Written for mathematicians, engineers, and researchers in experimental science, as well as anyone interested in fractals, this book explains the geometrical and analytical properties of trajectories, aggregate contours, geographical coastlines, profiles of rough surfaces, and other curves of finite and fractal length. The approach is by way of precise definitions from which properties are deduced and applications and computational methods are derived. Written without the traditional heavy symbolism of mathematics texts, this book requires two years of calculus while also containing material…mehr
Written for mathematicians, engineers, and researchers in experimental science, as well as anyone interested in fractals, this book explains the geometrical and analytical properties of trajectories, aggregate contours, geographical coastlines, profiles of rough surfaces, and other curves of finite and fractal length. The approach is by way of precise definitions from which properties are deduced and applications and computational methods are derived. Written without the traditional heavy symbolism of mathematics texts, this book requires two years of calculus while also containing material appropriate for graduate coursework in curve analysis and/or fractal dimension.
'I. Sets of Null Measure on the Line.- 1. Perfect Sets and Their Measure.- 1.1 Duality set-measure.- 1.2 Closed sets and contiguous intervals.- 1.3 Perfect sets.- 1.4 Binary trees and the power of perfect sets.- 1.5 Symmetrical perfect sets.- 1.6 Tree representation of perfect sets.- 1.7 Bibliographical notes.- 2. Covers and Dimension.- 2.1 What is a null measure?.- 2.2 Hierarchy of sets of null measure.- 2.3 Cantor-Minkowski measure.- 2.4 Space filling and the order of growth.- 2.5 Orders of growth and dimension.- 2.6 Equivalent definitions of the dimension.- 2.7 Examples of computing the dimension.- 2.8 Some properties of the dimension.- 2.9 Upper and lower dimensions.- 2.10 Bibliographical notes.- 3. Contiguous Intervals and Dimension.- 3.1 Borel's logarithmic rarefaction.- 3.2 Index of Besicovitch-Taylor.- 3.3 Equivalent orders of growth.- 3.4 The contiguous intervals and the fractal dimension.- 3.5 Algorithms to compute the dimension.- 3.6 Bibliographical notes.- II. Rectifiable Curves.- 4. What Is a Curve?.- 4.1 Some types of sets in the plane.- 4.2 Velocities, trajectories.- 4.3 The definition of a curve.- 4.4 Bibliographical notes.- 5. Polygonal Curves and Length.- 5.1 Rectifiability.- 5.2 Hausdorff distance.- 5.3 Polygonal approximations.- 5.4 The length of a curve.- 5.5 Two distinct notions.- 5.6 Measuring the length by compass.- 5.7 Bibliographical notes.- 6. Parameterized Curves, Support of a Measure.- 6.1 Parameterization by arc length.- 6.2 Image measure.- 6.3 Length by instantaneous velocity.- 6.4 The devil staircase.- 6.5 Length by the average of local velocity.- 6.6 Bibliographical notes.- 7. Local Geometry of Rectifiable Curves.- 7.1 Tangent, cone, convex hulls.- 7.2 Relations between local properties.- 7.3 Counterexamples.- 7.4 Tangent almost everywhere.- 7.5 Local length, almost everywhere.- 7.6 Rectifiability revisited.- 7.7 Bibliographical notes.- 8. Length, by Intersections with Straight Lines.- 8.1 Intersections, projections.- 8.2 The measure of families of straight lines.- 8.3 Family of lines intersecting a set.- 8.4 The case of convex sets.- 8.5 Length by secant lines.- 8.6 The length by projections.- 8.7 Application: practical computation of length.- 8.8 The length by random intersections.- 8.9 Buffon needle.- 8.10 Bibliographical notes.- 9. The Length by the Area of Centered Balls.- 9.1 Minkowski sausage.- 9.2 Length by the area of sausages.- 9.3 Convergence of the algorithm of the sausages.- 9.4 Reduction of balls to parallel segments.- 9.5 Bibliographical notes.- III. Nonrectifiable Curves.- 10. Curves of Infinite Length.- 10.1 What is infinite length?.- 10.2 Two examples.- 10.3 Dimension.- 10.4 Some examples of dimensions of curves.- 10.5 Classical covers: balls and boxes.- 10.6 Covers by figures of any kind.- 10.7 Covering curves by crosses.- 10.8 Bibliographical notes.- 11. Fractal Curves.- 11.1 What is a fractal curve?.- 11.2 A fractal curve is nowhere rectifiable.- 11.3 Diameter, size.- 11.4 Characterization of a fractal curve.- 12. Graphs of Nondifferentiable Functions.- 12.1 Curves parameterized by the abscissa.- 12.3 Size of local arcs.- 12.3 Variation of a function.- 12.4 Fractal dimension of a graph.- 12.5 Hölder exponent.- 12.6 Functions defined by series.- 12.7 Weierstrass function.- 12.8 Fractal dimension and the structure function.- 12.9 Functions constructed by diagonal affinities.- 12.10 Invariance under change of scale.- 12.11 The Weierstrass-Mandelbrot function.- 12.12 The spectrum of invariant functions.- 12.13 Computing the dimensions of the graphs.- 12.14 Bibliographical notes.- 13. Curves Constructed by Similarities.- 13.1 Similarities.- 13.2 Self-similar structure.- 13.3 Generator.- 13.4 Self-similar structure on [0,1].- 13.5 Parameterization of the generator.- 13.6 The limit curve ?.- 13.7 Simplicity criterion.- 13.8 Similarity and dimension exponent.- 13.9 Examples.- 13.10 Th
