Shape and energies of geometric objects / Louis H. Kauffman, Ljubica S. Velimirović, Marija S. Najdanović, Svetozar R. Rančić.

Format:

Book

Author/Creator:

Kauffman, Louis H., 1945- author.

Velimirović, Ljubica S., author.

Najdanović, Marija S., author.

Rančić, Svetozar R., author.

Series:

K & E series on knots and everything ; vol. 80.

Series on knots and everything ; vol. 80

Language:

English

Subjects (All):

Geometry, Differential.

Physical Description:

xvi, 275 pages : illustrations ; 24 cm.

Place of Publication:

Singapore : World Scientific, [2026]

Contents:

Intro

Contents

Preface

About the Authors

1. On Shape and Energies

1.1 Shape

1.2 Curvatures

1.3 Energies

1.3.1 Energies of Curves

1.3.2 Energies of Knots

1.3.3 Energies of Surfaces

1.4 Deformations

1.5 Visualization

2. Curves and Knots

2.1 Curves

2.2 Infinitesimal Bending of Curves

2.2.1 Infinitesimal Bending of the First Order

2.2.1.1 Rotational Field

2.2.2 Infinitesimal Bending of the Second Order

2.2.3 Some Special Classes of Curves

2.2.3.1 Plane Curves

2.2.3.2 Spherical Curves

2.2.3.3 Curves on Ruled Surfaces

2.3 Knots

2.3.1 Infinitesimal Bending of Knots

2.4 Surfaces Defined by Bending Curves

2.5 Curves on Surfaces with a Given Precision

2.5.1 Spherical Curves

2.5.2 Curves on Cylinder

2.5.3 Curves on Hyperbolic Paraboloid

2.5.4 Curves on Helicoid

2.5.5 Torus Knots

2.6 Variations of Geometric Magnitudes

2.6.1 The First Variation

2.6.1.1 Area of the Region Bounded by a Plane Curve

2.6.1.2 Curvature of a Curve on the Hyperbolic Paraboloid

2.6.2 The Second Variation

2.7 Willmore Energy

2.7.1 The First Variation of Willmore Energy

2.7.2 The Second Variation of Willmore Energy

2.7.3 Willmore Energy of Knots

2.8 Total Curvature

2.8.1 The First Variation of the Total Curvature

2.8.2 The Second Variation of the Total Curvature

2.8.3 Total Curvature of Knots

2.9 Torsional Energy

2.9.1 The First Variation of the Torsional Energy

2.9.2 The Second Variation of the Torsional Energy

2.9.3 Torsional Energy of Knots

2.10 Total Torsion

2.10.1 The First Variation of the Total Torsion

2.10.2 The Second Variation of the Total Torsion

2.10.3 Total Torsion of Knots

2.11 Total Normalcy

2.11.1 The First Variation of the Total Normalcy

2.11.2 Total Normalcy of Knots

2.12 Möbius Energy of Knots

3. Surfaces

3.1 Surfaces

3.2 Infinitesimal Deformations of Surfaces

3.3 Infinitesimal Bending of Surfaces

3.4 Infinitesimal Bending of the First Order

3.4.1 Infinitesimal Bending of the Surfaces at Descartes Coordinate System

3.4.2 Rotational Field

3.4.3 Kinematic System of Equations

3.4.4 Gaudi's Surfaces

3.5 Variation of Geometric Magnitudes

3.5.1 Variations of Some Geometric Magnitudes under Infinitesimal Deformations

3.5.2 Variation of Some Geometric Magnitudes under Infinitesimal Bending

3.6 Willmore Energy

3.6.1 Variation of the Willmore Energy under Infinitesimal Deformations

3.6.2 Variation of the Willmore Energy under Infinitesimal Bending

3.6.3 CMC Surfaces: Minimal Surfaces

3.7 Total Mean Curvature

3.8 Volume of Generalized Cone

3.8.1 An Example of Volume Variation Calculation

3.8.2 Deformed Swept Volume

4. Surfaces of Revolution

4.1 Infinitesimal Bending of the First Order of Surfaces of Revolution

4.2 Classification of Toroids Generated by Quadrangle with Respect to Flexibility.

Notes:

Includes bibliographical references (pages 265-271) and index.

ISBN:

9819819458

9789819819454

OCLC:

1533688878