General Theory of Irregular Curves

Inhaltsverzeichnis

• 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.
• 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.
• 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.