane
65
123 Curvature and Torsion of a Curve in Space
67
124 Natural Equations of a Curve
69
143 Composed BSpline Curves
95
144 BetaSpline Curves
97
145 Interpolation Using Cubic Hermite Curves
104
146 Composed CatmullRom Spline Curves
106
NonEuclidean Geometry in the HalfPlane
109
152 Examples of Visualization
111
Convex Hulls
125
Polyhedra with MAPLE
127
Regular Polyhedra
129
171 What Is a Polyhedron?
130
172 Platonic Solids
134
173 StarShaped Polyhedra
140
SemiRegular Polyhedra
146
181 What Are SemiRegular Polyhedra?
146
Surfaces with MAPLE
163
Surfaces in Space
165
192 Regular Parametrized Surface
169
194 Tangent Planes and Normal Vectors
177
195 The Osculating Paraboloid and a Type of Smooth Point
186
196 Singular Points on Surfaces
193
Some Classes of Surfaces
199
202 Surfaces of Revolution
202
203 Ruled Surfaces
209
204 Envelope of a OneParameter Family of Surfaces
218
Some Other Classes of Surfaces
225
212 Translation Surfaces
227
213 Twisted Surfaces
228
214 Parallel Surfaces Equidistants
229
216 Cissoidal and Conchoidal Maps
232
217 Inversion of a Surface
233
References
237
Index
241
Prawa autorskie

Inne wydania - Wyświetl wszystko

Kluczowe wyrazy i wyrażenia

Odniesienia do tej książki

Informacje bibliograficzne