Geometry of Curves and Surfaces with MAPLE

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.