SHAPE AND ENERGIES OF GEOMETRIC OBJECTS

In this book, we study the shape of different geometric objects (curves, surfaces, and knots) through small deformation: infinitesimal bending. The shape of geometric objects is determined by their geometric curvature as well as curvature-based functionals, which are referred to as energies. Different kinds of curvatures, their properties, and their influence on shape and energy are discussed. For the investigation of shape and energy, we use our own software visualization tools.

We aim to offer a mathematical way of considering shape. That is, all the geometrical information that is invariant to translations, rotations, and size changes. A more flexible definition of shape takes into consideration the fact that we often deal with deformable shapes in reality. By allowing also isometric (or near-isometric) deformations, such as infinitesimal bending, the intrinsic geometry of the object will stay the same, while sub-parts might be located at very different positions in space.

A special part of the book presents the construction of examples of unknots, as well as knots that can resist simplification by energy gradient and force evolution methods in programming models.

Contents:

• On Shape and Energies:
  • Shape
  • Curvatures
  • Energies
  • Deformations
  • Visualization
• Curves and Knots:
  • Curves
  • Infinitesimal Bending of Curves
  • Knots
  • Surfaces Defined by Bending Curves
  • Curves on Surfaces with a Given Precision
  • Variations of Geometric Magnitudes
  • Willmore Energy
  • Total Curvature
  • Torsional Energy
  • Total Torsion
  • Total Normalcy
  • Möbius Energy of Knots
• Surfaces:
  • Surfaces
  • Infinitesimal Deformations of Surfaces
  • Infinitesimal Bending of Surfaces
  • Infinitesimal Bending of the First Order
  • Variation of GeometricMagnitudes
  • Willmore Energy
  • TotalMean Curvature
  • Volume of Generalized Cone
• Surfaces of Revolution:
  • Infinitesimal Bending of the First Order of Surfaces of Revolution
  • Classification of Toroids Generated by Quadrangle with Respect to Flexibility
  • Variation of the Volume of Surface of Revolution
  • Variation of the Volume of a Surface of Revolution Generated by a Meridian Being Infinitesimally Bent
  • Infinitesimal Bending of Higher Order of Surfaces of Revolution
  • Examples of the Infinitesimal Bending of Toroids Generated by a Polygonal Meridian
• Shape and Energy of Knots: Topological Topics (Louis H Kauffman):
  • Knot Dynamics
  • Moving Down Energy Gradients
  • Hard Knots and Collapsing Tangles

Readership: Advanced undergraduates, graduate students, researchers, and practitioners in mathematics, biology, physics, mechanics, and architecture, especially those interested in the applications of differential geometry and topology.