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A mathematician, a real one, one for whom mathematical objects are abstract and exist only in his mind or in some remote Platonic universe, never "sees" a curve. A curve is infinitely narrow and invisible. Yet, we all have "seen" straight lines, circles, parabolas, etc. when many years ago (for some of us) we were taught elementary geometry at school. E. Mach wanted to suppress from physics everything that could not be perceived: physics and metaphysics must not exist together. Many a scientist was deeply influenced by his philosophy. In his book Claude Tricot tells us that a curve has a non-vanishing width. Its width is that of the pencil or of the pen on the paper, or of the chalk on the blackboard. The abstract curve which cannot be seen and which does not really concern us here is the intersection of all those thick curves that contain it. For Claude Tricot it is only the thick curves that are pertinent. He describes in detail the way bumps, peaks, and irregularities appear on the curve as its width decreases. This is not a new point of view. Indeed Hausdorff and Bouligand initiated the idea at the beginning of this century. However, Claude Tricot manages to refine the theory extensively and interestingly. His approach is both realistic and mathematically rigorous. Mathematicians who only feed on abstractions as well as engineers who tackle tangible problems will enjoy reading this book.
Table des matières
I. Sets of Null Measure on the Line.- 1. Perfect Sets and Their Measure.- 2. Covers and Dimension.- 3. Contiguous Intervals and Dimension.- II. Rectifiable Curves.- 4. What Is a Curve?.- 5. Polygonal Curves and Length.- 6. Parameterized Curves, Support of a Measure.- 7. Local Geometry of Rectifiable Curves.- 8. Length, by Intersections with Straight Lines.- 9. The Length by the Area of Centered Balls.- III. Nonrectifiable Curves.- 10. Curves of Infinite Length.- 11. Fractal Curves.- 12. Graphs of Nondifferentiable Functions.- 13. Curves Constructed by Similarities.- 14. Deviation, and Expansive Curves.- 15. The Constant-Deviation Variable-Step Algorithm.- 16. Scanning a Curve with Straight Lines.- 17. Lateral Dimension of a Curve.- 18. Dimensional Homogeneity.- IV. Annexes, References and Index.- A. Upper Limit and Lower Limit.- A.1 Convergence.- A.2 Nonconvergent sequences.- A.3 Nonconvergent functions.- A.5 Some applications.- B. Two Covering Lemmas.- B.1 Vitali's lemma.- B.2 Covers by homothetic convex sets.- C. Convex Sets in the Plane.- C.1 Convexity.- C.2 Size of a convex set.- C.3 Breadth of a convex set.- C.4 Area of a convex set.- C.5 Convex hull.- C.6 Perimeter of the convex hull.- C.7 Area of the convex hull of a curve.- References.
Résumé
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.
