Geometry of Curves and Surfaces with MAPLESpringer 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 |
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