'I. Sets of Null Measure on the Line.- 1. Perfect Sets and Their Measure.- 1.1 Duality set-measure.- 1.2 Closed sets and contiguous intervals.- 1.3 Perfect sets.- 1.4 Binary trees and the power of perfect sets.- 1.5 Symmetrical perfect sets.- 1.6 Tree representation of perfect sets.- 1.7 Bibliographical notes.- 2. Covers and Dimension.- 2.1 What is a null measure?.- 2.2 Hierarchy of sets of null measure.- 2.3 Cantor-Minkowski measure.- 2.4 Space filling and the order of growth.- 2.5 Orders of growth and dimension.- 2.6 Equivalent definitions of the dimension.- 2.7 Examples of computing the dimension.- 2.8 Some properties of the dimension.- 2.9 Upper and lower dimensions.- 2.10 Bibliographical notes.- 3. Contiguous Intervals and Dimension.- 3.1 Borel's logarithmic rarefaction.- 3.2 Index of Besicovitch-Taylor.- 3.3 Equivalent orders of growth.- 3.4 The contiguous intervals and the fractal dimension.- 3.5 Algorithms to compute the dimension.- 3.6 Bibliographical notes.- II. Rectifiable Curves.- 4. What Is a Curve?.- 4.1 Some types of sets in the plane.- 4.2 Velocities, trajectories.- 4.3 The definition of a curve.- 4.4 Bibliographical notes.- 5. Polygonal Curves and Length.- 5.1 Rectifiability.- 5.2 Hausdorff distance.- 5.3 Polygonal approximations.- 5.4 The length of a curve.- 5.5 Two distinct notions.- 5.6 Measuring the length by compass.- 5.7 Bibliographical notes.- 6. Parameterized Curves, Support of a Measure.- 6.1 Parameterization by arc length.- 6.2 Image measure.- 6.3 Length by instantaneous velocity.- 6.4 The devil staircase.- 6.5 Length by the average of local velocity.- 6.6 Bibliographical notes.- 7. Local Geometry of Rectifiable Curves.- 7.1 Tangent, cone, convex hulls.- 7.2 Relations between local properties.- 7.3 Counterexamples.- 7.4 Tangent almost everywhere.- 7.5 Local length, almost everywhere.- 7.6 Rectifiability revisited.- 7.7 Bibliographical notes.- 8. Length, by Intersections with Straight Lines.- 8.1 Intersections, projections.- 8.2 The measure of families of straight lines.- 8.3 Family of lines intersecting a set.- 8.4 The case of convex sets.- 8.5 Length by secant lines.- 8.6 The length by projections.- 8.7 Application: practical computation of length.- 8.8 The length by random intersections.- 8.9 Buffon needle.- 8.10 Bibliographical notes.- 9. The Length by the Area of Centered Balls.- 9.1 Minkowski sausage.- 9.2 Length by the area of sausages.- 9.3 Convergence of the algorithm of the sausages.- 9.4 Reduction of balls to parallel segments.- 9.5 Bibliographical notes.- III. Nonrectifiable Curves.- 10. Curves of Infinite Length.- 10.1 What is infinite length?.- 10.2 Two examples.- 10.3 Dimension.- 10.4 Some examples of dimensions of curves.- 10.5 Classical covers: balls and boxes.- 10.6 Covers by figures of any kind.- 10.7 Covering curves by crosses.- 10.8 Bibliographical notes.- 11. Fractal Curves.- 11.1 What is a fractal curve?.- 11.2 A fractal curve is nowhere rectifiable.- 11.3 Diameter, size.- 11.4 Characterization of a fractal curve.- 12. Graphs of Nondifferentiable Functions.- 12.1 Curves parameterized by the abscissa.- 12.3 Size of local arcs.- 12.3 Variation of a function.- 12.4 Fractal dimension of a graph.- 12.5 Hölder exponent.- 12.6 Functions defined by series.- 12.7 Weierstrass function.- 12.8 Fractal dimension and the structure function.- 12.9 Functions constructed by diagonal affinities.- 12.10 Invariance under change of scale.- 12.11 The Weierstrass-Mandelbrot function.- 12.12 The spectrum of invariant functions.- 12.13 Computing the dimensions of the graphs.- 12.14 Bibliographical notes.- 13. Curves Constructed by Similarities.- 13.1 Similarities.- 13.2 Self-similar structure.- 13.3 Generator.- 13.4 Self-similar structure on [0,1].- 13.5 Parameterization of the generator.- 13.6 The limit curve ?.- 13.7 Simplicity criterion.- 13.8 Similarity and dimension exponent.- 13.9 Examples.- 13.10 Th
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