General Theory of Irregular Curves

Main Authors: Alexandrov, V.V., Reshetnyak, Yu.G. (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Published: Dordrecht Springer Netherlands 1989, 1989
Edition:1st ed. 1989
Series:Mathematics and its Applications, Soviet Series
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • I: General Notion of a Curve
  • 1.1. Definition of a Curve
  • 1.2. Normal Parametrization of a Curve
  • 1.3. Chains on a Curve and the Notion of an Inscribed Polygonal Line
  • 1.4. Distance Between Curves and Curve Convergence
  • 1.5. On a Non-Parametric Definition of the Notion of a Curve
  • II: Length of a Curve
  • 2.1. Definition of a Curve Length and its Basic Properties
  • 2.2. Rectifiable Curves in Euclidean Spaces
  • 2.3. Rectifiable Curves in Lipshitz Manifolds
  • III: Tangent and the Class of One-Sidedly Smooth Curves
  • 3.1. Definition and Basic Properties of One-Sidedly Smooth Curves
  • 3.2. Projection Criterion of the Existence of a Tangent in the Strong Sense
  • 3.3. Characterizing One-Sidedly Smooth Curves with Contingencies
  • 3.4. One-Sidedly Smooth Functions
  • 3.5. Notion of c-Correspondence. Indicatrix of Tangents of a Curve
  • 3.6. One-Sidedly Smooth Curves in Differentiable Manifolds
  • IV: Some Facts of Integral Geometry
  • 8.3. Complete Two-Dimensional Indicatrix of a Curve of a Finite Complete Torsion
  • 8.4. Continuity and Additivity of Absolute Torsion
  • 8.5. Definition of an Absolute Torsion Through Triple Chains and Paratingences
  • 8.6. Right-Hand and Left-Hand Indices of a Point. Complete Torsion of a Curve
  • IX: Frenet Formulas and Theorems on Natural Parametrization
  • 9.1. Frenet Formulas
  • 9.2. Theorems on Natural Parametrization
  • X: Some Additional Remarks
  • References
  • 4.1. Manifold Gnk of k-Dimensional Directions in Vn
  • 4.2. Imbedding of Gnk into a Euclidean Space
  • 4.3. Existence of Invariant Measure of Gnk
  • 4.4. Invariant Measure in Gnk and Integral. Uniqueness of an Invariant Measure
  • 4.5. Some Relations for Integrals Relative to the Invariant Measure in Gnk
  • 4.6. Some Specific Subsets of Gnk
  • 4.7. Length of a Spherical Curve as an Integral of the Function Equal to the Number of Intersection Points
  • 4.8. Length of a Curve as an Integral of Lengths of its Projections
  • 4.9. Generalization of Theorems on the Mean Number of the Points of Intersection and Other Problems
  • V: Turn or Integral Curvature of a Curve
  • 5.1. Definition of a Turn. Basic Properties of Curves of a Finite Turn
  • 5.2. Definition of a Turn of a Curve by Contingencies
  • 5.3. Turn of a Regular Curve
  • 5.4. Analytical Criterion of Finiteness of a Curve Turn
  • 5.5. Basic Integra-Geometrical Theorem on a Curve Turn
  • 5.6. Some Estimates and Theorems on a Limiting Transition
  • 5.7. Turn of a Curve as a Limit of the Sum of Angles Between the Secants
  • 5.8. Exact Estimates of the Length of a Curve
  • 5.9. Convergence with a Turn
  • 5.10 Turn of a Plane Curve
  • VI: Theory of a Turn on an n-Dimensional Sphere
  • 6.1. Auxiliary Results
  • 6.2. Integro-Geometrical Theorem on Angles and its Corrolaries
  • 6.3. Definition and Basic Properties of Spherical Curves of a Finite Geodesic Turn
  • 6.4. Definition of a Geodesic Turn by Means of Tangents
  • 6.5. Curves on a Two-Dimensional Sphere
  • VII: Osculating Planes and Class of Curves with an Osculating Plane in the Strong Sense
  • 7.1. Notion of an Osculating Plane
  • 7.2. Osculating Plane of a Plane Curve
  • 7.3. Properties of Curves with an Osculating Plane in the Strong Sense
  • VIII: Torsion of a Curve in a Three-Dimensional Euclidean Space
  • 8.1. Torsion of a Plane Curve
  • 8.2. Curves of a Finite Complete Torsion