General Theory of Irregular Curves
Main Authors:  , 

Corporate Author:  
Format:  eBook 
Language:  English 
Published: 
Dordrecht
Springer Netherlands
1989, 1989

Edition:  1st ed. 1989 
Series:  Mathematics and its Applications, Soviet Series

Subjects:  
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 NonParametric 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 OneSidedly Smooth Curves
 3.1. Definition and Basic Properties of OneSidedly Smooth Curves
 3.2. Projection Criterion of the Existence of a Tangent in the Strong Sense
 3.3. Characterizing OneSidedly Smooth Curves with Contingencies
 3.4. OneSidedly Smooth Functions
 3.5. Notion of cCorrespondence. Indicatrix of Tangents of a Curve
 3.6. OneSidedly Smooth Curves in Differentiable Manifolds
 IV: Some Facts of Integral Geometry
 8.3. Complete TwoDimensional 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. RightHand and LeftHand 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 kDimensional 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 IntegraGeometrical 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 nDimensional Sphere
 6.1. Auxiliary Results
 6.2. IntegroGeometrical 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 TwoDimensional 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 ThreeDimensional Euclidean Space
 8.1. Torsion of a Plane Curve
 8.2. Curves of a Finite Complete Torsion